Home

2003 Jean-Pierre Serre

1. GACC Serre J P Groupes alg briques et corps de classes Hermann Paris 1959 2nd edn 1975 translated into English and Russian CL Serre J P Corps locaux Hermann Paris 1962 4th edn 2004 translated into English CG Serre J P Cohomologie galoisienne LNM vol 5 Springer Berlin 1964 5th edn revised and completed 1994 translated into English and Russian LALG Serre J P Lie Algebras and Lie Groups Benjamin New York 1965 2nd edn LNM vol 1500 Springer Berlin 1992 translated into Russian ALM Serre J P Alg bre locale Multiplicit s LNM vol 11 Springer Berlin 1965 written with the help of P Gabriel 3rd edn 1975 translated into English and Russian ALSC Serre J P Algebres de Lie semi simples complexes Benjamin New York 1966 translated into English and Russian RLGF Serre J P Repr sentations lin aires des groupes finis Hermann Paris 1967 trans lated into English German Japanese Polish Russian and Spanish McGill Serre J P Abelian adic Representations and Elliptic Curves Benjamin New York 1968 written with the help of W Kuyk and J Labute 2nd edn A K Peters 1998 translated into Japanese and Russian CA Serre J P Cours d arithm tique Presses Univ France Paris 1970 4th edn 1995 translated into Chinese English Japanese and Russian AA Serre J P Arbres amalgames SL2 Ast risque vol 46 Soc Math France Paris 19
2. In the philosophy of science there is a very strong current in favor of the concept of rupture Are there ruptures in mathematics For example the emergence of prob ability as a new way in which to represent the world What is its significance in mathematics Philosophers like to talk of rupture I suppose it adds a bit of spice to what they say I do not see anything like that in mathematics no catastrophe and no revolution Progress yes as I ve already said but that is not the same We work sometimes on old questions and sometimes on new ones There is no boundary between the two There is a deep continuity between the mathematics of two centuries ago and that 22 Jean Pierre Serre Mon premier demi si cle au Coll ge de France d 1l y a deux si cles et celles de maintenant Le temps des math maticiens est la longue dur e de feu mon coll gue Braudel Quant aux probabilit s elles sont utiles pour leurs applications a la fois math matiques et pratiques d un point de vue purement math matique elles constituent une branche de la th orie de la mesure Peut on vraiment parler a leur sujet de mani re nouvelle de se repr senter le monde S rement pas en math matique Est ce que les ordinateurs changent quelque chose la fa on de faire des math matiques On avait coutume de dire que les recherches en math matiques taient peu co teuses des crayons et du papier et voil nos besoins satisfait
3. mais employaient des notations tr s lourdes J ai r dig pour elle un expos adapt ses besoins et je l ai ensuite publi dans un livre intitul Repr sentations Lin aires des Groupes Finis J ai fait mon travail de math maticien et de mari mis des choses sur les rayons Le vrai en math matiques a t il le m me sens qu ailleurs Non C est un vrai absolu C est sans doute ce qui fait l impopularit des math ma tiques dans le public L homme de la rue veut bien tol rer l absolu quand il s agit de religion mais pas quand il s agit de math matique Conclusion croire est plus facile que d montrer Jean Pierre Serre My First Fifty Years at the Coll ge de France 29 useful for the building of computers Cryptography should also be mentioned it is a source of interesting problems in number theory As for the place of mathematics in relation to other sciences mathematics can be seen as a big warehouse full of shelves Mathematicians put things on the shelves and guarantee that they are true They also explain how to use them and how to reconstruct them Other sciences come and help themselves from the shelves math ematicians are not concerned with what they do with what they have taken This metaphor is rather coarse but it reflects the situation well enough Of course one does not choose to do mathematics just for putting things on shelves one does math ematics for the fun of it Here is a persona
4. resentations associated to elliptic curves The main theorem is that if E is an el liptic curve over an algebraic number field k without complex multiplication then ge End Ve The proof turns out to be a clever combination of a finiteness theorem due to Shafarevich together with the above mentioned results on abelian and locally algebraic adic representations of k Jean Pierre Serre An Overview of His Work 67 Serre also proves that if E is an elliptic curve over an algebraic number field k such that its j invariant is not an algebraic integer of k then the group G Imp where p pe is open in GL2 Zz Later Serre would eliminate the condition regarding the modular invariant j see below It is also proved in McGill that if E E are elliptic curves defined over an algebraic number field k whose invariants j E j E are not algebraic integers and whose Gal k k modules Ve E Ve E are isomorphic then E and E are isogenous over k The result is a special case of the Tate conjecture proved later by G Faltings 1983 13 2 5 The above results were improved in the seminal paper Propri t s galoisi ennes des points d ordre fini des courbes elliptiques S154 GE 94 1972 which is dedicated to Andr Weil The main theorem states that if E is an elliptic curve defined over an algebraic number field k which does not have complex multiplica tion then Gg Aut 7 for almost all In particula
5. 10 1 1 Previously Bass Lazard and Serre S108 61 1964 had proved the con gruence subgroup conjecture for SL Z n gt 3 and Sp Z for n gt 2 every arith metic subgroup is a congruence subgroup The same result had been obtained inde pendently by J Mennicke Bass Lazard Serre s proof is by induction on n gt 3 It relies on a computation of the cohomology of the profinite groups SL2 Z SP Zp with coefficients in Q Z and in Q Zp this computation is made possi ble by Lazard s results see above on the cohomology of p adic Lie groups 10 1 2 In S126 GE 74 1967 GE 103 1975 Bass Milnor and Serre prove the congruence subgroup conjecture when k is an algebraic number field G SL for n gt 3 and G Sp forn gt 2 In order to do this they determine the corresponding universal Mennicke symbols associated to these groups and to the ring of integers of a totally imaginary algebraic number field k 10 1 3 The solution of the congruence subgroup problem in the case where G is a split simply connected simple group of rank gt 1 was obtained by H Matsumoto 1966 1969 by using the known cases SL3 and Sp The congruence subgroup problem as well as its connection to Moore s theory of universal coverings of G A is discussed in S minaire Bourbaki S123 77 1967 10 1 4 The paper S143 GE 86 1970 is about the S congruence subgroup problem for SL If gt 2 the answer is almost posi
6. 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 List of Publications for Jean Pierre Serre 1985 A b 4ac Math Medley 13 1 1 10 La vie et l uvre de Ivan Matveevich Vinogradov C R Acad Sci S r G n Vie Sci 2 6 667 669 Sur la lacunarit des puissances de 7 Glasgow Math J 27 203 221 R sum des cours de 1984 1985 Annuaire du Coll ge de France 85 90 1986 uvres Vol I 1949 1959 Springer Verlag Berlin uvres Vol II 1960 1971 Springer Verlag Berlin uvres Vol IH 1972 1984 Springer Verlag Berlin R sum des cours de 1985 1986 Annuaire du Coll ge de France 95 99 1987 Lettre a J F Mestre In Current trends in arithmetical algebraic geometry Arcata Calif 1985 Contemp Math No 67 263 268 Amer Math Soc Providence RI Une relation dans la cohomologie des p groupes C R Acad Sci Paris S r I Math 304 20 587 590 Sur les repr sentations modulaires de degr 2 de Gal Q Q Duke Math J 54 1 179 230 1988 Groupes de Galois sur Q In S minaire Bourbaki 1987 1988 Expos 689 337 350 Ast risque Nos 161 162 R sum des cours de 1987 1988 Annuaire du Coll ge de France 79 82 1989 F Klein Lektsti ob ikosaedre i reshenii uravnenii pyatoi stepeni Nauka Moscow Trans lated from the German by A L Gorodentsev and A A Kirillov Tr
7. The general theory of local rings was the subject of the lecture course S79 42 1958 which was later published in book form Alg bre Locale Multiplici t s S111 ALM 1965 Its topics include the general theory of noetherian modules and their primary decomposition Hilbert polynomials integral extensions Krull Samuel dimension theory the Koszul complex Cohen Macaulay modules and the homological characterization of regular local rings mentioned above The book cul minates with the celebrated Tor formula which gives a homological definition for intersection multiplicities in algebraic geometry in terms of Euler Poincar char acteristics This led Serre to several conjectures on regular local rings of mixed characteristic most of them but not all were later proved by P Roberts 1985 H Gillet C Soul 1985 and O Gabber The book had a profound influence on a whole generation of algebraists 5 Projective Modules Given an algebraic vector bundle E over an algebraic variety V let S E denote its sheaf of sections As pointed out in FAC one gets in this way an equivalence be tween vector bundles and locally free coherent O sheaves When V is affine with coordinate ring A T V this may be viewed as a correspondence between vec tor bundles and finitely generated projective A modules under this correspondence trivial bundles correspond to free modules 5 0 1 The above considerations apply when V is the af
8. chelon n gt 3 In S minaire H Cartan E N S 1960 61 Append a Expos 17 3 pp Groupes proalg briques Inst Hautes Etudes Sci Publ Math 7 5 68 Analogues k hl riens de certaines conjectures de Weil Ann of Math 2 71 392 394 Sur la rationalit des repr sentations d Artin Ann of Math 2 72 405 420 R sum des cours de 1959 1960 Annuaire du Coll ge de France 41 43 1961 Formes bilin aires sym triques enti res discriminant 1 In S minaire H Cartan E N S 1961 62 Expos 14 16 pp Sur les corps locaux corps r siduel alg briquement clos Bull Soc Math France 89 105 154 Exemples de vari t s projectives en caract ristique p non relevables en caract ristique z ro Proc Nat Acad Sci U S A 47 108 109 R sum des cours de 1960 1961 Annuaire du Coll ge de France 51 52 List of Publications for Jean Pierre Serre 87 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 1962 Cohomologie galoisienne des groupes alg briques lin aires In Collog Th orie des Groupes Alg briques Bruxelles 1962 53 68 Librairie Universitaire Louvain Cohomologie galoisienne Lecture Notes in Mathematics Vol 5 Springer Verlag Berlin Translated into English and Russian Corps locaux Publications de I Institut de Math matique de l Universit de Nancago
9. re de rationalit pour les surfaces alg briques d apr s K Kodaira In S minaire Bour baki 1956 1957 Expos 146 14 pp with S S Chern and F Hirzebruch On the index of a fibered manifold Proc Amer Math Soc 8 587 596 with S Lang Sur les rev tements non ramifi s des vari t s alg briques Amer J Math 79 319 330 Erratum Amer J Math 81 279 280 1959 Sur la cohomologie des vari t s alg briques J Math Pures Appl 9 36 1 16 R sum des cours de 1956 1957 Annuaire du Coll ge de France 61 62 86 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 List of Publications for Jean Pierre Serre 1958 Classes des corps cyclotomiques d apres K Iwasawa In S minaire Bourbaki 1958 1959 Expos 174 11 pp Espaces fibr s alg briques In S minaire C Chevalley Anneaux de Chow et applications Expos 1 37 pp Rev tements Groupe fondamental In Structures alg briques et structures topologiques Monographies de l Enseignement Math matique No 7 97 136 Institut de Math matiques Universit de Gen ve Modules projectifs et espaces fibr s fibre vectorielle In S minaire P Dubreil M L Dubreil Jacotin et C Pisot 1957 58 Expos 23 18 pp with A Borel Le th or me de Riemann Roch Bull Soc Math France 86 97 136 Quelques propri t
10. where x is the cyclotomic character the weight k p is given by a rather sophis ticated formula which depends only on the ramification at p The paper contains numerical examples for p 2 3 7 in support of the conjecture they were implemented with the help of J F Mestre Some months after the appear ance of this paper and after examining the examples more closely Serre slightly modified the conjectures for the primes p 2 3 in the case of Galois representa tions of dihedral type Since their publication Serre s conjectures have generated abundant literature They imply Fermat s Last Theorem and variants thereof as well as the Shimura Taniyama Weil Conjecture and generalizations of it As G Frey 1986 and Serre pointed out a weak form of conjecture 3 2 49 known as conjecture epsilon is suf ficient to prove that Fermat s Last Theorem follows from the Shimura Taniyama Weil conjecture in the semistable case Conjecture epsilon was proved by K Ribet 1990 in a brilliant study in which he made use of the arithmetical properties of modular curves Shimura curves and their Jacobians Once Ribet s theorem was proved the task of proving Shimura Taniyama Weil conjecture in the semistable Jean Pierre Serre An Overview of His Work 71 case would be accomplished five years later by A Wiles 1995 and R Taylor and A Wiles 1995 Serre s modularity conjecture may be viewed as a first step in the direct
11. 1 if k has characteristic zero 7 1 2 The paper S93 51 1961 is a sequel to the one mentioned above and is also its motivation It deals with local class field theory in the geometric set ting in an Oberwolfach lecture Serre once described it as reine geometrische Klassenk rpertheorie im Kleinen Let K be a field which is complete with respect to a discrete valuation and suppose that its residue field k is algebraically closed By using a construction of M Greenberg the group of units Ux of K may be viewed as a commutative pro algebraic group over k so that the fundamental group Jean Pierre Serre An Overview of His Work 47 T1 UK is well defined The reciprocity isomorphism takes the simple form X ab Ti UK Gx where Gi denotes the Galois group of the maximal abelian extension of K This isomorphism is compatible with the natural filtration of x UK and the filtration of Gi given by the upper numbering of the ramification groups Hence there is a conductor theory related to Artin representations see below 7 1 3 An Artin representation a has a Z valued character In the paper S90 CE 46 1960 it is shown that a is rational over Qp provided is different from the residue characteristic but it is not always rational over Q The same paper conjec tures the existence of a conductor theory for regular local rings of any dimension analogous to the one in dimension 1 a few results have been obtained on this re ce
12. 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 List of Publications for Jean Pierre Serre Problem section In Proceedings of the Second International Conference on the Theory of Groups Australian Nat Univ Canberra 1973 733 740 Lecture Notes in Math Vol 372 Springer Verlag Berlin Fonctions z ta p adiques In Journ es Arithm tiques Grenoble 1973 Bull Soc Math France M m No 37 157 160 Valeurs propres des endomorphismes de Frobenius d apr s P Deligne In S minaire Bour baki 1973 1974 Expos 446 190 204 Lecture Notes in Math Vol 431 Springer Verlag Berlin R sum des cours de 1973 1974 Annuaire du Coll ge de France 43 47 1975 Divisibilit de certaines fonctions arithm tiques In S minaire Delange Pisot Poitou 1974 75 Th orie des nombres Expos 20 28 pp Valeurs propres des op rateurs de Hecke modulo L In Journ es Arithm tiques de Bordeaux 1974 Ast risque Nos 24 25 109 117 with B Mazur Points rationnels des courbes modulaires XO N d apr s B Mazur et A Ogg In S minaire Bourbaki 1974 1975 Expos 469 238 255 Lecture Notes in Math Vol 514 Springer Verlag Berlin R sum des cours de 1974 1975 Annuaire du Coll ge de France 41 46 1976 with A Borel Cohomologie d immeubles et de groupes S arithm tiques Topology 15 3 211 23
13. 1971 Annuaire du Coll ge de France 51 55 1972 Propri t s galoisiennes des points d ordre fini des courbes elliptiques Invent Math 15 4 259 331 Congruences et formes modulaires d apr s H P F Swinnerton Dyer In S minaire Bour baki 1971 1972 Expos 416 319 338 Lecture Notes in Math Vol 317 Springer Verlag Berlin R sum des cours de 1971 1972 Annuaire du Coll ge de France 55 60 1973 W Kuyk and J P Serre editors Modular functions of one variable IIT Lecture Notes in Math Vol 350 Springer Verlag Berlin Formes modulaires et fonctions z ta p adiques In Modular functions of one variable III Lecture Notes in Math Vol 350 191 268 Springer Verlag Berlin Correction Lecture Notes in Math Vol 476 149 150 with A Borel Corners and arithmetic groups Comment Math Helv 48 436 491 Avec un appendice Arrondissement des vari t s a coins par A Douady et L H rault R sum des cours de 1972 1973 Annuaire du Coll ge de France 51 56 1974 Divisibilit des coefficients des formes modulaires de poids entier C R Acad Sci Paris S r 279 679 682 with P Deligne Formes modulaires de poids 1 Ann Sci cole Norm Sup 4 7 507 530 Amalgames et points fixes In Proceedings of the Second International Conference on the Theory of Groups Australian Nat Univ Canberra 1973 633 640 Lecture Notes in Math Vol 372 Springer Verlag Berlin 90 164
14. Archim de serait un interlocuteur plus indiqu C est lui le grand math maticien de Antiquit Il a fait des choses extraordinaires aussi bien en math matique qu en physique En philosophie des sciences il y a un courant tr s fort en faveur d une pens e de la rupture N y a t il pas de ruptures en math matiques On a d crit par exemple l mergence de la probabilit comme une mani re nouvelle de se repr senter le monde Quelle est sa signification en math matiques Les philosophes aiment bien parler de rupture Je suppose que cela ajoute un peu de piment leurs discours Je ne vois rien de tel en math matique ni catastrophe ni r volution Des progr s oui je lai d j dit ce n est pas la m me chose Nous travaillons tant t de vieilles questions tant t des questions nouvelles Il n y a pas de fronti re entre les deux Il y a une grande continuit entre les math matiques Jean Pierre Serre My First Fifty Years at the Coll ge de France 21 Bourbaki had nothing to do with it its books are meant for mathematicians not for students and even less for teen agers Note that Bourbaki was careful not to write anything on this topic Its doctrine was simple one does what one chooses to do one does it the best one can but one does not explain why I very much like this attitude which favors work over discourse too bad if it sometimes lead to misun derstandings How would you describe the de
15. Collected Papers American Mathematical Society Providence RI Edited and with a foreword by J P Serre R partition asymptotique des valeurs propres de l op rateur de Hecke T J Amer Math Soc 10 1 75 102 1998 La distribution d Euler Poincar d un groupe profini In Galois representations in arith metic algebraic geometry Durham 1996 London Math Soc Lecture Note Ser Vol 254 461 493 Cambridge Univ Press Cambridge J L Nicolas I Z Ruzsa and A S rk zy On the parity of additive representation functions J Number Theory 13 2 292 317 With an appendix by J P Serre Robert L Griess Jr and A J E Ryba Embeddings of PGL2 31 and SL2 32 in Eg C Duke Math J 94 1 181 211 With appendices by M Larsen and J P Serre Moursund Lectures arXiv 0305 257 1999 La vie et l uvre d Andr Weil L Enseign Math 2 45 1 2 5 16 Sous groupes finis des groupes de Lie In S minaire Bourbaki 1998 99 Expos 864 415 430 Ast risque No 266 2000 uvres Collected papers Vol IV 1985 1998 Springer Verlag Berlin 2001 Expos s de s minaires 1950 1999 Documents Math matiques Paris 1 Soci t Math matique de France Paris Pierre Colmez and Jean Pierre Serre editors Correspondance Grothendieck Serre Docu ments Math matiques Paris 2 Soci t Math matique de France Paris Also available as Correspondence Grothendieck Serre bilingual edition Amer Ma
16. Expos 1 6 pp Groupes d homotopie In S minaire H Cartan E N S 1949 1950 Expos 2 7 pp Groupes d homotopie relatifs Applications aux espaces fibr s In S minaire H Cartan E N S 1949 1950 Expos 9 8 pp Homotopie des espaces fibr s Applications In S minaire H Cartan E N S 1949 1950 Expos 10 7 pp Extensions de corps ordonn s C R Acad Sci Paris 229 576 577 Compacit locale des espaces fibr s C R Acad Sci Paris 229 1295 1297 1950 Extensions de groupes localement compacts d apr s Iwasawa et Gleason In S minaire Bourbaki 1950 1951 Expos 27 6 pp Applications alg briques de la cohomologie des groupes I In S minaire H Cartan E N S 1950 1951 Expos 5 7 pp Applications alg briques de la cohomologie des groupes II Th orie des alg bres simples In S minaire H Cartan E N S 1950 1951 Expos s 6 7 20 pp La suite spectrale des espaces fibr s Applications In S minaire H Cartan E N S 1950 1951 Expos 10 9 pp Espaces avec groupes d op rateurs Compl ments In S minaire H Cartan E N S 1950 1951 Expos 13 12 pp La suite spectrale attach e une application continue In S minaire H Cartan E N S 1950 51 Expos 21 8 pp Sur un th or me de T Szele Acta Univ Szeged Sect Sci Math 13 190 191 Trivialit des espaces fibr s Applications C R Acad Sci Paris 230 916 918 with A Borel Impossibilit de fibrer
17. Jacobians Jm form a projective system which is the geometric ana logue of the 1d le class group Class field theory for function fields in one variable over finite fields is dealt with in Chap VI The reciprocity isomorphism is proved and explicit computations of norm residue symbols are made Chapter VII contains a general cohomological treatment of extensions of commutative algebraic groups 46 P Bayer 7 1 1 Based on the lecture course S91 GE 47 1960 a theory of commutative pro algebraic groups is developed in S88 49 1960 Its application to geometric class field theory can be found in S93 51 1961 Let k be an algebraically closed field A commutative quasi algebraic group over k is defined as a pure inseparable isogeny class of commutative algebraic groups over k If G is such a group and is connected then it has a unique connected linear subgroup L such that the quotient G L is an abelian variety As for the group L it is the product of a torus T by a unipotent group U The group T is a product of groups isomorphic to G and the group U has a composition series whose quotients are isomorphic to Gg it is isogenous to a product of truncated Witt vector groups The isomorphism classes of the groups Ga Gm cyclic groups of prime order and sim ple abelian varieties are called the elementary commutative quasi algebraic groups The commutative quasi algebraic groups form an abelian category Q The finite commutative quasi
18. Lie subgroup of the adic Lie group GL T The Lie algebra gg of G4 is a subalgebra of gl V and does not change under finite extensions of the ground field it acts on Vy When k is finitely generated over Q it is known that the rank of g is independent of but it is not known whether the same is true for the dimension of gs Serre has written several papers and letters on the properties of Ge and g see below 13 4 1 The first occurrence of the Ge and the ge in Serre s papers can be found in S109 62 1964 it was complemented a few years later by S151 89 1971 When k is a number field the Mordell Weil theorem says that the group A k of the k rational points of A is a finitely generated abelian group J W S Cassels had asked whether it is true that every subgroup of finite index of A k contains a con gruence subgroup at least when A is an elliptic curve In the first paper Serre trans formed the problem into another one relative to the cohomology of the Gz s namely the vanishing of H ge Ve for every and he solved it when dim A 1 In the second paper he solved the general case by proving a vanishing criterion for the cohomology of Lie algebras which implies that H ge Ve 0 for every n and every 13 4 2 In 1968 Serre and Tate published the seminal paper Good reduction of abelian varieties S134 79 1968 Let K be a field v a discrete valuation O the valuation ring of v and k it
19. Then examples of cohomological invariants are provided by the Stiefel Whitney classes w H k O 4 gt H k Z 2Z Arason s invariant a H k Spin g gt H k Z 2Z Merkurjev Suslin s invariant ms H k SLp gt H k u8 Rost s invariants 52 P Bayer 9 3 2 The presentation of recent work on Galois cohomology and the formulation of some open problems was the purpose of the expos S246 166 1994 at the S minaire Bourbaki To every connected semisimple group G whose root system over k is irreducible one associates a set of prime numbers S G which plays a spe cial role in the study of the cohomology set H k G For example all the divisors of the order of the centre of the universal covering G of G are included in S G Tits theorem 1992 proves that given a class x H l k G there exists an exten sion k k of S G primary degree that kills x that is to say such that x maps to zero in H k G Serre asks whether it is true that given finite extensions k k whose degrees are coprime to G the mapping H l k G 4 k G is in jective assuming that G is connected In this lecture he also gives extensions and variants of Conjectures I and II which deal with an imperfect ground field or for which one merely assumes that cd G lt 1 for every p S G He gives a list of cases in which Conjecture IT has been proved namely groups of type SLp associ
20. Tp p acting on F amp M are finite in number In particular there exists a weight k such that each system of eigen values can be realized by a form of weight lt k the precise value for k was found by Tate As an illustration Serre gives a complete list of all the systems ap which occur for the primes lt 23 He also raises a series of problems and conjec tures which lead twelve years later to his own great work S216 143 1987 on modular Galois representations As is well known this became a key ingredient in the proof of Fermat s Last Theorem 13 3 5 In S178 113 1977 Serre and H Stark prove that each modular form of weight 1 2 is a linear combination of theta series in one variable thus answering a question of G Shimura 1973 The proof relies on the bounded denominator property of modular forms on congruence subgroups 13 3 6 In the long paper entitled Quelques applications du th or me de densit de Chebotarey S197 GE 125 1981 one finds a number of precise estimates both for elliptic curves and for modular forms These estimates are of two types either they are unconditional or they depend on the Generalized Riemann Hypothesis GRH The work in question is essentially analytic It uses several different ingredients explicit forms of Chebotarev theorem due to J C Lagarias H L Montgomery A M Odlyzko with applications to infinite Galois extensions with an adic Lie
21. VIII Actualit s Sci Indust No 1296 Hermann Paris Translated into English Endomorphismes compl tement continus des espaces de Banach p adiques Inst Hautes Etudes Sci Publ Math 12 69 85 with A Frohlich and J Tate A different with an odd class J reine angew Math 209 6 7 R sum des cours de 1961 1962 Annuaire du Coll ge de France 47 51 1963 Structure de certains pro p groupes d apr s DemuSkin In S minaire Bourbaki 1962 1963 Expos 252 11 pp G om trie alg brique In Proc Internat Congr Mathematicians Stockholm 1962 190 196 Inst Mittag Leffler Djursholm R sum des cours de 1962 1963 Annuaire du Coll ge de France 49 53 1964 Groupes analytiques p adiques d apr s Michel Lazard In S minaire Bourbaki 1963 1964 Expos 270 10 pp with A Borel Th or mes de finitude en cohomologie galoisienne Comment Math Helv 39 111 164 Exemples de vari t s projectives conjugu es non hom omorphes C R Acad Sci Paris 258 4194 4196 with H Bass and M Lazard Sous groupes d indice fini dans SL n Z Bull Amer Math Soc 70 385 392 Sur les groupes de congruence des vari t s ab liennes Zzv Akad Nauk SSSR Ser Mat 28 3 20 1965 Lie algebras and Lie groups Lectures given at Harvard University W A Benjamin Inc New York Amsterdam Translated into Russian Alg bre locale Multiplicit s Cours au Coll ge de France 1957 1958 r dig par P
22. a contractible symmetric space X on which N acts properly and a study of X N shows that N is infinite with a few exceptions In the case of characteristic p gt 0 X is the Bruhat Tits tree In the case k Q X is Poincar s half plane In the case where k is an imaginary quadratic field X is the hyperbolic 3 space and the quotient manifold X N can be compactified by adding to it a finite set of 2 tor1 which correspond to elliptic curves with complex multiplication by k this is a special case of the general compactifications intro duced a few years later by Borel and Serre see Sect 10 2 2 10 2 Cohomology of Arithmetic Groups A locally algebraic group A over a per fect field k is called by Borel Serre a k group of type ALA if it is an extension of an arithmetic Gal k k group by a linear algebraic group over k In a joint pa per with Borel S106 1964 included in A Borel uvres Collected Papers it is proved that for k a number field and S a finite set of places of k the mapping H k A gt IL gs H k A is proper 1 e the inverse image of any point is finite This result applied to A Aut G implies the finiteness of the number of classes of k torsors of a linear group G which are isomorphic locally everywhere to a given k torsor 10 2 1 Let k be a global field and S a finite set of places of k Let L be a linear reductive algebraic group defined over k In S139 83 1969 S148 1971 and S
23. authors and introduces a systematic method to obtain more precise bounds based on Weil s explicit formula Let Ng g be the maximum value of N C as C runs through all curves of genus g defined over F The value Ng 1 was already known for most q s it is equal to q 1 2q for the others it is q 2q Serre obtains the exact value of Ng 2 for every q It is not very different from Weil s bound If A q limsup _ oo Ng q q then Drinfeld Vladut proved that A q lt qi 1 for every q and Ihara showed that A q g 1 if q is a square Serre proves that A g gt O for all q more precisely A g gt c log qg for some absolute c gt 0 His proof uses class field towers like in Golod Shafarevich These papers generated considerable interest in determining the actual maximum and minimum of the number of points for a given pair g g 11 3 1 Kristin Lauter obtained improvements on the bounds for the number of ra tional points of curves over finite fields along the lines of S201 GE 128 1983 and S200 129 1983 In her papers S265 2001 S267 2002 we find ap pendices written by Serre The appendix in S267 2002 is particularly appealing It gives an equivalence between the category of abelian varieties over F which are isogenous over F to a product of copies of an ordinary elliptic curve E defined over F and the category of torsion free Rg modules of finite type where Ra denote
24. auxquels ils versaient une partie de leur salaire Quant mon enseignement voici ce que j en disais dans une interview de 1986 Enseigner au Coll ge est un privil ge merveilleux et redoutable Merveilleux cause de la libert dans le choix des sujets et du haut niveau de l auditoire chercheurs au CNRS visiteurs trangers coll gues de Paris et d Orsay beau coup sont des habitu s qui viennent r guli rement depuis cinq dix ou m me vingt ans Redoutable aussi il faut chaque ann e un sujet de cours nouveau soit sur ses propres recherches ce que je pr fere soit sur celles des autres comme un cours annuel dure environ vingt heures cela fait beaucoup Comment s est pass e votre lecon inaugurale A mon arriv e au Coll ge j tais un jeune homme de trente ans La le on inaugu rale m apparaissait presque comme un oral d examen devant professeurs famille coll gues math maticiens journalistes etc J ai essay de la pr parer Au bout d un mois J avais r ussi en crire une demi page Arrive le jour de la le on un moment assez solennel J ai commenc par lire la demi page en question puis j ai improvis Je ne sais plus tr s bien ce que j ai dit je me souviens seulement avoir parl de l Alg bre et du r le ancillaire qu elle joue en G om trie et en Th orie des Nombres D apr s le compte rendu paru dans le journal Combat j ai pass mon temps essuyer machinale
25. be the normalized Killing form We should point out that the bound they obtain has now been superseded especially for the Spin groups 10 Discrete Subgroups The study of discrete subgroups of Lie groups goes back to F Klein and H Poincar Let us consider a global field k and a finite set S of places of k containing the set Soo of all the archimedean places Let O be the ring of S integers of k and let us denote by Ax and A the ring of adeles and of S adeles of k respectively We write Al for the ring of finite adeles of k obtained by taking S Soo Given a linear algebraic group G defined over k we shall consider a fixed faithful representation G GL Let Ir G k NGL O In G k we may distinguish two types of subgroup namely the S arithmetic subgroups and the S congruence subgroups A subgroup I of G k is said to be S arithmetic if T A T is of finite index in both F and T Let q C O be an ideal and GL O q ker GL O GL O q We define Ta r NGL O q A subgroup I of G k is said to be an S congruence sub group if it is S arithmetic and it contains a subgroup Iq for some non zero ideal q The S congruence subgroup problem is the question is every S arithmetic sub group an S congruence subgroup If S Soo one just refers to the congruence subgroup problem Since an S congruence subgroup is S arithmetic there is a homomorphism of topological groups x G k G k where G
26. de France lorsque j explique des math matiques de parler un ami Devant un ami on n a pas envie de lire un texte Si l on a oubli une formule on en donne la structure cela suffit Pendant l expos j ai en t te une quantit de choses qui me permettraient de parler bien plus longtemps que pr vu Je choisis suivant I auditoire et l inspiration du moment Seule exception le s minaire Bourbaki o l on doit fournir un texte suffisam ment l avance pour qu il puisse tre distribu en s ance C est d ailleurs le seul s minaire qui applique une telle r gle tr s contraignante pour les conf renciers Quel est la place de Bourbaki dans les math matiques fran aises d aujourd hui C est le s minaire qui est le plus int ressant Il se r unit trois fois par an en mars mai et novembre Il joue un r le la fois social occasion de rencontres et math matique expos de r sultats r cents souvent sous une forme plus claire que celle des auteurs 1l couvre toutes les branches des math matiques Les livres Topologie Alg bre Groupes de Lie sont encore lus non seule ment en France mais aussi l tranger Certains de ces livres sont devenus des clas siques je pense en particulier celui sur les syst mes de racines J ai vu r cemment dans le Citations Index de AMS que Bourbaki venait au 6 rang par nombre de citations parmi les math maticiens fran ais de plus au
27. de sph res In S minaire H Cartan E N S 1954 1955 Expos No 20 7 pp Tores maximaux des groupes de Lie compacts In S minaire Sophus Lie E N S 1954 1955 Expos 23 8 pp Sous groupes ab liens des groupes de Lie compacts In S minaire Sophus Lie E N S 1954 1955 Expos 24 8 pp 1955 Le th or me de Brauer sur les caract res In S minaire Bourbaki 1954 1955 Expos 111 7 pp Faisceaux alg briques coh rents Ann of Math 2 61 197 278 G om trie alg brique et g om trie analytique Ann Inst Fourier Grenoble 6 1 42 Un th or me de dualit Comment Math Helv 29 9 26 Une propri t topologique des domaines de Runge Proc Amer Math Soc 6 133 134 Notice sur les travaux scientifiques In uvres Collected papers Vol I 1949 1959 394 401 Springer Verlag Berlin 1956 Th orie du corps de classes pour les rev tements non ramifi s de vari t s alg briques d apr s S Lang In S minaire Bourbaki 1955 1956 Expos 133 9 pp Correspondence Amer J Math 78 898 Cohomologie et g om trie alg brique In Proceedings of the International Congress of Mathematicians 1954 Amsterdam Vol III 515 520 Erven P Noordhoff N V Gronin gen Sur la dimension homologique des anneaux et des modules noeth riens In Proceedings of the international symposium on algebraic number theory Tokyo amp Nikko 1955 175 189 Science Council of Japan Tokyo 1957 Crit
28. des groupes formels de dimension 1 Bull Sci Math 2 91 113 115 R sum des cours de 1966 1967 Annuaire du Coll ge de France 51 52 1968 Abelian l adic representations and elliptic curves McGill University lecture notes Writ ten with the collaboration of W Kuyk and J Labute W A Benjamin Inc New York Amsterdam Translated into Japanese and Russian with J Tate Good reduction of abelian varieties Ann of Math 2 88 492 517 F G Frobenius Gesammelte Abhandlungen B nde I IT II Edited by J P Serre Springer Verlag Berlin Groupes de Grothendieck des sch mas en groupes r ductifs d ploy s Inst Hautes Etudes Sci Publ Math 34 37 52 R sum des cours de 1967 1968 Annuaire du Coll ge de France 47 50 1969 Une interpr tation des congruences relatives a la fonction t de Ramanujan In S minaire Delange Pisot Poitou 1967 68 Th orie des Nombres Expos 14 17 pp Cohomologie des groupes discrets C R Acad Sci Paris S r A B 268 A268 A271 Travaux de Baker In S minaire Bourbaki 1969 70 Expos 368 73 86 Lecture Notes in Math Vol 180 Springer Verlag Berlin R sum des cours de 1968 1969 Annuaire du Coll ge de France 43 46 List of Publications for Jean Pierre Serre 89 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 1970 p torsio
29. group as Galois group properties of adic varieties such as the following the number of points mod of an adic analytic variety of dimension d is O 4 for n oo general theorems on adic representations Let us mention two applications Given a non zero modular form f X anq which is an eigenvalue of all the Hecke operators and is not of type CM complex multiplication Serre proves that the series f is not lacunary more precisely if M s x denotes the number of integers n lt x such that a 0 then there exists a constant gt 0 such that M f x ax for x oo On the other hand if f 40 has complex multiplication then there exists a constant gt 0 such that M f x ax log x 2 for x oo In concrete examples Serre provides estimates for Furthermore if E Q is an elliptic curve without complex multiplication and if we assume GRH then there exists an absolute constant c such that the Galois group Ge of the points of the division of E is isomorphic to GL2 F for every 70 P Bayer prime gt c log Ng log log 2Nz where Ng denotes the product of all the primes of bad reduction of E 13 3 7 The Dedekind 7 function is a cusp form of weight 1 2 In S208 139 1985 a paper which Serre dedicated to R Rankin he studies the lacunar ity of the powers 7 when r is a positive integer If r is odd it was known that 7 is lacunary if r 1 3 If r is even it wa
30. is A has no coho mological torsion Moreover he gives an example of an abelian variety for which H A W is nota finitely generated module over the Witt vectors thus contradict ing an imprudent conjecture sic he had made in S76 38 1958 2 3 3 In S76 38 1958 the influence of Andr Weil is clear as it is in many of Serre s other papers The search for a good cohomology for varieties defined over finite fields was motivated by the Weil conjectures 1948 on the zeta function of these varieties As is well known such a cohomology was developed a few years later by Grothendieck using tale topology 2 3 4 In S89 45 1960 Serre shows that Weil s conjectures could be proved easily if a big if some basic properties of the cohomology of complex Kahler varieties could be extended to projective varieties over a finite field This was the starting point for Grothendieck s formulation of the so called standard conjectures on motives which are still unproved today 2 3 5 In S94 GE 50 1961 Serre constructs a non singular projective variety in characteristic p gt 0 which cannot be lifted to characteristic zero He recently 2005 improved this result by showing that if the variety can be lifted as a flat scheme to a local ring A then p A 0 The basic idea consists in transposing the problem to the context of finite groups 2 3 6 In his lecture S103 56 1963 at the Intern
31. le probl me des 4 couleurs coloriage des cartes avec seulement quatre couleurs et le probl me de K pler empilement des sph res dans l espace 3 dimensions Cela conduit des d monstrations qui ne sont pas r ellement v rifiables autrement dit ce ne sont pas de vraies d monstrations mais seulement des faits exp rimen taux tr s vraisemblables mais que personne ne peut garantir Vous avez voqu l augmentation du nombre des math maticiens Quelle est au jourd hui la situation Ou vont les math matiques L augmentation du nombre des math maticiens est un fait important On pouvait craindre que cela se fasse au d triment de la qualit En fait il n y a rien eu de tel Il y a beaucoup de tr s bons math maticiens en particulier parmi les jeunes fran ais un tr s bon augure Ce que je peux dire concernant l avenir c est qu en d pit de ce grand nombre de math maticiens nous ne sommes pas court de mati re Nous ne manquons pas de probl mes alors qu il y a un peu plus de deux si cles la fin du XVIII Lagrange tait pessimiste il pensait que la mine tait tarie qu il n y avait plus grand chose trouver Lagrange a crit cela juste avant que Gauss ne relance les math matiques de mani re extraordinaire lui tout seul Aujourd hui il y a beaucoup de terrains prospecter pour les jeunes math maticiens et aussi pour les moins jeunes J esp re Selon un lieu c
32. niveau mondial les n 1 et 3 sont des Fran ais et s appellent tous deux Lions un bon point pour le Coll ge J ai gard un tr s bon souvenir de ma collaboration Bourbaki entre 1949 et 1973 Elle m a appris beaucoup de choses la fois sur le fond en me for ant r diger des choses que je ne connaissais pas et sur la forme comment crire de fa on tre compris Elle m a appris aussi ne pas trop me fier aux sp cialistes La m thode de travail de Bourbaki est bien connue distribution des r dactions aux diff rents membres et critique des textes par lecture haute voix ligne ligne c est lent mais efficace Les r unions les congr s avaient lieu 3 fois par an Les discussions taient tr s vives parfois m me passionn es En fin de congr s on distribuait les r dactions de nouveaux r dacteurs Et l on recommengait Le m me chapitre tait souvent r dig quatre ou cinq fois La lenteur du processus explique que Bourbaki n ait publi finalement qu assez peu d ouvrages en quarante ann es d existence depuis les ann es 1930 1935 jusqu la fin des ann es 1970 o la production a d clin En ce qui concerne les livres eux m mes on peut dire qu ils ont rempli leur mission Les gens ont souvent cru que ces livres traitaient des sujets que Bourbaki trouvait int ressants La r alit est diff rente ses livres traitent de ce qui est utile pour faire des choses int r
33. of Siegel s theorem Hilbert s irreducibility theorem and its applications to the inverse Galois problem construction of elliptic curves of large rank sieve methods Davenport Halberstam s theorem asymptotic formulas for the number of integral points on affine varieties defined over number fields and the solution to the class number 1 problem by using integral points on modular curves 11 4 2 The paper S189 GE 122 1979 is an appendix to a text by M Waldschmidt on transcendental numbers 1970 It contains several useful properties of con nected commutative algebraic groups defined over a field k of characteristic zero They concern the following the existence of smooth projective compactifications quadratic growth at most of the height function when k is an algebraic extension of Q and uniformization by entire functions of order lt 2 when k C 11 4 3 The publication S194 1980 reproduces two letters addressed to D Masser The questions concern some of Masser s results on the linear independence of periods and pseudo periods of elliptic functions 1977 In the first letter Serre studies independence properties of the fields of division points of elliptic curves defined over an algebraic number field and with complex multiplication by differ ent quadratic imaginary number fields In the second letter Serre proves that under some reasonable assumptions the degree of the field generated by the division poin
34. other hand in particular in biology there are situations involving very many elements that have to be processed collectively There are branches of math ematics that deal with such questions They meet a need Another branch logic is 28 Jean Pierre Serre Mon premier demi si cle au Coll ge de France l informatique pour la fabrication des ordinateurs Il faut mentionner aussi la cryp tographie qui est une source de probl mes int ressants relatifs la th orie des nom bres En ce qui concerne la place des math matiques par rapport aux autres sciences on peut voir les math matiques comme un grand entrep t empli de rayonnages Les math maticiens d posent sur les rayons des choses dont ils garantissent qu elles sont vraies ils en donnent aussi le mode d emploi et la mani re de les reconstituer Les autres sciences viennent se servir en fonction de leurs besoins Le math maticien ne s occupe pas de ce qu on fait de ses produits Cette m taphore est un peu triviale mais elle refl te assez bien la situation Bien entendu on ne choisit pas de faire des math matiques pour mettre des choses sur les rayons on fait des math matiques pour le plaisir d en faire Voici un exemple personnel Ma femme Josiane tait sp cialiste de chimie quan tique Elle avait besoin d utiliser les repr sentations lin aires de certains groupes de sym tries Les ouvrages disponibles n taient pas satisfaisants ils taient corrects
35. par la th orie des cordes Son aspect le plus positif est de fournir aux math maticiens un grand nombre d nonc s qu il leur faut d montrer ou ventuellement d molir Par ailleurs notamment en biologie il y a tout ce qui rel ve de syst mes com portant un grand nombre d l ments qu il faut traiter collectivement Il existe des branches des math matiques qui s occupent de ces questions Cela r pond une demande Il y a aussi des demandes qui concernent la logique c est le cas de Jean Pierre Serre My First Fifty Years at the Coll ge de France 27 the loop space I couldn t help from waking up my wife who was sleeping in the bunk below Tve got it I said My thesis and many other things originated from that idea Of course these sudden discoveries are rare they have only happened to me twice in sixty years But they are illuminating moments truly exceptional Are there exchanges between the disciplines at the Coll ge de France No not for me There is no collective work even between the mathematicians at the College We work on quite different things This is not a bad thing The College is not supposed to be a club Many commonplace sayings such as collective work interdisciplinarity and team work do not apply to us What do you think about the dialogue between the neuroscientist Jean Pierre Changeux and the mathematician Alain Connes recorded in the book Mati re pens e This boo
36. report on Galois groups over Q presented to the S minaire Bourbaki S217 147 1988 Serre provides a summary of the status of the inverse Galois problem that is of the question of whether all finite groups are Galois groups of an equation with rational coefficients He mentions the solution of the problem in the solvable case due to I Shafarevich 1954 and its improvements by J Neukirch 1979 He gives Hilbert s realizations of the symmetric and alternating groups as Galois groups over Q by means of Hilbert s irreducibility theorem And in the 62 P Bayer most detailed part of the exposition he explains the rigidity methods of H Matzat 1980 and J G Thompson 1984 and presents a list of the simple groups known at that moment to be Galois over Q As another type of example he considers the realization of certain central extensions of simple groups which had recently been obtained thanks to his Tr x formula S204 131 1984 12 0 3 The papers S226 151 1990 and S227 152 1990 are related to the results of Mestre mentioned above The first one is about lifting elements of odd order from to An if one has several elements and their product is equal to 1 what is the product of their liftings 1 or 1 In the second paper which may be viewed as a geometrization of the first one Serre considers a ramified covering x Y X of curves in which all the ramification indices are odd He gives a formula r
37. s des vari t s ab liennes en caract ristique p Amer J Math 80 715 739 Sur la topologie des vari t s alg briques en caract ristique p In Symposium internacional de topolog a algebraica 24 53 Universidad Nacional Aut noma de M xico and UNESCO Mexico City Morphismes universels et vari t d Albanese In S minaire C Chevalley 1958 59 Vari t s de Picard Expos 10 22 pp Morphismes universels et diff rentielles de troisi me esp ce In S minaire C Chevalley 1958 59 Vari t s de Picard Expos 11 8 pp R sum des cours de 1957 1958 Annuaire du Coll ge de France 55 58 1959 Corps locaux et isog nies In S minaire Bourbaki 1958 1959 Expos 185 9 pp On the fundamental group of a unirational variety J London Math Soc 34 48 1 484 Groupes alg briques et corps de classes Publications de l institut de math matique de l universit de Nancago VII Hermann Paris Translated into English and Russian R sum des cours de 1958 1959 Annuaire du Coll ge de France 67 68 1960 Rationalit des fonctions amp des vari t s alg briques d apr s Bernard Dwork In S minaire Bourbaki 1959 1960 Expos 198 11 pp Rev tements ramifi s du plan projectif d apr s S Abhyankar In S minaire Bourbaki 1959 1960 Expos 204 7 pp Groupes finis a cohomologie p riodique d apr s R Swan In S minaire Bourbaki 1960 1961 Expos 209 12 pp Rigidit du foncteur de Jacobi d
38. studied by J F Mestre of conductor 5077 are isogenous Let K be an algebraic number field and A an abelian variety defined over K of dimension d Let pe Gal K K GL Ty A be the adic representation de fined by the Tate module Let Gat be the closure of Ge under the Zariski topology which is a Q algebraic subgroup of the general linear group GL2g Mumford and Tate conjectured that given A and K the group Gye is essentially independent of and more precisely that the connected component G 3 could be deduced from the Mumford Tate group by extension of scalars of Q to Q In the second part of the course S209 135 1985 Serre proves a series of results in this direction He shows for instance that the finite group Gee G78 is independent of 13 4 4 In the course S213 136 1986 Serre studies the variation with of the adic Lie groups associated to abelian varieties Let us keep the previous notation Given the homomorphism p Gal K K gt Ge c Auto Serre proves that if K is sufficiently large the image of p is open in the prod uct Ge i e the pg are almost independent In the case where n is odd or is equal to 2 or 6 and if End A Z he shows that the image of p is open in the product of the groups of symplectic similitudes GSp 7 es Here es is the al ternating form over 7 A deduced from a polarization e of A The ingredients of the proof are many the above
39. the Fields Medals With respect to Serre the committee acknowledged the new insights he had provided in topology and algebraic geometry At not quite 28 Serre became the youngest mathematician to receive the distinction a record that still stands today In his presentation of the Fields medalists H Weyl perhaps a little worried by Serre s youth recommended the two laureates to carry on as you began In the following sections we shall see just how far Serre followed Weil s advice In his address at the International Congress S63 27 1956 Serre describes the extension of sheaf theory to algebraic varieties defined over a field of any char acteristic see FAC Sect 2 1 One of the highlights is the algebraic analogue of the analytic duality theorem mentioned above it was soon vastly generalized by Grothendieck He also mentions the following problem If X is a non singular projective variety is it true that the formula By X dim H4 X Q p q n yields the n th Betti numbers of X occurring in Weil conjectures This is so for most varieties but there are counterexamples due to J Igusa 1955 2 Sheaf Cohomology The paper FAC 1955 by Serre the paper Tohoku 1957 by Grothendieck and the book by R Godement on sheaf theory 1964 were publications that did the most to stimulate the emergence of a new methodology in topology and abstract algebraic geometry This new approach emerged in the next twenty y
40. the extension L but only on the structure of G The proof uses unpublished results of his own on the cohomology of unitary groups in characteristic 2 9 5 Essential Dimension Let G be a simple algebraic group of adjoint type de fined over a field k The essential dimension of G is by definition the essential dimension of the functor of G torsors F H F G which is defined over the Jean Pierre Serre An Overview of His Work 53 category of field extensions F of k in loose terms it is the minimal number of parameters one needs in order to write a generic G torsor Here H 1 F G de notes the non abelian Galois cohomology set of G In S277 2006 Serre and V Chernousov give a lower bound for the essential dimension at a prime p 2 ed G 2 and for the essential dimension ed G of G It is proved in the paper that ed G 2 gt r 1 and thus ed G gt r 1 with r rankG Lower bounds for ed G p had been obtained earlier by Z Reichstein and B Youssin 2000 In their proof these authors made use of resolution of singularities so that their results were only valid for fields k of characteristic zero however a recent paper of P Gille and Z Reichstein has removed this restriction The proof of Chernousov Serre for p 2 is valid in every characteristic different from 2 It makes use of the exis tence of suitable orthogonal representations of G attached to quadratic forms The quadratic forms involved turn out to
41. the hypersurface no hypothesis of irreducibility or of non singularity is made Serre shows that the number N of zeros of a ho mogeneous polynomial f f Xo Xn in P F of degree d lt q 1 is at most dg py 2 where p q q 1 is the number of points in the projective space P F The result has been widely used in coding the Ory 11 4 Diophantine Problems Certain classical methods of transcendence based on the study of the solutions of differential equations mainly due to Th Schneider were transposed to the p adic setting by S Lang 1962 1965 In the S minaire Delange Pisot Poitou S124 1967 Serre studies the dependence of p adic expo nentials avoiding the use of differential equations 11 4 1 The book Lectures on the Mordell Weil Theorem S203 MW 1984 arose from the notes taken by M Waldschmidt of a course taught by Serre 1980 1981 which were translated and revised with the help of M Brown The content of the lectures was heights N ron Tate heights on abelian vari eties the Mordell Weil theorem on the finiteness generation of the rational points of any abelian variety defined over a number field Belyi theorem characterizing the non singular projective complex curves definable over Q Chabauty and Manin Demjanenko theorems on the Mordell conjecture previous to Faltings theorem 1983 Siegel s theorem on the integral points on affine curves Baker s effective forms
42. the ring of the finite adeles of Q The paper ends with a statement of the Sato Tate conjecture for arbitrary motives 14 Group Theory In response to a question raised by Olga Taussky 1937 on class field towers Serre proves in S145 85 1970 that for a finite p group G the knowledge of G does not in general imply the triviality of any term D G of its derived series More precisely for every n gt 1 and for every non cyclic finite abelian p group P of order 4 there exists a finite p group G such that D G 1 and G P 14 1 Representation Theory Serre s popular book Repr sentations lin aires des groupes finis S130 RLGF 1967 gives a reader friendly introduction to represen tation theory It also contains less elementary chapters on Brauer s theory of mod ular representations explained in terms of Grothendieck K groups the highlight being the cde triangle The text is well known to physicists and chemists and its first chapters are a standard reference in undergraduate or graduate courses on the subject 14 1 1 The paper S245 164 1994 is about the semisimplicity of the tensor product of group representations A theorem of Chevalley 1954 states that if k is a field of characteristic zero G is a group and V and V gt are two semisimple lIndeed the first part of the book was written by Serre for the use of his wife Josiane who was a quantum chemist and needed character theory in her
43. 149 88 1971 Serre undertakes the study of the cohomology of the S arithmetic subgroups I which are contained in L k For this purpose the group I is viewed as a discrete subgroup of a finite product G Ga of real or ultramet ric Lie groups The main tool is provided by the Bruhat Tits buildings associated to the v adic Lie groups L k for v S Soo The most important contributions include bounds for the cohomological dimension cd I finiteness properties and several results relating to the Euler Poincar characteristic x I and its relations with the values of zeta functions at negative integers generalizing the well known formula x SL2 Z 1 1 12 10 2 2 Borel and Serre S144 90 1970 prove that if G is a connected reduc tive linear algebraic group defined over Q which does not have non trivial charac ters it is possible to associate to G a contractible manifold with corners X whose interior X is a homogeneous space of G R isomorphic to a quotient G R K for a maximal compact subgroup K of G R Its boundary 3X has the same ho motopy type as the Tits building X of G i e the simplicial complex whose faces 56 P Bayer correspond to the k parabolic subgroups of G An arithmetic subgroup l C G Q acts properly on X and the quotient X T is compact this gives a compactification of X T which is often called the Borel Serre compactification If F is torsion free then the cohomol
44. 2 Divisibilit de certaines fonctions arithm tiques L Enseignement Math 2 22 3 4 227 260 Repr sentations lin aires des groupes finis alg briques d apr s Deligne Lusztig In S minaire Bourbaki 1975 76 Expos 487 256 273 Lecture Notes in Math Vol 567 Springer Verlag Berlin R sum des cours de 1975 1976 Annuaire du Coll ge de France 43 50 1977 Arbres amalgames SL2 R dig avec la collaboration de Hyman Bass Ast risque No 46 Translated into English and Russian Repr sentations adiques In Algebraic number theory Kyoto 1976 177 193 Japan Soc Promotion Sci Tokyo with H M Stark Modular forms of weight 1 2 In Modular functions of one variable VI Bonn 1976 Lecture Notes in Math Vol 627 27 67 Springer Verlag Berlin Modular forms of weight one and Galois representations In Algebraic number fields L functions and Galois properties Durham 1975 193 268 Academic Press London Majorations de sommes exponentielles In Journ es Arithm tiques de Caen Univ Caen Caen 1976 111 126 Ast risque No 41 42 J P Serre and D B Zagier editors Modular functions of one variable V Lecture Notes in Mathematics Vol 601 Springer Verlag Berlin J P Serre and D B Zagier editors Modular functions of one variable VI Lecture Notes in Mathematics Vol 627 Springer Verlag Berlin Points rationnels des courbes modulaires X N d apr s Barry Mazur In S
45. 2003 Jean Pierre Serre Jean Pierre Serre Mon premier demi si cle au College de France Jean Pierre Serre My First Fifty Years at the Coll ge de France Marc Kirsch Ce chapitre est une interview par Marc Kirsch Publi pr c demment dans Lettre du Coll ge de France n 18 d c 2006 Reproduit avec autorisation This chapter is an interview by Marc Kirsch Previously published in Lettre du Coll ge de France no 18 d c 2006 Reprinted with permission M Kirsch BX Coll ge de France 11 place Marcelin Berthelot 75231 Paris Cedex 05 France e mail marc kirsch college de france fr H Holden R Piene eds The Abel Prize 15 DOI 10 1007 978 3 642 01373 7_3 Springer Verlag Berlin Heidelberg 2010 16 Jean Pierre Serre Mon premier demi si cle au Coll ge de France Jean Pierre Serre Professeur au Coll ge de France titulaire de la chaire d Alg bre et G om trie de 1956 1994 Vous avez enseign au Coll ge de France de 1956 1994 dans la chaire d Alg bre et G om trie Quel souvenir en gardez vous J ai occup cette chaire pendant 38 ans C est une longue p riode mais il y a des pr c dents si l on en croit l Annuaire du Coll ge de France au XIX si cle la chaire de physique n a t occup e que par deux professeurs l un est rest 60 ans l autre 40 Il est vrai qu il n y avait pas de retraite cette poque et que les pro fesseurs avaient des suppl ants
46. 498 524 On the values of the characters of compact Lie groups Oberwolfach Reports 1 666 667 Compl te r ductibilit In S minaire Bourbaki 2003 2004 Expos 932 195 217 Ast risque No 299 2005 L Illusie Grothendieck s existence theorem in formal geometry In Fundamental algebraic geometry 179 233 Math Surveys Monogr Vol 123 Amer Math Soc Providence RI 2005 With a letter by J P Serre BL bases and unitary groups in characteristic 2 Oberwolfach Reports 2 37 40 Groupes Finis arXiv math 0503 154 2006 with V Chernousov Estimating essential dimensions via orthogonal representations J Algebra 305 2 1055 1070 with M Rost and J P Tignol La forme trace d une alg bre simple centrale de degr 4 C R Math Acad Sci Paris 342 2 83 87 Coordonn es de Kac Oberwolfach Reports 3 1787 1790 2007 Bounds for the orders of the finite subgroups of G k In Group representation theory 405 450 EPFL Press Lausanne 2008 Three letters to Walter Feit on group representations and quaternions J Algebra 319 549 557 Two letters to Jaap Top In Algebraic Geometry and its Applications 84 87 World Sci Publ Co Le groupe de Cremona et ses sous groupes finis In S minaire Bourbaki 2008 2009 Expos 1000 24 pp 2009 A Minkowski style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field Moscow Math J 9 1 193 208 Ho
47. 77 written with the help of H Bass 3rd edn 1983 translated into English and Russian MW Serre J P Lectures on the Mordell Weil Theorem Vieweg Wiesbaden 1989 3rd edn 1997 translated and edited by Martin Brown from notes of M Waldschmidt French edition Publ Math Univ Pierre et Marie Curie 1984 TGT Serre J P Topics in Galois Theory Jones amp Bartlett Boston 1992 written with the help of H Darmon 2nd edn AK Peters 2008 SEM Serre J P Expos s de s minaires 1950 1999 Documents Math matiques Paris vol 1 Soc Math France Paris 2001 2nd edn augmented 2008 GRSE Colmez P Serre J P eds Correspondance Grothendieck Serre Documents Math matiques Paris vol 2 Soc Math France Paris 2001 bilingual edn AMS 2004 CI Garibaldi S Merkurjev A Serre J P Cohomological Invariants in Galois Cohomol ogy Univ Lect Ser vol 28 Am Math Soc Providence 2003 List of Publications for Jean Pierre Serre 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1948 Groupes d homologie d un complexe simplicial In S minaire H Cartan E N S 1948 49 Expos 2 9 pp with H Cartan Produits tensoriels In S minaire H Cartan E N S 1948 49 Expos 11 12 pp 1949 Extensions des applications Homotopie In S minaire H Cartan E N S 1949 1950 Espaces fibr s et homotopie
48. 8 Serre S18 GE 3 1950 and Hochschild Serre S46 GE 15 1953 go further Given a discrete group G a nor mal subgroup K of G and a G module A they construct a spectral sequence H G K H K A gt H G A If H K A 0 for 0 lt r lt q the spectral sequence gives rise to the exact se quence 0 H4 G K A gt H4 G A gt H4 K ASE HIH G K AK HI G A This sequence became a key ingredient in many proofs Similar results hold for Lie algebras as shown in S47 16 1953 1 3 Sheaf Cohomology of Complex Manifolds In his seminar at the Ecole Nor male Sup rieure Cartan showed in 1952 1953 that earlier results of K Oka and himself can be reinterpreted and generalized in the setting of Stein manifolds by using analytic coherent sheaves and their cohomology he thus obtained his well known Theorems A and B In S42 GE 23 1953 see also the letters to Cartan reproduced in S231 1991 Serre gives several applications of Cartan s theorems he shows for instance that the Betti numbers of a Stein manifold of complex dimen sion n can be computed la de Rham using holomorphic differential forms in particular they vanish in dimension gt n But Serre soon became more interested in compact complex manifolds and especially in algebraic ones A first step was the theorem obtained in collaboration with Cartan see S41 24 1953 that the cohomology groups H4 X F associated to a com
49. 8 contains a report by Serre on Deligne s work upon request by the Fields Medal Committee Deligne was awarded the Fields Medal in 1978 A Grothendieck was awarded the Fields Medal in 1966 together with M Atiyah P Cohen and S Smale Serre s original report written in 1965 on the work of Grothendieck and addressed to the Fields Medal Committee was reproduced much later in S220 1989 Among many other applications Weil conjectures imply good estimations for certain exponential sums since these sums can be viewed as traces of Frobenius endomorphisms acting on the cohomology of varieties over finite fields The results of Deligne on this subject were explained by Serre in S180 111 1977 11 3 Number of Points of Curves Over Finite Fields Let C be an absolutely irreducible non singular projective curve of genus g defined over F After the proof of the Riemann hypothesis for curves due to Weil 1940 1948 it was known that the number N N C of the rational points of C over F satisfies the inequality IN q 1 lt 2gq Several results due to H Stark 1973 Y Ihara 1981 and V G Drinfeld and S G Vladut 1983 showed that Weil s bound can often be improved On the other hand it was of interest for coding theory to have curves of low genus with many points Jean Pierre Serre An Overview of His Work 59 In the papers S201 128 1983 and S200 129 1983 Serre expands the results of the previous
50. A Ogg on the cuspidal group of the Jacobian of the modular curve Xo N and some of the results of Mazur on the rational points of this curve In it we find the modular interpretation of the modular curve the definition of the Hecke operators as correspondences acting on it the study of the Eisenstein ideal a study of the N ron model of the Jacobian of the modular curve a study of the regular model of the modular curve and so on 11 1 3 The third lecture S183 1977 explains the results of Mazur on the Eisen stein ideal and the rational points of modular curves 1978 and on the rational isogenies of prime degree 1978 The main theorem is that if N is a prime not be longing to the set 2 3 5 7 11 13 17 19 37 43 67 163 then the modular curve Xo N has no rational points other than the cusps As an application one obtains the possible structures for the rational torsion groups Eror Q of the elliptic curves defined over Q In many aspects the above work paved the way for G Faltings proof of the Mordell Weil theorem 1983 11 1 4 The paper S237 159 1993 written with T Ekedahl gives a long list of curves of high genus whose Jacobian is completely decomposable 1 e isogenous to a product of elliptic curves They ask whether it is true that for every genus g gt 0 there exists a curve of genus g whose Jacobian is completely decomposed or on the contrary whether the genus of a curve whose Jacobian is comple
51. P F Swinnerton Dyer 1973 on congruences Swinnerton Dyer s results on this topic were presented by Serre at the S minaire Bourbaki S155 95 1972 13 3 2 In the papers S161 100 1974 S168 1975 and S173 108 1976 it is proved that given a modular form f XP o cne i z M with respect to a con gruence subgroup of the full modular group SL2 Z and of integral weight k gt 1 for each integer m gt 1 the set of integers n which satisfy the congruence c 0 mod m is of density 1 The proof uses adic representations combined with an analytic argument due to E Landau Given a cusp form f anq q ernie without complex multiplication of weight k gt 2 normalized eigenvector of all the Hecke operators and with coefficients in Z Serre shows that the set of integers n such that a 0 has a density which is gt 0 in particular the series f is not lacunary 13 3 3 Deligne and Serre in the paper S162 GE 101 1974 dedicated to H Cartan prove that every cusp form of weight 1 which 1s an eigenfunction of the Hecke op erators corresponds by Mellin s transform to the Artin L function of an irreducible complex linear representation p Gal Q Q GL 2 C Moreover the Artin con ductor of p coincides with the level of the cusp form provided it is a newform In order to prove the theorem Serre and Deligne construct Galois representations mod for each prime of sufficiently small imag
52. accessible to the general Lie theory This elementary remark opened up many possibilities since it is much easier to classify Lie algebras than profinite groups 3 0 3 The booklet Alg bres de Lie semi simples complexes S119 ALSC 1966 reproduces a series of lectures given in Algiers in 1965 It gives a concise introduc tion mostly with proofs to complex semisimple Lie algebras and thus supplements Lie Algebras and Lie Groups The main chapters are those on Cartan subalgebras Jean Pierre Serre An Overview of His Work 43 representation theory for 5l2 root systems and their Weyl groups structure theo rems for semisimple Lie algebras linear representations of semisimple Lie algebras Weil s character formula without proof and the dictionary between compact Lie groups and reductive algebraic groups over C without proof but see G bres S239 160 1993 The book also gives a presentation of semisimple Lie algebras by generators and relations including the so called Serre relations which as he says should be called Chevalley relations because of their earlier use by Chevalley 4 Local Algebra Serre s work in FAC and GAGA made him introduce homological methods in local algebra such as flatness and the characterization of regular local rings as the only noetherian local rings of finite homological dimension completing an earlier result of A Auslander and D Buchsbaum 1956 cf S64 33 1956 4 0 1
53. ace that he introduces is more general than the one that was usual at the time allowing him to deal with loop spaces as follows given a pathwise connected topological space X and a point x X the loop space Q at x is viewed as the fiber of a fiber space E over the base X The elements of E are the paths of X starting at x The crucial fact is that the map which assigns to each path its endpoint is a fibering f E X in the above sense The space E is contractible and once Leray s theory is suitably adapted it turns to be very useful for relating the homology of Q to that of X Jean Pierre Serre An Overview of His Work 37 Serre s thesis contains several applications For example by combining Morse s theory 1938 with his own results Serre proves that on every compact connected Riemannian manifold there exist infinitely many geodesics connecting any two dis tinct points But undoubtedly the most remarkable application is the one concern ing the computation of the homotopy groups of spheres 7 Sn 1 1 Homotopy Groups of Spheres Earlier studies by H Hopf H Freudenthal 1938 and others had determined the groups 7 S for i lt n 2 L Pontrjagin and G W Whitehead 1950 had computed the groups z 42 S and H Hopf had proved that the group 72 _1 S for n even has Z as a quotient and hence is infinite Thanks also to Freudenthal s suspension theorem it was known that the group Tn Kk S depends only on k i
54. algebraic groups form a subcategory Qo of Q If G is a com mutative quasi algebraic group and G is its connected component the quotient ro G G G isa finite abelian group The category of commutative pro algebraic groups is defined as P Pro Q Let Po Pro Qo be the subcategory of abelian profinite groups The category P has projective limits and enough projective ob jects Every projective object of P is a product of indecomposable projectives and the indecomposable projective groups coincide with the projective envelopes of the elementary commutative quasi algebraic groups The functor xo P Po is right exact its left derived functors are denoted by G 7 G One of the main results of the paper is that 7 G 0 if i gt 1 The group x G is called the fundamental group of G The connected and simply connected commutative pro algebraic groups form a subcategory S of P For each object G in P there exists a unique group G in S and a morphism u G G whose kernel and cokernel belong to Po so that one obtains an exact sequence 1 71 G gt G gt G gt xo G 1 By means of the universal covering functor the categories P Po and S become equivalent After computing the homotopy groups of the elementary commutative pro algebraic groups it is shown in S88 GE 49 1960 that every commutative pro algebraic group has cohomological dimension lt 2 if k has positive characteristic and has cohomological dimension lt
55. ally interested in the p elementary abelian subgroups of G for p a prime They defined the p rank p of G as the largest integer n such that G contains such a subgroup with order p the theorem above shows that lt G lt W where is the rank of G and W the p rank of its Weyl group W N T They show that if G is con nected and its p rank is greater than its rank then G has homological p torsion As a corollary the compact Lie groups of type G2 F4 E7 and Eg have homolog ical 2 torsion The proof uses results on the cohomology algebra modulo p of the classifying space Bg of G which had been studied in Borel s thesis 3 0 2 Serre s book Lie Algebras and Lie Groups S110 LALG 1965 was based on a course at Harvard As indicated by its title it consists of two parts the first one gives the general theory of Lie algebras in characteristic zero including the standard theorems of Lie Engel Cartan and Whitehead but not including root systems The second one is about analytic manifolds over a complete field k either real complex or ultrametric It is in this context that Serre gives the standard Lie dictionary Lie groups Lie algebras assuming that k has characteristic zero His interest in the p adic case arose when he realized around 1962 that the rather mysterious Galois groups associated to the Tate modules of abelian varieties see Sect 13 are p adic Lie groups so that their Lie algebras are
56. amified representations of GL2 Q in characteristic p 4 which are universal with the property of containing an eigen vector of the Hecke operator Te with a given eigenvalue a In an appendix to the paper R Livn mentions further developments of these questions For example a general study of the representations of GL2 Q z in characteristic p 4 was done later by Marie France Vign ras 1989 the case p has recently been studied by several people 13 3 10 The general strategy of the work of Wiles 1995 and Taylor Wiles 1995 on modular elliptic curves and Fermat s Last Theorem was presented by Serre at the S minaire Bourbaki S248 168 1995 The proof that any semistable elliptic curve defined over Q is modular is long and uses results of Ribet Mazur Lang lands Tunnell Diamond among others On this occasion Serre said that he did not claim to have verified all the technical details of the proof qui sont essentiels bien entendu 13 4 Abelian Varieties and adic Representations Let A be an abelian variety of dimension d defined over a field k Given a prime char k the Tate module Tr A lim A k is a free Zy module of rank 2d Let V A T A Qx lt The action of the absolute Galois group of k on the Tate module of A gives an adic representation pg Gal ks k GL T A GL24 Ze its image Ge 72 P Bayer is a compact subgroup of GL Z4 hence it is a
57. anslation edited and with a preface by A N Tyurin With appendices by V I Arnol d J P Serre and A N Tyurin Rapport au comit Fields sur les travaux de A Grothendieck K Theory 3 3 199 204 Y Ihara K Ribet and J P Serre editors Galois groups over Q Mathematical Sciences Research Institute Publications Vol 16 Springer Verlag New York J G Thompson Hecke operators and noncongruence subgroups In Group theory Singa pore 1987 de Gruyter Berlin Including a letter from J P Serre R sum des cours de 1988 1989 Annuaire du Coll ge de France 75 78 1990 Construction de rev tements tales de la droite affine en caract ristique p C R Acad Sci Paris S r I Math 311 6 341 346 Sp cialisation des l ments de Br2 Q T1 1h C R Acad Sci Paris S r I Math 311 7 397 402 Rel vements dans A C R Acad Sci Paris S r I Math 311 8 477 482 Rev tements a ramification impaire et th ta caract ristiques C R Acad Sci Paris S r I Math 311 9 547 552 R sum des cours de 1989 1990 Annuaire du Coll ge de France 81 84 List of Publications for Jean Pierre Serre 93 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 1991 Motifs Journ es Arithm tiques Luminy 1989 Ast risque 198 200 11 333 349 Lettre M Tsfasman Journ es Arithm tiq
58. ated to elements of norm 1 of a central simple k algebra D of rank n by A S Merkurjev and A Suslin 1983 1985 Spin groups in particular all those of type Bn by A S Merkurjev classical groups except those of triality type D4 by Eva Bayer and Raman Parimala 1995 groups of type G2 and F4 In conclusion Conjecture II remains open for the types E6 E7 Eg and triality type D4 9 4 Self dual Normal Basis E Bayer Fluckiger and H W Lenstra 1990 defined the notion of a self dual normal base in a G Galois algebra L K and proved the existence of such a base when G has odd order When G has even order existence criteria were given by E Bayer and Serre S244 163 1994 in the special case where the 2 Sylow subgroups of G are elementary abelian if 24 is the order of such a Sylow group they associate to L K ad Pfister form qz and show that a self dual normal base exists if and only if qz is hyperbolic Thanks to Voevodsky s proof of Milnor s conjecture this criterion can also be stated as the vanishing of a specific element of H4 K Z 2Z 9 4 1 In the Oberwolfach announcement S275 2005 Serre gives a criterion for the existence of a self dual normal base for a finite Galois extension L K of a field K of characteristic 2 He proves that such a base exists if and only if the Galois group of L K is generated by squares and by elements of order 2 Note that the criterion does not depend on K nor on
59. atic forms over k of rank n k Pfister k the isomorphism classes of n Pfister forms over k of rank equal ton Examples of functors H are provided by the abelian Galois cohomology groups HH k C and their direct sum H k C where C denotes a discrete Gal k ko module or by the functor H k W k where W k stands for the Witt ring of non degenerate quadratic forms on k The aim of the lectures is to determine the group of invariants Inv A H In the background material of the book concerning Galois cohomology we find the notion of the residue of a cohomology class of H K C at a discrete valuation v of a field K as well as the value at v for those cohomology classes with residue equal to zero Basic properties of restriction and corestriction of invariants are ob tained in perfect analogy with the case of group cohomology An important tool is the notion of versal torsor which plays an analogous role to that of the universal bundle in topology an invariant is completely determined by its value on a ver sal G torsor These techniques allow the determination of the mod 2 invariants for quadratic forms hermitian forms rank n tale algebras octonions or Albert alge bras when char ko 4 2 In particular the mod 2 invariants of rank n tale algebras make up a free H ko module whose basis consists of the Stiefel Whitney classes wi for O lt i lt n 2 This gives a new proof of Serre s earlier formula on this sub
60. ational Congress of Math ematicians held in Stockholm in 1962 Serre offered a summary of scheme the ory After revising Grothendieck s notion of Grassmannian Hilbert scheme Picard scheme and moduli scheme for curves of a given genus he goes on to schemes over complete noetherian local rings where he mentions the very interesting and in those days recent results of N ron Although the language of schemes would become usual in Serre s texts it is worth saying that he has never overused it 42 P Bayer 3 Lie Groups and Lie Algebras Serre s interest in Lie theory was already apparent in his complementary thesis S28 14 1952 which contains a presentation of the results on Hilbert s fifth problem up to 1951 i e just before it was solved by A M Gleason D Montgomery and L Zippin 3 0 1 In 1953 Borel and Serre began to be interested in the finite subgroups of compact Lie groups a topic to which they would return several times in later years see e g S250 167 1996 and S260 174 1999 In S45 1953 reproduced in A Borel uvres Collected Papers no 24 they prove that every supersolvable finite subgroup of a compact Lie group G is contained in the normalizer N of a maximal torus T of G this is a generalization of a theorem of Blichfeldt relative to G U In particular the determination of the abelian subgroups of G is reduced to that of the abelian subgroups of N Borel and Serre were especi
61. auxquelles Dieu a d ob ir C est ce qu il a dit La Knesset a appr ci M Schmidt Hommes de Science 218 227 Hermann Paris 1990 AMS American Mathematical Society JE Littlewood A Mathematician s Miscellany Methuen and Co 1953 This book of fers a very good description of the unconscious aspect of creative work A few years ago my friend R Bott and myself went to receive a prize in Israel the Wolf prize awarded by the Knesset in Jerusalem Bott had to say a few words on mathematics He asked me what he should say I replied It s very simple all you have to explain is this other sciences seek to discover the laws that God has chosen mathematics seeks to discover the laws which God has to obey And that is what he said The Knesset appreciated it Serre and Henri Cartan Prix Julia 1970 Jean Pierre Serre My First Fifty Years at the College de France Anatole Abragam Serre and Jaques Tits Serre and Yuichiro Taguchi 31 32 Jean Pierre Serre Mon premier demi si cle au Coll ge de France Serre Ge W A LA i i Serre May 9 2003 photo by Chino Hasebe Jean Pierre Serre My First Fifty Years at the College de France Serre 2003 The Abel Lecture Oslo 2003 33 Jean Pierre Serre An Overview of His Work Pilar Bayer Introduction The work of Jean Pierre Serre represents an important breakthrough in at least four mathematical areas algebraic topology alg
62. can bring great joy Poincar Hadamard and Little wood have explained it very well As for myself I still have the memory of an idea that contributed to unlocking homotopy theory It happened one night while traveling home from vacation in 1950 in the sleeping car of a train I had been looking for a fiber space with such and such properties Then the answer came 26 Jean Pierre Serre Mon premier demi si cle au Coll ge de France l espace des lacets Je n ai pas pu m emp cher de r veiller ma femme qui dormait dans la couchette du dessous pour lui dire a y est Ma th se est sortie de l et bien d autres choses encore Bien s r ces d couvertes soudaines sont rares cela m est arriv peut tre deux fois en soixante ans Mais ce sont des moments lumineux vraiment exceptionnels Le Coll ge de France est il un endroit ou l on change avec d autres disciplines Non pas pour moi M me entre les math maticiens du Coll ge il n y a pas de tra vail collectif Il faut pr ciser que nous travaillons dans des branches souvent tr s s par es Ce n est pas un mal le Coll ge n est pas cens tre un club Un certain nom bre de lieux communs modernes comme le travail collectif V interdisciplinarit et le travail en quipe ne s appliquent pas nous Qu avez vous pens du dialogue entre un sp cialiste de neurosciences Jean Pierre Changeux et le math maticien Alain Connes qui est restitu
63. d Sci Paris 236 2475 2477 with H Cartan Un th or me de finitude concernant les vari t s analytiques compactes C R Acad Sci Paris 237 128 130 Quelques probl mes globaux relatifs aux vari t s de Stein In Colloque sur les fonctions de plusieurs variables 1953 57 68 Georges Thone Li ge Cohomologie modulo 2 des complexes d Eilenberg MacLane Comment Math Helv 27 198 232 with A Borel Groupes de Lie et puissances r duites de Steenrod Amer J Math 75 409 448 with A Borel Sur certains sous groupes des groupes de Lie compacts Comment Math Helv 27 128 139 with G P Hochschild Cohomology of group extensions Trans Amer Math Soc 14 110 134 List of Publications for Jean Pierre Serre 85 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 with G P Hochschild Cohomology of Lie algebras Ann of Math 2 57 591 603 Groupes d homotopie et classes de groupes ab liens Ann of Math 2 58 258 294 1954 Faisceaux analytiques In S minaire Bourbaki 1953 1954 Expos 95 6 pp Repr sentations lin aires et espaces homog nes k hl riens des groupes de Lie compacts d apr s Armand Borel et Andr Weil In S minaire Bourbaki 1953 1954 Expos 100 8 pp Les espaces K zt n In S minaire H Cartan E N S 1954 1955 Expos 1 7 pp Groupes d homotopie des bouquets
64. d by R Remmert and J P Serre Un exemple de s rie de Poincar non rationnelle Nederl Akad Wetensch Indag Math 41 4 469 471 1980 Deux lettres Abelian functions and transcendental numbers Collog Ecole Polytech Palaiseau 1979 M m Soc Math France 2 95 102 Extensions icosa driques In Seminar on Number Theory 1979 1980 Expos 19 7 pp Univ Bordeaux I Talence R sum des cours de 1979 1980 Annuaire du Coll ge de France 65 72 1981 Quelques applications du th or me de densit de Chebotarev Inst Hautes tudes Sci Publ Math 54 323 401 R sum des cours de 1980 1981 Annuaire du Coll ge de France 67 73 1982 R sum des cours de 1981 1982 Annuaire du Coll ge de France 81 89 1983 Nombres de points des courbes alg briques sur F In Seminar on number theory 1982 1985 Exp No 22 8 pp Univ Bordeaux I Talence Sur le nombre des points rationnels d une courbe alg brique sur un corps fini C R Acad Sci Paris S r I Math 296 9 397 402 R sum des cours de 1982 1983 Annuaire du Coll ge de France 81 86 1984 Autour du Th or me de Mordell Weil I et IT Publ Math Univ Pierre et Marie Curie Notes de cours r dig es par M Waldschmidt Translated into English L invariant de Witt de la forme Tr x Comment Math Helv 59 4 65 1 676 R sum des cours de 1983 1984 Annuaire du Coll ge de France 79 83 92 206 207 208 209
65. d the analytic aspects of the com plex projective varieties The main result is the following Assume that X is a projective variety over C and let X be the complex analytic space associated to X Then the natural functor algebraic coherent sheaves on X analytic coherent sheaves on X is an equivalence of categories which preserves cohomology As applications we mention the invariance of the Betti numbers under automor phisms of the complex field C when X is non singular as well as the comparison of principal algebraic fiber bundles of base X and principal analytic fiber bundles of base X with the same structural group G GAGA contains an appendix introducing the notion of flatness and applying it to compare the algebraic and the analytic local rings of X and X at a given point Flatness was to play an important role in Grothendieck s later work 2 2 1 Let X be a normal analytic space and S a closed analytic subset of X with codim gt 2 at every point In S120 GE 68 1966 Serre studies the extendibility of coherent analytic sheaves F on X S He shows that it is equivalent to the coherence of the direct image i 7 where i X S X denotes the inclusion When X is projective this implies that the extendible sheaves are the same as the algebraic ones 2 3 Cohomology of Algebraic Varieties The paper S68 GE 35 1957 gives a cohomological characterization of affine varieties similar to that
66. dans le livre Mati re pens e Ce livre est un bel exemple de dialogue de sourds Changeux ne comprend pas ce que dit Connes et inversement C est assez tonnant Personnellement je suis du c t de Connes Les v rit s math matiques sont ind pendantes de nous Notre seul choix porte sur la fa on de les exprimer Si on le d sire on peut le faire sans introduire aucune terminologie Consid rons par exemple une troupe de soldats Leur g n ral aime les arranger de deux fa ons soit en rectangle soit en 2 carr s C est au sergent de les placer Il s aper oit qu il n a qu les mettre en rang par 4 s il en reste 1 qu il n a pas pu placer ou bien il arrivera a les mettre tous en rectangle ou bien il arrivera a les r partir en deux carr s Traduction technique le nombre n des soldats est de la forme 4k 1 Sin n est pas premier on peut arranger les soldats en rectangle Sin est premier un th or me d a Fermat dit que n est somme de deux carr s Quelle est la place des math matiques par rapport aux autres sciences Y a t il une demande nouvelle de math matiques venant de ces sciences Sans doute mais il faut s parer les choses Il y a d une part la physique th orique qui est tellement th orique qu elle est cheval entre math matique et physique les physiciens consid rant que ce sont des math matiques tandis que les math mati ciens sont d un avis contraire Elle est symbolis e
67. e this allows them to lift these representations to characteristic zero and to obtain from them the desired complex representation the proof also uses an average bound on the eigenvalues of the Hecke operators due to Rankin Note that here the existence of adic representations as sociated to modular forms of weight k gt 2 is used to deduce an existence theorem for complex representations associated to weight k 1 modular forms The paper became a basic reference on the subject since it represents a small but non trivial step in the direction of the Langlands conjectures In particular it shows that certain Artin L functions are entire Another consequence is that the Ramanujan Petersson conjecture holds for weight k 1 For weight k gt 2 its truth follows from Deligne s results on Weil s conjectures and on the existence of Jean Pierre Serre An Overview of His Work 69 adic representations associated to cusp forms Not long after the study of mod ular forms of weight k 1 was illustrated by Serre in S179 110 1977 with many numerical examples due to Tate 13 3 4 In 1974 Serre opened the Journ es Arithm tiques held in Bordeaux with a lecture on Hecke operators mod S169 104 1975 in those days a fairly new subject for most of the audience Consider the algebra M of modular forms mod with respect to the modular group SL2 Z Serre proves that the systems of eigenvalues ap of the Hecke operators
68. e k Q Schur s bound is optimal Both results were recalled by Serre in his lectures with almost complete proofs The S bound for any reductive group and any finite subgroup of G k is obtained in terms of vg W the adic valuation of the order of the Weyl group of G and cer tain cyclotomic invariants of the field k defined ad hoc The Minkowski bound is more precise but in order to obtain it Serre needs to assume that the group G is semisimple of inner type the action of Gal k k on its Dynkin diagram is trivial If r is its rank then its Weyl group W has a natural linear representation of degree r The ring of invariants Q x1 x isa polynomial algebra Q P1 P where P are homogenous polynomials of degrees dj lt d2 lt lt d Under the as sumption that G is semisimple of inner type with root system R Serre gives a Minkowski style bound M k R for G which depends only on the cyclotomic invariants of the field k and the degrees d i 1 r Moreover it is optimal except when 2 and 1 does not belong to W As an illustration let us mention that if G is a Q group of type Eg then the order of any finite subgroup of G Q divides M Q Eg 2 3 59 74 11 137 19 31 and that this bound is sharp 15 Miscellaneous Writings Serre has written an endless number of impeccable letters over the years They are now found as appendices of books in papers or simply carefully saved
69. e n gt 1 and with T path connected then there exists a continuous map of the circle S in S which cannot be lifted to T The second application is arithmetical and concerns the number of zeros N f in the finite field F of a polynomial f Z X Serre shows that if the degree of f is n gt 1 and f is irreducible then the set Po f of the primes p such that V f gt 0 has a natural density gt 1 n The proof consists of combining Cameron Cohen s theorem with Chebotarev s density theorem The paper is illus trated with the computation of N f for f x x 1 and n 2 3 4 In these three cases it is shown how the numbers N f can be read from the coefficients of suitable cusp forms of weight 1 82 P Bayer And finally let us mention the preprint How to use finite fields for problems concerning infinite fields a mathematical entertainment just written by Serre S284 2009 in which he discusses old results of PA Smith 1934 M Lazard 1955 and A Grothendieck 1966 and shows how to prove them and sometimes improve them either with elementary tools or with topological techniques Acknowledgements The author acknowledges the valuable help provided by Jean Pierre Serre and by the editors of the book during the preparation of the manuscript References GE Serre J P uvres Collected Papers vol I 1949 1959 vol IT 1960 1971 vol HI 1972 1984 vol IV 1985 1998 Springer Berlin 1986 2000
70. e problem of the determination of the finite subgroups of a Lie group has received a great amount of attention Embedding questions of finite simple groups and their non split cen tral extensions in Lie groups of exceptional type have been solved by the work of many mathematicians The paper S250 167 1996 contains embeddings of some of the groups PSL F into simple Lie groups Let G denote a semisim ple connected linear algebraic group over an algebraically closed field k which is simple of adjoint type let h be its Coxeter number The purpose of the paper is to prove that if p h 1 is a prime then the group PGL2 F can be embedded into G k except if char k 4 2 and h 2 and that if p 2h 1 is a prime then the group PSL F can be embedded into G k Since for G PGL 2 one has h 2 the theorem generalizes the classical result that the groups A4 PSL2 F3 S4 PGL2 F3 and As PSL2 F5 can be embedded into PGL2 C The result for p 2h 1 was known in the case of characteristic zero it was part of a con jecture by B Kostant 1983 and it had been verified case by case with the aid of computers Moreover the values h 1 or 2h 1 for p are maximal in the sense that if PGL2 F respectively PSL2 F are embedded in G C then p lt h 1 respectively p lt 2h I In his paper Serre proves the two results in a unified way One starts from a certain principal homomorphism PGL2 F G F if p 2 h If p
71. ears through the momentous work EGA 1960 1964 SGA 1968 1971 1977 accomplished by Grothendieck and his collaborators 2 1 FAC In his foundational paper entitled Faisceaux alg briques coh rents S56 29 1955 known as FAC Serre introduces coherent sheaves in the setting of 40 P Bayer algebraic varieties over an algebraically closed field k of arbitrary characteristic There are three chapters in FAC Chapter I is devoted to coherent sheaves and general sheaf theory Chapter I starts with a sheaf style definition of what an algebraic variety is with the re striction that its local rings are reduced and then shows that the theory of affine algebraic varieties is similar to Cartan s theory of Stein manifolds the higher coho mology groups of a coherent sheaf are zero Chapter III is devoted to projective varieties The cohomology groups of coherent sheaves are usually non zero but they are finite dimensional it is shown that they can be computed algebraically using the Ext functors which had just been defined by Cartan Eilenberg this was their first application to algebraic geometry there would be many others 2 2 GAGA In 1956 Serre was appointed Professeur at the Coll ge de France in the chair Alg bre et G om trie The same year he published the paper G om trie alg brique et g om trie analytique S57 32 1955 1956 usually known as GAGA in which he compares the algebraic an
72. ebraic geometry algebra and number theory His outstanding mathematical achievements have been a source of inspi ration for many mathematicians His contributions to the field were recognized in 2003 when he was awarded the Abel Prize by the Norwegian Academy of Sciences and Letters presented on that occasion for the first time To date four volumes of Serre s work uvres Collected Papers II III 1986 IV 2000 S210 S211 S212 S261 have been published by Springer Verlag These volumes include 173 papers from 1949 to 1998 together with com ments on later developments added by the author himself Some of the papers that he coauthored with A Borel are to be found in A Borel uvres Collected Pa pers Springer 1983 Serre has written some twenty books which have been fre quently reprinted and translated into several languages mostly English and Russian but sometimes also Chinese German Japanese Polish or Spanish He has also delivered lectures in many seminars Bourbaki Cartan Chevalley Delange Pisot Poitou Grothendieck Sophus Lie etc some of them have been gathered in the books S262 SEM 2001 2008 Summarizing his work is a difficult task especially because his papers present a rich web of interrelationships and hence can hardly be put in linear order Here we have limited ourselves to a presentation of their contents with only a brief dis cussion of their innovative character Researc
73. elating several cohomological invariants of this covering here the behaviour of the theta characteristics of X under z plays an essential role He also asks whether there is a general formula including those in S204 131 1984 and S227 152 1990 This was done later by H Esnault B Kahn and E Viehweg 1993 12 0 4 The course given by Serre S236 GE 156 1992 focuses on the Galois co homology of pure transcendental extensions Suppose that K is a field endowed with a discrete valuation v of residue field k Let C denote a discrete Gal K K module unramified at v and such that nC O for some integer n gt O coprime to the characteristic of K Given a cohomology class a H K C one defines the notions of a residue of at v a pole of amp at v and a value a v When K k X is the function field of a smooth connected projective curve defined over k there is a residue formula and an analogue of Abel s theorem The theory is applied to the solution of specialization problems of the Brauer group of K in the Brauer group of k If x e X K and a Br K then a x Bra k whenever x is not a pole of a Serre looks at the function a x and in particular at its vanishing set V In S225 150 1990 he deals with the case K Q 7 T for n 2 The results are completed with asymptotic estimations on the number of zeros of ob tained by sieving arguments they depend on the number of Q irreduc
74. erious role of v 35 in Feit s counterexample Let K Q JN for N a positive square free integer N 3 mod 8 Let Ox denote its ring of integers The field K splits the quaternion algebra 1 1 hence there exists an irreducible representation V of degree 2 over K of Qg In the first letter Serre proves that there exists an Ox free lattice of V which is stable under the group Qg if and only if the integer N can be represented by the binary quadratic form x 2y In order to prove this equivalence Serre makes use of Gauss genus theory any lattice L C V stable under Qs yields an invariant c L which lies in the genus group Cx C 7 of the quadratic field K It turns out that L is free as Og module if and only if c L 1 The exact evaluation of the genus characters on c L yields the criterion above Another version of the computation of the invariant c L is explained in the sec ond letter Serre uses the fact due to Gauss that for a positive square free inte ger N N 3 mod 8 any representation of N as a sum of three squares yields an Ox module of rank 1 which lies in a well defined genus and moreover every class in that genus is obtainable by a suitable representation of N as a sum of three squares If D 1 1 denotes the standard quaternion algebra over Q and R is its Hurwitz maximal order Serre embeds the ring of integers Ox in R by map ping V N to ai bj ck where a b c N It turns out that t
75. essantes Prenez l exemple de la th orie des nombres Les publications de Bourbaki en parlent tr s peu Pourtant ses membres l appr ciaient beaucoup mais ils jugeaient que cela ne faisait pas partie des l ments il fallait d abord avoir compris beaucoup d alg bre de g om trie et d analyse Par ailleurs on a souvent imput Bourbaki tout ce que l on n aimait pas en math matiques On lui a reproch notamment les exc s des maths modernes dans les programmes scolaires Il est vrai que certains responsables de ces programmes se Jean Pierre Serre My First Fifty Years at the Coll ge de France 19 mathematics I feel I am speaking to a friend You don t want to read a text out to a friend if you have forgotten a formula you give its structure that s enough During the lecture I have a lot of possible material in my mind much more than possible in the allotted time What I actually say depends on the audience and my inspiration Only exception the Bourbaki seminar for which one has to provide a text suffi ciently in advance so that it can be distributed during the meeting This is the only seminar that applies this rule it is very restrictive for lecturers What is Bourbaki s place in French mathematics now Its most interesting feature is the Bourbaki seminar It is held three times a year in March May and November It plays both a social role an occasion for meeting other people and a mathematical
76. evich For these results Faltings was awarded the Fields Medal in 1986 In the same paper S118 GE 70 1966 Serre shows that the set of places of k at which a curve E without complex multiplication has a supersingular reduction is of density zero in the set of all the places of k This does not preclude the set of these places being infinite On the contrary Serre thought that this could well be the case Indeed N D Elkies 1987 proved that for every elliptic curve E defined over a real number field there exist infinitely many primes of supersingular reduction in agreement with Serre s opinion Note that the case of a totally imaginary ground field remains open S Lang and H Trotter 1976 conjectured an asymptotic for mula which is still unproved for the frequency of the supersingular primes in the reduction of an elliptic curve E without complex multiplication and defined over Q 13 2 4 The results of S118 70 1966 were completed in the lecture course S122 71 1966 and in S133 McGill 1968 In Chap I of McGill Serre considers adic representations of the absolute Galois group Gal k k of a field k For k an algebraic number field he defines the concepts of a rational adic representation and of a compatible system of rational adic representations these notions go back to Y Taniyama 1957 He relates the equidistribution of conjugacy classes of Frobenius elements to the existence of some analytic
77. f a projective A module the rank of which does not exceed the dimension of the maximal spectrum of A When dim A 1 one recovers the theorem of Steinitz Chevalley on the structure of the torsion free modules over Dedekind rings 6 Algebraic Number Fields The S minaire Bourbaki report S70 GE 41 1958 contains an exposition of Iwa sawa s theory for the p cyclotomic towers of number fields and the p components of their ideal class groups The main difference with wasawa s papers is that the structure theorems for the so called l modules are deduced from general statements on regular local rings of dimension 2 this viewpoint has now become the standard approach to such questions 6 0 1 Cours d Arithm tique S146 CA 1970 arose as a product of two lecture courses taught in 1962 and 1964 at the Ecole Normale Sup rieure The book which in its first edition had the format 11 cm x 18 cm and cost only 12 francs has been frequently translated and reprinted and has been the most accessible introduction to certain chapters of number theory for many years The first part which is purely algebraic gives the classification of quadratic forms over Q We find there equa tions over finite fields two proofs of the quadratic reciprocity law an introduction to p adic numbers and properties of the Hilbert symbol The quadratic forms are studied over Qp over Q as well as over Z in the case of discriminant 1 The second part of the bo
78. f n gt k 1 However it was not even known that the 7 Sn are finitely generated groups Serre shows that they are what is more he shows that the groups 7 for i gt n are finite except for 772 _1 S when n is even which is the direct sum of Z and a finite group Given a prime p he also shows that the p primary component of 77 Sn is zero ifi lt n 2p 3 andn 23 and that the p primary component of 7742p 3 Sn is cyclic he proved later that it has order p 1 1 1 The study of the homotopy groups was pursued by Serre for about two years the results were published in several papers S31 GE 12 1952 S32 GE 13 1952 S48 18 1953 S43 19 1953 S40 22 1953 He also wrote two Comptes rendus notes with Cartan on the technique of killing homotopy groups S29 10 1952 S30 11 1952 and two papers with Borel on the use of Steenrod operations S26 GE 8 1951 S44 1951 a consequence being that the only spheres which have an almost complex structure are So S2 and Se whether S6 has a complex structure or not is still a very interesting open question despite several attempts to prove the opposite 1 1 2 Soon after his thesis Serre was invited to Princeton During his stay January February 1952 he realized that some kind of localization process is possible in the computation of homotopy groups More precisely the paper S48 CE 18 1953 introduces a mod C termin
79. f the Collected Works of H Cartan S192 1979 also in three volumes He was the editor of the Collected Works of R Steinberg S253 1997 15 0 2 We should also mention expository papers that Serre likes to call math ematical entertainment where he takes a rather simple looking fact as a starting point for explaining a variety of deeper results One such paper is S206 GE 140 1985 whose title is just the high school dis criminant formula A b 4ac Given an integer A one wants to classify the quadratic polynomials ax bx c with discriminant A up to SL2 Z conjugation This is a classical problem started by Euler Legendre and Gauss Serre explains the results which were obtained in the late 1980s by combining Goldfeld s ideas 1976 with a theorem of Gross Zagier 1986 and Mestre s proof 1985 of the modularity of a certain elliptic curve of conductor 5077 and rank 3 Another such paper is S268 2002 SEM 2008 By an elementary theorem of C Jordan 1872 if G is a group acting transitively on a finite set of n gt 1 elements the subset Go of the elements of G which act without fixed points is non empty Moreover P J Cameron and A M Cohen 1992 have refined this result by proving that the ratio Go G gt 1 n and that itis gt 1 n if n is not a prime power Serre gives two applications The first one is topological and says that if f T S is a finite covering of a topological space S of degre
80. fine n space over a field k in which case A is the polynomial ring k X 1 Xn Serre mentions in FAC that he does not know of any finitely generated projective k X1 X module which is not free This gave rise to the so called Serre conjecture although it had been 4A P Bayer stated as a problem and not as a conjecture Much work was done on it See e g the book by T Y Lam called Serre s Conjecture in its 1977 edition and Serre s Prob lem in its 2006 one The case n 2 was solved by C S Seshadri 1979 see Serre s report on it in S minaire Dubreil Pisot 48 1960 61 this report also gives an in teresting relation between the problem for n 3 and curves in affine 3 space which are complete intersections Twenty years after the publication of FAC and after partial results had been obtained by several authors especially in dimension 3 D Quillen 1976 and A Suslin 1976 independently and simultaneously solved Serre s problem in any dimension 5 0 2 In his contribution S73 GE 39 1958 at the S minaire Dubreil Dubreil Jacotin Pisot Serre applies the projective modules vector bundles idea to an arbitrary commutative ring A Guided by transversality arguments of topology he proves the following splitting theorem Assume A is commutative noetherian and that Spec A is connected Then every finitely generated projective A module is the direct sum of a free A module and o
81. garde surtout le souvenir d une id e qui a contribu d bloquer la th orie de l homotopie Cela s est pass une nuit de retour de vacances en 1950 dans une couchette de train Je cherchais un espace fibr ayant telles et telles propri t s La r ponse est venue Jean Pierre Serre My First Fifty Years at the Coll ge de France 25 On the other hand some proofs do need a computer in order to check a series of cases that would be impossible to do by hand Two classic examples are the four color problem shading maps using only four colors and the Kepler conjecture packing spheres into three dimensional space This leads to proofs which are not really verifiable in other words they are not genuine proofs but just experimental facts very plausible but nobody can guarantee them You mentioned the increasing number of mathematicians today But where is math ematics going The increase in the number of mathematicians is an important fact One could have feared that this increase in size was to the detriment of quality But in fact this is not the case There are many very good mathematicians in particular young French mathematicians a good omen for us What I can say about the future is that despite this huge number of mathemati cians we are not short of subject matter There is no lack of problems even though just two centuries ago at the end of the 18th century Lagrange was pessimistic he thought that the
82. group G is open in Aut 7 This applies in particular to formal groups of dimension 1 without formal complex multiplication 13 1 1 The topic of Hodge Tate decompositions was also considered in the book S133 McGill 1968 which we will discuss in a moment 13 1 2 In S191 119 1979 it is shown that the inertia subgroup of a Galois group acting on a Hodge Tate module V over a local field is almost algebraic in the sense that it is open in a certain algebraic subgroup Hy of the general linear group GLy In two important cases Serre determines the structure of the connected com ponent H of Hy In the commutative case H is a torus If the weights of V are reduced to 0 and 1 the simple factors of H are of classical type An Bn Cn Dn 13 2 Elliptic Curves and adic Representations Over the years Serre has given several courses on elliptic curves Three of these were at the Coll ge de France Jean Pierre Serre An Overview of His Work 65 S115 67 1965 S122 71 1966 S153 93 1971 and one at McGill University Montreal in 1967 Abundant material was presented in these lectures Most of it was published soon after in the papers S118 70 1966 S154 94 1972 and in the book Abelian adic representations and elliptic curves S133 McGill 1968 The course S115 67 1965 covered general properties of elliptic curves theorems on the structure of their endomorphism ring reduction of e
83. h 1 this homomorphism can be lifted to a homomorphism PGL2 Fp gt G Zp A key point is that the Lie algebra L of Gp turns out to be cohomologically trivial as a PGL2 F module through the adjoint representation This is not the case if p 2h 1 since then H 2 PGL F p L has dimension 1 lifting to Zp is not possible one has to use a quadratic extension of Zp Once this is done the case where char k 0 is settled An argument based on the Bruhat Tits theory gives the other cases As a corollary of the theorem one obtains that PGL2 Fj9 and PSL2 F37 can be embedded in the adjoint group E7 C and that PGL F3 and PSL2 F 6 can be embedded in Eg C 14 3 1 In his lecture S260 1999 SEM 2008 delivered at the S minaire Bour baki 1998 1999 Serre describes the state of the art techniques in the classifica tion of the finite subgroups of a connected reductive group G over an algebraically closed field k of characteristic zero He begins by recalling several important results For example if p is a prime which does not divide the order of the Weyl group W of G then every p group A of G is contained in a torus of G and hence is abelian The torsion set Tor G is by definition the set of prime numbers p for which there exists an abelian p subgroup of G which cannot be embedded in any torus of G The sets Tor G for G simply connected and quasi simple are well known for in stance Tor G 2 3 5 if G is of type Eg moreo
84. h supported in part by MTM2006 04895 P Bayer lt Departament d Algebra i Geometria Facultat de Matem tiques Universitat de Barcelona Gran Via de les Corts Catalanes 585 08007 Barcelona Spain e mail bayer ub edu url http atlas mat ub es personals bayer H Holden R Piene eds The Abel Prize 35 DOI 10 1007 978 3 642 01373 7_4 Springer Verlag Berlin Heidelberg 2010 36 P Bayer Broadly speaking the references to his publications are presented thematically and chronologically In order to facilitate their location the number of a paper corre sponding to the present List of Publications is followed by its number in the uvres if applicable and by the year of its publication Thus a quotation of the form S216 GE 143 1987 will refer to paper 216 of the List included in the uvres as number 143 Serre s books will be referred to in accordance with the List and the References at the end of this manuscript The names of other authors followed by a date will denote the existence of a publication but for the sake of simplicity no explicit mention of it will be made A significant part of Serre s work was given in his annual courses at the Coll ge de France When we mention one of these it will be understood to be a course held at this institution unless otherwise stated 1 The Beginnings The mathematical training of J P Serre can be seen as coming from two closely related sources On one hand i
85. he invariant c R is the same as the one obtained before In the third letter and more gener ally given any quaternion algebra D over Q and an imaginary quadratic field K which splits D if we choose an embedding K D the Ox invariant c Op of a maximal order Op containing Ox does not depend on the choice of Op Serre determines c Op c D K Cx Cz in terms of D and K By making use of the Hilbert symbol the genus group Cx C a can be embedded in the 2 component of the Brauer group Br2 Q moreover the image of c D K in Br2 Q is equal to D dp d In this formula D denotes the element of the Brauer group defined by the quaternion algebra D dp is the signed discriminant of D d with d gt 0 is the discriminant of K and dp d stands for the Hilbert symbol In the special case D 1 1 and Qg the formula tells us that there exists a free 78 P Bayer Ox module of rank 2 which gives the standard irreducible representation of Qg over K Q d if and only if either 2 d 0 or 1 d 0 that is if and only if d is representable either by x 2y7 or by x y For example if d 8p p 3 mod 8 p prime non free lattices exist 14 2 Algebraic Groups The lecture course S141 84 1969 focused on dis crete groups Some of its contents would be published in S139 GE 83 1969 and S149 GE 88 1971 Another part was published in the book Arbres amalgames SL S176 AA 1977 w
86. he second one states that Tr Ad g gt rank G for all g G the bound being optimal if and only if there is an element c G such that ctc t7 for every t T where T is a maximal torus of G the proof is a case by case ex plicit computation in the E6 case the computation was not made by Serre himself but by A Connes 14 1 4 In another Oberwolfach report S279 2006 Serre defines the so called Kac coordinates in such a way that they can be used to classify the finite subgroups Jean Pierre Serre An Overview of His Work FT of G which are isomorphic to un without having to assume that n is prime to the characteristic 14 1 5 In 1974 Serre had asked W Feit whether given a linear representation p G GL K of a finite group G over a number field K it could be realized over the ring of integers OK Although he did not expect a positive answer he did not know of any counterexample Given p there are Ox lattices which are stable under the action of G but the point is that as Ox is a Dedekind ring these lattices need not be free as Ox modules There is an invariant attached to them which lies in the ideal class group Cx Pic Ox of K Feit provided the following counterexample if G Qs is the quaternion group of order 8 and K Q 35 the answer to the question is no The paper S281 2008 reproduces three letters of Serre to Feit about this question written in 1997 Their purpose was to clarify the myst
87. ian is working on a proof but needs a tech nical lemma then through a search engine such as Google he will track down colleagues who have worked on the question and send them an e mail In this way in just a few days or even hours he may be able to find somebody who has proved the required lemma Of course this only applies to easy problems those for which you want to use a reference rather than to reconstruct a proof For really difficult questions a mathematician would have little chance of finding someone to help him Computer and Internet are thus the tools which speed up our work They allow us to make our manuscripts accessible to everybody without waiting for publication in a journal That is very convenient But this acceleration also has its disadvantages E mail produces informal correspondence which is less likely to be kept than the paper one It is unusual to throw letters away but one can easily delete or lose e mails when one changes computers for example Recently a bilingual version French on one page and English on the other of my correspondence with A Grothendieck between 1955 and 1987 has been published That would not have been possible if the correspondence had been by e mail 24 Jean Pierre Serre Mon premier demi si cle au Coll ge de France Par ailleurs certaines d monstrations font appel a l ordinateur pour v rifier une s rie de cas qu il serait impraticable de traiter la main Deux cas classiques
88. ible compo nents of the polar divisor of a One of the questions raised in this paper how often does a conic have a rational point was solved later by C Hooley 1993 and C R Guo 1995 the upper bound given by the sieve method has the right order of magnitude 12 0 5 Cohomological Invariants in Galois Cohomology S269 CI 2003 is a book co authored by S Garibaldi A Merkurjev and J P Serre The algebraic in variants discussed in it are the Galois cohomology analogues of the characteris tic classes of topology but here the topological spaces are replaced by schemes Spec k for k a field The text is divided in two parts The first one consists of an expanded version of a series of lectures given by Serre at UCLA in 2001 with notes by S Garibaldi The second part is due to Merkurjev with a section by Garibaldi we shall not discuss it here In Chap I Serre defines a quite general notion of invariant to be applied Jean Pierre Serre An Overview of His Work 63 throughout the book to several apparently disparate situations Given a ground field ko and two functors A Fields x Sets and H Fields Abelian Groups an H invariant of A is defined as a morphism of functors a H Here Fields denotes the category of field extensions k of ko Examples of functors A are k Et k the isomorphism classes of tale k algebras of rank n k Quad k the isomorphism classes of non degenerate quadr
89. ierre Gabriel Lecture Notes in Mathematics Vol 11 Springer Verlag Berlin Translated into English and Russian Zeta and L functions In Arithmetical Algebraic Geometry Proc Conf Purdue Univ 1963 82 92 Harper amp Row New York Classification des vari t s analytiques p adiques compactes Topology 3 409 412 Sur la dimension cohomologique des groupes profinis Topology 3 413 420 R sum des cours de 1964 1965 Annuaire du Coll ge de France 45 49 1966 Groupes p divisibles d apres J Tate In S minaire Bourbaki 1966 1967 Expos 318 14 Pp Existence de tours infinies de corps de classes d apr s Golod et Safarevi In Les Tendances G om triques en Alg bre et Th orie des Nombres 231 238 Editions du Centre National de la Recherche Scientifique 88 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 List of Publications for Jean Pierre Serre Groupes de Lie adiques attach s aux courbes elliptiques In Les Tendances G om triques en Alg bre et Th orie des Nombres 239 256 ditions du Centre National de la Recherche Scientifique Alg bres de Lie semi simples complexes W A Benjamin Inc New York Amsterdam Translated into English and Russian Prolongement de faisceaux analytiques coh rents Ann Inst Fourier 16 1 363 374 with A Borel S Ch
90. in the draw ers of mathematicians As mentioned above some of these letters are included in the uvres Some were collected in the text edited by S S Chern and F Hirze bruch in Wolf Prize in Mathematics vol 2 World Scientific 2001 a letter to John Jean Pierre Serre An Overview of His Work 81 McCleary 1997 two letters to David Goss 1991 2000 a letter to Pierre Deligne 1967 and a letter to Jacques Tits 1993 One finds in them comments on his the sis on the writing of FAC on adic representations as well as historical data on the Shimura Taniyama Weil modularity conjecture In the letter to Tits there is an account of the themes on which Serre was working in the years around 1993 Ga lois representations inverse Galois problem Abhyankar s problem trace forms and Galois cohomology The Grothendieck Serre correspondence S263 GRSE 2001 published more recently is another invaluable resource for understanding the origins of the concepts and tools of current algebraic geometry 15 0 1 Serre has written essays on the work of other mathematicians for exam ple a publication of historical character on a prize delivered to J S Smith and H Minkowski S238 1993 or publications about the life and work of A Weil S259 1999 and that of A Borel S270 2004 S271 2004 He was the editor of the Collected Works of EG Frobenius S135 1968 in three volumes He and R Remmert were the editors o
91. ion of a mod p analogue of the Langlands program Many people have worked on it A general proof was presented at a Summer School held at Luminy France July 9 20 2007 a survey of this work can be found in the expository paper Ch Khare 2007 1 According to the Citation Database MathSciNet S216 GE 143 1987 is Serre s second most frequently cited paper the first one being fittingly the one he dedi cated to A Weil S154 GE 94 1972 13 3 9 A summary of Serre s lecture course on Galois representations mod p and modular forms mod p can be found in S218 145 1988 In it Serre relates modular forms modulo p with quaternions Two letters on this subject ad dressed to J Tate and D Kazhdan are collected in S249 169 1996 In the letter to Tate Serre formulates a quaternion approach to modular forms modulo a prime p through quaternion algebras Let D be the quaternion field over Q ram ified only at p and at ov and let D A be the group of the adelic points of the multiplicative group D viewed as an algebraic group over Q The main result of the letter to Tate is that the systems of eigenvalues az with ag F provided by the modular forms mod p coincide with those obtained under the natural Hecke action on the space of locally constant functions f D A Do gt Fa The result is proved by evaluating the modular forms at supersingular elliptic curves In the letter to Kazhdan Serre studies certain unr
92. iprocity isomorphisms Sometimes the reciprocity isomorphisms can be made explicit by means of a symbol computation The first historical example is the quadratic reciprocity law of Legendre and Gauss The cohomological treat ment of class field theory started with papers of G P Hochschild E Artin J Tate A Weil and T Nakayama 7 1 Geometric Class Field Theory Groupes Alg briques et Corps de Classes S82 GACC 1959 was the first book that Serre published It evolved from his first course at the Coll ge de France S69 GE 37 1957 and its content is mainly based on earlier papers by S Lang 1956 and M Rosenlicht 1957 Chapter I is a r sum of the book Chapter II gives the main theorems on al gebraic curves including Riemann Roch and the duality theorem with proofs Chapters III IV are devoted to a theorem of Rosenlicht stating that every ratio nal function f X G from a non singular irreducible projective curve X to a commutative algebraic group G factors through a generalized Jacobian Jm A gen eralized Jacobian is a commutative algebraic group which is an extension of an abelian variety the usual Jacobian J by an algebraic linear group Lm depending on a modulus m The groups Lm provide the local symbols in class field theory In Chap V it is shown that every abelian covering of an irreducible algebraic curve is the pull back of a separable isogeny of a generalized Jacobian When m varies the generalized
93. is 233 680 682 1952 Cohomologie et fonctions de variables complexes In S minaire Bourbaki 1952 1953 Ex pos 71 6 pp Le cinquieme probleme de Hilbert Etat de la question en 1951 Bull Soc Math France 80 1 10 with H Cartan Espaces fibr s et groupes d homotopie I Constructions g n rales C R Acad Sci Paris 234 288 290 with H Cartan Espaces fibr s et groupes d homotopie IT Applications C R Acad Sci Paris 234 393 395 Sur les groupes d Eilenberg MacLane C R Acad Sci Paris 234 1243 1245 Sur la suspension de Freudenthal C R Acad Sci Paris 234 1340 1342 1953 Cohomologie et arithm tique In S minaire Bourbaki 1952 1953 Expos 77 7 pp Espaces fibr s alg briques d apr s Andr Weil In S minaire Bourbaki 1952 1953 Expos 82 7 pp Travaux d Hirzebruch sur la topologie des vari t s In S minaire Bourbaki 1953 1954 Ex pos 88 6 pp Fonctions automorphes d une variable application du th or me de Riemann Roch In S minaire H Cartan E N S 1953 1954 Expos s 4 5 15 pp Deux th or mes sur les applications compl tement continues In S minaire H Cartan E N S 1953 1954 Expos 16 7 pp Faisceaux analytiques sur l espace projectif In S minaire H Cartan E N S 1953 1954 Expos s 18 19 17 pp Fonctions automorphes In S minaire H Cartan E N S 1953 1954 Expos 20 23 pp Quelques calculs de groupes d homotopie C R Aca
94. istable reduction theorem was also proved by D Mumford except that his proof based on the use of theta functions did not include the case where the residue characteristic is equal to 2 Suppose that A has complex multiplication by F over the field K where F denotes an algebraic number field of degree 2d d dim A In the same work Serre and Tate prove that every abelian variety defined over an algebraic number Jean Pierre Serre An Overview of His Work 73 field K and with complex multiplication over this field has potential good reduction at all the places of K and that it has good reduction at the places of K outside the support of its Gr ssencharakter This result generalizes some earlier ones of M Deuring 1955 in the case of elliptic curves The exponent of the conductor at v is given by 2dn where n is the smallest integer such that the Gr ssencharakter is zero when restricted to the ramification group v in the upper numbering 13 4 3 In S209 GE 135 1985 Serre explains how the theorems obtained by G Faltings 1983 in his paper on the proof of Mordell s conjecture allow a better understanding of the properties of the adic representations associated to abelian varieties In the first part of the lectures Serre gives an effective criterion for showing that two adic representations are isomorphic the m thode des corps quartiques This criterion was applied to prove that two elliptic curves
95. it was quite a tense moment I started by read ing the half page I had prepared and then I improvised I can no longer remember what I said I only recall that I spoke about algebra and the ancillary role it plays in geometry and number theory According to the report that appeared in the newspa per Combat I spent most of the time mechanically wiping the table that separated me from my audience I did not feel at ease until I had a piece of chalk in my hand and I started to write on the blackboard the mathematician s old friend A few months later the Secretary s Office told me that all inaugural lectures were written up but they had not received the transcript of mine As it had been improvised I offered to repeat it in the same style mentally putting myself back in the same situation One evening I was given a tape recorder and I went into an office at the Coll ge I tried to recall the initial atmosphere and to make up a lecture as close as possible to the original one The next day I returned the tape recorder to the Secretary s Office They told me that the recording was inaudible I decided that I had done all I could and left it there My inaugural lecture is still the only one that has not been written up As a rule I don t write my lectures I don t consult notes and often I don t have any I like to do my thinking in front of the audience When I am explaining 18 Jean Pierre Serre Mon premier demi si cle au Coll ge
96. ject S204 GE 131 1984 as well as its generalization by B Kahn 1984 Sim ilarly the W ko module Inv S W is free of finite rank with basis given by the Witt classes of the first 1 2 exterior powers of the trace form Among other re sults one finds an explicit description of all possible trace forms of rank lt 7 and an application of trace forms to the study of Noether s problem which we recall in what follows Given a finite group G the property Noe G ko means that there exists an embedding p G GL ko such that if K is the subfield of ko X1 Xn fixed by G then Kp is a pure transcendental extension of k Deciding whether Noe G ko is true is the Noether problem for G and ko Serre proves that Noether s problem has a negative answer for SL 2 F7 2 A6 or the quaternion group Q16 of order 16 The book includes also several letters one of these is a letter from Serre to R S Garibaldi dated in 2002 in which he explains his motivations 64 P Bayer 12 0 6 Let k be a field of characteristic different from 2 The norm form of any quaternion algebra defined over k is a 2 fold Pfister form In S278 2006 M Rost J P Serre and J P Tignol study the trace form g4 x Trd x of a central sim ple algebra A of degree 4 over k under the assumption that k contains a primitive 4 root of unity They prove that g4 q2 q4 in the Witt ring of quadratic forms over k were q2 and q4 are uniquely deter
97. k denotes the completion of G k in the topology defined by the S congruence subgroups and G k the completion in that of the S arithmetic subgroups The group G k can be identified with the clo sure of G k in G al Let C G denote the kernel of x The group C G which coincides with the ker x restricted to I is profinite and z is an epimorphism The S congruence subgroup problem has a positive answer if and only if the congruence kernel C 5 G is trivial i e x is an isomorphism It is so when G is a torus Chevalley 1951 or is unipotent When G is semisimple and not simply 54 P Bayer connected the problem has a negative answer Hence the most interesting case is when G is semisimple and simply connected 10 1 Congruence Subgroups Recall that a semisimple group over k is said to be split or to be a Chevalley group if it has a maximal torus which splits over k Split groups provide a suitable framework for the study of the congruence sub group problem In S126 GE 74 1967 GE 103 1975 H Bass J Milnor and Serre formulate the S congruence subgroups conjecture precisely in the following form if G is split of rank gt 2 simply connected and quasi simple then the group ex tension 1 gt C G k gt GX gt G k 1 is central and moreover C5 G is trivial unless k is totally imaginary in the latter case C S G i k is the finite subgroup consisting of the roots of unity of k
98. k is a good example of dialogue of the deaf Changeux does not under stand what Connes says and vice versa It is quite astonishing Personally I am on Connes side Mathematical truths are independent of us Our only choice is in the way in which we express them If you want you can do this without introducing any terminology Consider for example a company of soldiers The general likes to arrange them in two ways either in a rectangle or in two squares It is up to the sergeant to put them in the correct positions He realizes that he only has to put them in rows of four if there is one left over that he cannot place either he will manage to put them all in a rectangle or manage to arrange them in two squares Technical translation the number n of soldiers is congruent to 1 mod 4 If n is not a prime the soldiers can be arranged in a rectangle If n is a prime a theorem of Fermat shows that n is the sum of two squares What is the place of mathematics in relation to other sciences Is there a renewed demand for mathematics from these sciences Probably but there are different cases Some theoretical physics is so theoretical that itis half way between mathematics and physics Physicists consider it mathematics while mathematicians have the opposite view String theory is a good example The most positive aspect is to provide mathematicians with a large number of statements which they have to prove or maybe disprove On the
99. l example My wife Josiane was a specialist in quantum chem istry She needed linear representations of certain symmetry groups The books she was working with were not satisfactory they were correct but they used very clumsy notation I wrote a text that suited her needs and then published it in book form as Linear Representations of Finite Groups I thus did my duty as a mathe matician and as a husband putting things on the shelves Does truth in mathematics have the same meaning as elsewhere No It s an absolute truth This is probably what makes mathematics unpopular with the public The man in the street accepts the absolute in religion but not in mathe matics Conclusion to believe is easier than to prove 30 Jean Pierre Serre Mon premier demi si cle au Coll ge de France M Schmidt Hommes de Science 218 227 Hermann Paris 1990 AMS American Mathematical Society J E Littlewood A Mathematician s Miscellany Methuen and Co 1953 Ce livre ex plique bien la part inconsciente du travail cr atif Il y a quelques ann es mon ami R Bott et moi m me allions recevoir un prix isra lien le prix Wolf remis dans la Knesset a Jerusalem Bott devait dire quelques mots sur les math matiques Il m a demand que dire Je lui ai dit C est bien simple tu n as qu expliquer ceci les autres sciences cherchent a trouver les lois que Dieu a choisies les math matiques cherchent trouver les lois
100. lliptic curves Tate modules complex multiplication and so on Let be an elliptic curve de fined over a field k and let be a prime different from char k The Tate mod ules T E lim E ks are special cases of the adic homology groups associ ated to algebraic varieties The Galois group Gal k k acts on T E and on the Q vector space V E Qe T E We may consider the associated Galois adic representation pg Gal k k Aut Ts GL Z4 The image Ge of pe is an adic Lie subgroup of Aut 7 We shall denote by ge the Lie algebra of Ge The Galois extension associated to Gg is obtained by adding the coordinates of the points of E k of order a power of to the field k Suppose that k is an algebraic number field and that the elliptic curve E has complex multiplication Thus there exist an imaginary quadratic field F and a ring homomorphism F Q End E Then the Galois group Gy is abelian whenever F Ck and is non abelian otherwise If F C k the action of Gal k k on T E is given by a Gr ssencharakter whose conductor has its support in the set of places of bad reduction of E this result is due to M Deuring The usefulness of elliptic curves with complex multiplication consists in the fact that they provide an explicit class field theory for imaginary quadratic fields 13 2 1 A short account of the classical theory of complex multiplication can be found in S128 76 1967 13 2 2 By using fiber
101. ly if the class of H in the Brauer group of k equals the sum of 1 1 and the Witt invariant of the quadratic form Tr z on the subspace of k of elements of trace zero Moreover these conditions are equivalent to the fact that k can be generated by the roots of a quintic equation of the form X aX bX c 0 which is consistent with old results of Hermite and Klein 12 0 1 The obstruction associated to a Galois embedding problem defined by a Galois extension L K and by a central extension of the group Gal L K is given by a cohomology class the vanishing of which characterizes the solvability of the problem When the kernel of the central extension is the cyclic group C2 of order 2 the cohomology class can be identified with an element of Bro K H 2 K C2 assuming that the characteristic is not 2 In a paper dedicated to J C Moore S204 131 1984 Serre gives a formula relating the obstruction to certain Galois em bedding problems to the second Stiefel Whitney class of the trace form Tr x Through the use of Serre s formula N Vila 1984 1985 proved that the non trivial double covering An 2 A of the alternating group is the Galois group of a regular extension of Q T for infinitely many values of n gt 4 The result was ex tended to all n gt 4 by J F Mestre 1990 cf S235 TGT 1992 Explicit solutions to solvable embedding problems of this type were later obtained by T Crespo 12 0 2 In his
102. ment la table qui me s parait du public je ne me suis senti l aise que lorsque j ai pris en main un baton de craie et que J ai commenc crire sur le tableau noir ce vieil ami des math maticiens Quelques mois plus tard le secr tariat m a fait remarquer que toutes les le ons inaugurales taient r dig es et que la mienne ne l tait pas Comme elle avait t im provis e j ai propos de la recommencer dans le m me style en me remettant men talement dans la m me situation Un beau soir on m a ouvert un bureau du College et l on m a pr t un magn tophone Je me suis efforc de recr er atmosphere ini tiale et J ai refait une le on sans doute peu pr s semblable a I originale Le lende main J ai apport le magn tophone au secr tariat on m a dit que l enregistrement tait inaudible J ai estim que j avais fait tout mon possible et je m en suis tenu 1a Ma le on inaugurale est rest e la seule qui n ait jamais t r dig e En r gle g n rale je n cris pas mes expos s je ne consulte pas mes notes et souvent je n en ai pas J aime r fl chir devant mes auditeurs J ai le sentiment Jean Pierre Serre My First Fifty Years at the Coll ge de France 17 Jean Pierre Serre Professor at the Coll ge de France held the Chair in Algebra and Geometry from 1956 to 1994 You taught at the Coll ge de France from 1956 to 1994 holding the Chair in Alge bra and Geo
103. metry What are you memories of your time there I held the Chair for 38 years That is a long time but there were precedents According to the Yearbook of the Coll ge de France the Chair in Physics was held by just two professors in the 19th century one remained in his post for 60 years and the other for 40 It is true that there was no retirement in that era and that professors had deputies to whom they paid part of their salaries As for my teaching career this is what I said in an interview in 1986 Teaching at the Coll ge is both a marvelous and a challenging privilege Marvelous because of the freedom of choice of subjects and the high level of the audience CNRS Centre national de la recherche scientifique researchers visiting foreign academics col leagues from Paris and Orsay many regulars who have been coming for 5 10 or even 20 years It is challenging too new lectures have to be given each year either on one s own research which I prefer or on the research of others Since a series of lectures for a year s course is about 20 hours that s quite a lot Can you tell us about your inaugural lecture I was a young man about 30 when I arrived at the College The inaugural lecture was almost like an oral examination in front of professors family mathematician colleagues journalists etc I tried to prepare it but after a month I had only managed to write half a page When the day of the lecture came
104. minaire Bour baki 1977 78 Expos 511 89 100 Lecture Notes in Math Vol 710 Springer Verlag Berlin R sum des cours de 1976 1977 Annuaire du Coll ge de France 49 54 1978 Ch E Picard uvres de Ch E Picard Tome I Editions du Centre National de la Recherche Scientifique Paris With a foreword by J Leray J P Serre and M Herv List of Publications for Jean Pierre Serre 91 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 Une formule de masse pour les extensions totalement ramifi es de degr donn d un corps local C R Acad Sci Paris S r A B 286 22 A1031 A1036 Sur le r sidu de la fonction z ta p adique d un corps de nombres C R Acad Sci Paris S r A B 287 4 A183 A188 R sum des cours de 1977 1978 Annuaire du Coll ge de France 67 70 1979 M Waldschmidt Nombres transcendants et groupes alg briques With appendices by D Bertrand and J P Serre Ast risque Nos 69 70 Arithmetic groups In Homological group theory Proc Sympos Durham 1977 London Math Soc Lecture Note Ser Vol 36 105 136 Cambridge Univ Press Cambridge Groupes alg briques associ s aux modules de Hodge Tate In Journ es de G om trie Al g brique de Rennes Rennes 1978 Vol IIT Ast risque No 65 155 188 H Cartan uvres Vol I Il II Springer Verlag Berlin Edite
105. mine was exhausted and that there was nothing much more to discover Lagrange wrote this just before Gauss relaunched mathematics in an extraordinary way all by himself Today there are many fields to explore for young mathematicians and even for those who are not so young I hope It is often said in the philosophy of science that major mathematical discoveries are made by young mathematicians Was this the case for you I don t believe that the term major discovery applies to me I have rather done things that are useful for other mathematicians When I was awarded the Abel prize in 2003 most of the work cited by the jury had been done before I was 30 But if I had stopped then it would probably not have awarded me the prize I have done other things after that if only some conjectures that have kept many people busy Of my generation several of my colleagues have continued working beyond the age of 80 For example my old friends Armand Borel and Raoul Bott who both recently died aged 82 There is no reason to stop as long as health allows it But the subject matter has to be there When you are dealing with very broad subjects there is always something to do but if you are too specialized you can find yourself blocked for long periods of time either because you have proved everything that can be proved or to the contrary because the problems are too difficult It is very frustrating Discoveries in mathematics
106. mined 2 fold and 4 fold Pfister forms respectively The form q2 corresponds to the norm form of the quaternion algebra which is equivalent to A in the Brauer group of k Moreover A is cyclic if and only if q4 is hyperbolic The images of the forms q in H k Z 2Z yield cohomo logical invariants of PGLa since the set H k PGL4 classifies the central simple k algebras of degree 4 13 Galois Representations Serre has studied Galois representations especially adic representations in sev eral books and papers His pioneering contributions to these topics have broken new ground and have profoundly influenced their research in the last decades 13 1 Hodge Tate Modules The paper S129 72 1967 is based on a lecture delivered by Serre at a Conference on Local Fields held in Driebergen The Nether lands Take as the ground field a local field of characteristic zero whose residue field is of characteristic p gt 0 and let C be the completion of an algebraic closure K of K IfT is the Tate module associated to a p divisible group defined over the ring of integers of K a deep result of Tate states that Cp T has a decomposition analogous to the Hodge decomposition for complex cohomology This gives strong restrictions on the image G of Gal K K in Aut T For instance if the action of G is semisimple then the Zariski closure of G contains a p adic Mumford Tate group Under some additional hypotheses Serre shows that the
107. n 1948 and just after having finished his studies at the Ecole Normale Sup rieure he starts working at the S minaire Cartan this was a very active collaboration that was continued for about 6 years giving and writing lectures in homological algebra topology and functions of several complex variables On the other hand since 1949 and for about 25 years he works with Bourbaki In the fifties Serre publishes his first papers some of them coauthored with H Cartan A Borel and G P Hochschild and submits a doctoral dissertation un der the supervision of Cartan In an early publication Borel and Serre S17 GE 2 1950 prove the impossibility of fibering an Euclidean space with compact fibers not reduced to one point 1 0 1 Serre s thesis was entitled Homologie singuli re des espaces fibr s Applica tions S25 9 1951 and was followed by several publications S19 4 1950 S20 5 1951 S21 6 1951 Its initial purpose was to compute the coho mology groups of the Eilenberg MacLane complexes K II n by induction on n using the fact that the loop space of K II n is K I1 n 1 combined with the loop fibration see below The homotopy lifting property imposed by Serre on fiber spaces allows him to construct a spectral sequence in singular homology analogous to the one obtained by J Leray 1950 in the setting of Cech theory A dual spectral sequence also exists for cohomology The concept of fiber sp
108. n des courbes elliptiques d apr s Y Manin In S minaire Bourbaki 1969 70 Ex pos 380 281 294 Lecture Notes in Math Vol 180 Springer Verlag Berlin Le probleme des groupes de congruence pour SL2 Ann of Math 2 92 489 527 with A Borel Adjonction de coins aux espaces sym triques applications a la cohomolo gie des groupes arithm tiques C R Acad Sci Paris S r A B 271 A1156 A1158 Sur une question d Olga Taussky J Number Theory 2 235 236 Cours d arithm tique Presses Universitaires de France Paris Translated into Chinese English Japanese Russian 1971 Groupes discrets Compactifications In Colloque sur les Fonctions Sph riques et la Th orie des Groupes Expos 6 4 pp Inst Elie Cartan Univ de Nancy Nancy Cohomologie des groupes discrets In S minaire Bourbaki 1970 1971 Expos 399 337 350 Lecture Notes in Math Vol 244 Springer Verlag Berlin Cohomologie des groupes discrets In Prospects in mathematics Princeton Univ Press Princeton N J Ann of Math Studies No 70 77 169 Conducteurs d Artin des caract res r els Invent Math 14 173 183 Sur les groupes de congruence des vari t s ab liennes I Izv Akad Nauk SSSR Ser Mat 35 731 737 with A Borel Cohomologie supports compacts des immeubles de Bruhat Tits appli cations la cohomologie des groupes S arithm tiques C R Acad Sci Paris S r A B 272 A110 A113 R sum des cours de 1970
109. ntly by K Kato and his school but the general case is still open 7 1 4 An example in the geometric case of a separable covering of curves with a relative different whose class is not a square was given in a joint paper with A Fr h lich and J Tate S100 GE 54 1962 Such an example does not exist for number fields by a well known result of E Hecke 7 1 5 In S150 92 1971 Serre considers a Dedekind ring A of field of frac tions K a finite Galois extension L K with Galois group G and a real valued virtual character x of G Under the assumption that either the extension L K is tamely ramified or that x can be expressed as the difference of two characters of real linear representations he proves that the Artin conductor f f x L K is a square in the group of ideal classes of A 7 2 Local Class Field Theory Group cohomology and more specifically Galois cohomology is the subject of the lecture course S83 GE 44 1959 The content of this course can be found in Corps Locaux S98 CL 1962 The purpose of CL was to provide a cohomological presentation of local class field theory for valued fields which are complete with respect to a discrete valu ation with finite residue field In the first part one finds the structure theorem of complete discrete valuation rings In the second part Hilbert s ramification theory is given with the inclusion of the upper numbering of the ramification groups due to J Herbrand and
110. od g 1 is an invariant of the manifold two n manifolds with the same invariant are isomorphic 9 Group Cohomology By definition a profinite group is a projective limit of finite groups special cases are the pro p groups 1 e the projective limits of finite p groups The most inter esting examples of profinite groups are provided by the Galois groups of algebraic extensions and by compact p adic Lie groups 9 1 Cohomology of Profinite Groups and p adic Lie Groups To each profinite group G lim G acting in a continuous way on a discrete abelian group A one can lt associate cohomology groups H4 G A by using continuous cochains The main properties of the cohomology of profinite groups were obtained by Tate and also by Grothendieck in the early 1960s but were not published They are collected in the first chapter of Cohomologie Galoisienne S97 CG 1962 In the first chapter of CG given a prime p and a profinite group G the concepts of cohomological p dimension denoted by cd G and cohomological dimension denoted by cd G are defined Some pro p groups admit a duality theory they are called Poincar pro p groups Those of cohomological dimension 2 are the De mushkin groups They are especially interesting since they can be described by one explicit relation Demushkin Serre Labute they appear as Galois groups of the maximal pro p extension of p adic fields cf S104 58 1963 and CG Chapter
111. of Stein manifolds cf S42 23 1953 2 3 1 In his lecture S76 GE 38 1958 at the International Symposium on Alge braic Topology held in Mexico City Serre associates to an algebraic variety X defined over an algebraically closed field k of characteristic p gt 0 its cohomology Jean Pierre Serre An Overview of His Work 41 groups H X W with values in a sheaf of Witt vectors W Although this did not provide suitable Betti numbers the paper contains many ideas that paved the way for the birth of crystalline and p adic cohomology We stress the treatment given in this work to the Frobenius endomorphism F as a semilinear endomorphism of H X when X is a non singular projective curve The space H X may be identified with a space of classes of repartitions or ad les over the function field of X and its Frobenius endomorphism gives the Hasse Witt matrix of X By using Cartier s operator on differential forms Serre proves that H 1 X W is a free mod ule of rank 2g s over the Witt ring W k where g denotes the genus of the curve and p is the number of divisor classes of X killed by p 2 3 2 The above results were completed in the paper S75 GE 40 1958 dedicated to E Artin Given an abelian variety A Serre shows that the cohomology algebra H A is the exterior algebra of the vector space H A as in the classical case He also shows that the Bockstein s operations are zero that
112. of the 1970s when production faded away As for the books themselves one may say that they have fulfilled their mission People often believe that these books deal with subjects that Bourbaki found inter esting The reality is different the books deal with what is useful in order to do interesting things Take number theory for example Bourbaki s publications hardly mention it However the Bourbaki members liked it very much it but they consid ered that it was not part of the Elements it needed too much algebra geometry and analysis Besides Bourbaki is often blamed for everything that people do not like about mathematics especially the excesses of modern math in school curricula It is true that some of those responsible for these curricula claimed to follow Bourbaki But 20 Jean Pierre Serre Mon premier demi si cle au Coll ge de France sont r clam s de Bourbaki Mais Bourbaki n y tait pour rien ses crits taient des tin s aux math maticiens pas aux tudiants encore moins aux adolescents Notez que Bourbaki a vit de se prononcer sur ce sujet Sa doctrine tait simple on fait ce que l on choisit de faire on le fait du mieux que l on peut mais on n explique pas pourquoi on le fait J aime beaucoup ce point de vue qui privil gie le travail par rapport au discours tant pis s il pr te parfois des malentendus Comment analysez vous l volution de votre discipline depuis l poque de vo
113. ogy of I is isomorphic to that of X T and certain duality relations are fulfilled In particular the cohomological dimension of F is given by cd T dim X IgQ G If G SL the space X is essentially a space already defined by C L Siegel it is obtained by attaching boundary points and ideal points to X by means of the reduction theory of quadratic forms 10 2 3 Borel and Serre investigated in S152 91 1971 the cohomology of S arithmetic groups Let G be a semisimple algebraic group over an algebraic number field k and let C G k be an S arithmetic subgroup The group F is a discrete subgroup of G k Let Xs be the space defined by ves Xs Xoo X I X VES Soo Here X is the Bruhat Tits building of G over k and Xoo is the variety with corners associated to the algebraic group Res Q G obtained by restriction of scalars see above The group G k and a fortiori the group I acts on Xs Moreover I acts properly and the quotient Xs T is compact The study of the cohomology of I is thus reduced to that of Xs T In order to go further Borel and Serre need some information on the cohomology with compact support of each X they obtain it by compactifying X the boundary being the Tits building of G k endowed with a suitable topology Their main results may be summarized as follows Let d dim X m dim Xs s L d a TA L where denote the rank of G over k and ky respectively Ass
114. ok uses analytic methods It contains a chapter on L functions culminating in the standard proof of Dirichlet s theorem on primes in arithmetic progressions and a chapter on modular forms of level 1 together with their rela tions with elliptic curves Eisenstein series Hecke operators theta functions and Jean Pierre Serre An Overview of His Work 45 Ramanujan s t function In 1995 Serre was awarded the Leroy P Steele Prize for Mathematical Exposition for this delightful text 6 0 2 The notion of a p adic modular form was introduced by Serre in the pa per S158 GE 97 1973 which is dedicated to C L Siegel Such a form is defined as a limit of modular forms in the usual sense By using them together with pre vious results on modular forms mod p due to Swinnerton Dyer and himself he constructs the p adic zeta function of a totally real algebraic number field K This function interpolates p adically the values at the negative integers of the Dedekind zeta function x s after removal of its p factors these numbers were already known to be rational thanks to a theorem of Siegel 1937 Serre s results general ize the one obtained by Kubota Leopoldt in the sixties when K is abelian over Q They were completed later by D Barsky 1978 Pierrette Cassou Nogues 1979 and P Deligne K Ribet 1980 7 Class Field Theory Class field theory describes the abelian extensions of certain fields by means of what are known as rec
115. ology in which a class of objects C is treated as zero as is done in arithmetic mod p For instance he proves that the groups Ti S n even are C isomorphic to the direct sum of 7 1 1 and 7 S2n_1 where C denotes the class of the finite 2 groups The paper S48 GE 18 1953 also shows that every connected compact Lie group is homotopically equivalent to a product of spheres modulo certain exceptional prime numbers for classical Lie groups they are those which are lt h where h is the Coxeter number In the paper S43 GE 19 1953 Serre determines the mod 2 cohomology algebra of an Eilenberg MacLane complex K IT q in the case where the abelian group IT is finitely generated For this he combines results from both Borel s thesis and his own He also determines the asymptotic behaviour of the Poincar series of that algebra by analytic arguments similar to those used in the 38 P Bayer theory of partitions and deduces that for any given n gt 1 there are infinitely many i s such that 7r Sn has even order 1 1 3 In the same paper he computes the groups 7 4 Sn for i lt 4 and in S32 13 1952 and S40 22 1953 he goes up to 7 lt 8 These groups are now known for larger values of i but there is very little information on their asymptotic behaviour for i co 1 2 Hochschild Serre Spectral Sequence The first study of the cohomology of group extensions was R Lyndon s thesis 194
116. ommun de la philosophie des sciences les grandes d couvertes math matiques sont le fait de math maticiens jeunes Est ce votre cas Je ne crois pas que le terme de grande d couverte s applique moi J ai surtout fait des choses utiles pour les autres math maticiens En tout cas lorsque j ai eu le prix Abel en 2003 la plupart des travaux qui ont t cit s par le jury avaient t faits avant que je n aie 30 ans Mais si je m tais arr t ce moment l on ne m aurait sans doute pas donn ce prix j ai fait aussi d autres choses par la suite ne serait ce que des conjectures sur lesquelles beaucoup de gens ont travaill et travaillent encore Dans ma g n ration plusieurs de mes coll gues ont continu au del de 80 ans par exemple mes vieux amis Armand Borel et Raoul Bott morts tous deux r cem ment 82 ans Il n y a pas de raison de s arr ter tant que la sant le permet Encore faut il que le sujet s y pr te Quand on a des sujets tr s larges il y a toujours quelque chose faire mais si l on est trop sp cialis on peut se retrouver bloqu pendant de longues p riodes soit parce que l on a d montr tout ce qu il y avait d montrer soit au contraire parce que les probl mes sont trop difficiles C est tr s frustrant Les d couvertes math matiques donnent de grandes joies Poincar Hadamard Littlewood lont tr s bien expliqu En ce qui me concerne je
117. ond chapter of CG while the third chapter of CG is about non abelian cohomology After thirty years Serre returned to both topics in a series of three courses S234 153 1991 S236 156 1992 S247 165 1994 9 3 Galois Cohomology of Linear Algebraic Groups In his lecture delivered at Brussels in the Colloquium on Algebraic Groups S96 GE 53 1962 Serre pre Jean Pierre Serre An Overview of His Work 51 sented two conjectures on the cohomology of linear algebraic groups known as Conjecture I CI and Conjecture II CI Given an algebraic group G defined over a field k we may consider the coho mology group H k G G k and the cohomology set H k G isomorphism classes of G k torsors In what follows we will suppose that the ground field k is perfect and we will denote by cd k the cohomological dimension of Gal k k The above conjectures State CI If cd k lt 1 and G is a connected linear group then H lk G 0 CH If cd k lt 2 and G is a semisimple simply connected linear group then H k G 0 At the time the truth of Conjecture I was only known in the following cases when k is a finite field S Lang when k is of characteristic zero and has property C1 T Springer for G solvable connected and linear for G aclassical semisimple group Conjecture I was proved a few years later in a beautiful paper by R Steinberg 1965 which Serre included in
118. one the presentation of recent results often in a form that is clearer than that given by the authors It covers all branches of mathematics Bourbaki s books Topology Algebra Lie Groups etc are still widely read not just in France but also abroad Some have become classics I m thinking in partic ular about the book on root systems I recently saw in the AMS Citations Index that Bourbaki ranked sixth by number of citations among French mathematicians What s more at the world level numbers 1 and 3 are French and both are called Lions a good point for the Coll ge I have very good memories of my collabora tion with Bourbaki from 1949 to 1973 Bourbaki taught me many things both on background making me write about things which I did not know very well and on style how to write in order to be understood Bourbaki also taught me not to rely on specialists Bourbaki s working method is well known the distribution of drafts to the var ious members and their criticism by reading them aloud line by line slow but ef fective The meetings congr s were held three times a year The discussions were very lively sometimes passionate At the end of each congr s the drafts were distributed to new writers And so on A chapter could often be written four or five times The slow pace of the process explains why Bourbaki ended up publish ing with relatively few books over the 40 years from 1930 1935 till the end
119. othendieck ring R4 of Com Let K be the field of fractions of A Serre proves that the natural morphism i Ra Rx Et E K is an isomorphism if A is principal and under the assumption that all decomposition homomorphisms defined as in Brauer s theory for finite groups are surjective If M is an abelian group and Ty denotes the A group scheme whose character group is M the bialge bra C M can be identified with the group algebra A M If A is principal one has an isomorphism ch R4 Ty ane Z M provided by the rank Next Serre considers a split reductive group G and a split torus T of G which exists by hypothesis By composing ch with the restriction homomorphism Res Rx G Rx T a ho momorphism chg Rx G Z M is obtained Serre proves that chg is injective and that its image equals the subgroup Z M of the elements of Z M which are invariant under the Weyl group W of G relative to T As an illustration of this re sult the paper gives the following example if G GL M Z and W S is the symmetric group on n elements then Z M Z X1 Xn Mees ees and R4 GL Rx GL Z M Z Aq Ana Where A1 An denote the elementary symmetric functions in X1 Xn and the subscript stands for lo Jean Pierre Serre An Overview of His Work 79 calization with respect to This was what Grothendieck needed for his theory of rings 14 3 Finite Subgroups of Lie Groups and of Algebraic Groups Th
120. over a field k The main idea is to use the Tits build ing T of G A subgroup I of G is called completely reducible in G if for every maximal parabolic subgroup P of G containing I there exists a maximal parabolic subgroup P of G opposite to P which contains I There is a corresponding no tion of G irreducibility F is called G irreducible if it is not contained in any proper parabolic subgroup of G 1 e if it does not fix any point of the building X There is also a notion of G indecomposability These different notions behave very much like the classical ones 1 e those relative to G GL for instance there is an analogue of the Jordan H6lder theorem and also of the Krull Schmidt theo rem The proofs are based on Tits geometric theory of spherical buildings As one of the concrete applications given in the paper we only mention the following if r c G k G is of type Eg and V 1 lt i lt 8 denote the 8 fundamental irreducible representations of G and if one of them is a I module semisimple then all the others are also semisimple provided that char k gt 270 14 1 3 The Oberwolfach report S272 2004 states without proof two new re sults on the characters of compact Lie groups The first one is a generalization of a theorem of Burnside for finite groups given an irreducible complex character x of a compact Lie group G of degree gt 1 there exists an element x G of finite order with x x 0 T
121. owla C S Herz and K Iwasawa Seminar on complex multiplication Institute for Advanced Study Princeton N J 1957 58 Lecture Notes in Math Vol 21 Springer Verlag Berlin R sum des cours de 1965 1966 Annuaire du Coll ge de France 49 58 1967 Groupes de congruence d apr s H Bass H Matsumoto J Mennicke J Milnor C Moore In S minaire Bourbaki 1966 1967 Expos 330 17 pp D pendance d exponentielles p adiques In S minaire Delange Pisot Poitou 1965 66 Th orie des nombres Expos 15 14 pp Groupes finis d automorphismes d anneaux locaux r guliers r dig par M J Bertin In Colloque d Alg bre Paris 1967 Expos 8 11 pp with H Bass and J Milnor Solution of the congruence subgroup problem for SL n gt 3 and Sp n gt 2 Inst Hautes tudes Sci Publ Math 33 59 137 Erratum On a functorial property of power residue symbols Inst Hautes Etudes Sci Publ Math 44 241 244 1974 Local class field theory In Algebraic Number Theory Brighton 1965 128 161 Thomp son Washington D C Complex multiplication In Algebraic Number Theory Brighton 1965 292 296 Thomp son Washington D C Sur les groupes de Galois attach s aux groupes p divisibles In Proc Conf Local Fields Driebergen 1966 118 131 Springer Verlag Berlin Repr sentations lin aires des groupes finis Hermann Paris Translated into English Ger man Japanese Polish Spanish Commutativit
122. pact complex manifold X and with values in an analytic coherent sheaf F are finite dimensional vector spaces the proof is based on a result due to L Schwartz on completely continuous maps between Fr chet spaces This finiteness result played an essential role in GAGA see Sect 2 2 1 3 1 In a paper dedicated to H Hopf Serre S58 28 1955 proves a duality theorem in the setting of complex manifolds The proof is based on Schwartz s theory of distributions a distribution can be viewed either as a generalized function Jean Pierre Serre An Overview of His Work 39 or as a linear form on smooth functions hence distribution theory has a built in self duality 1 3 2 Previously in a letter GE 20 1953 addressed to Borel Serre had conjectured a generalization of the Riemann Roch theorem to varieties of higher dimension This generalization was soon proved by F Hirzebruch in his well known Habilita tionsschrift and presented by Serre at the S minaire Bourbaki The more general ver sion of the Riemann Roch theorem due to Grothendieck 1957 was the topic of a Princeton seminar by Borel and Serre A detailed account appeared in S74 1958 a paper included in A Borel uvres Collected Papers no 44 and which was for many years the only reference on this topic 1 4 The Amsterdam Congress At the International Congress of Mathematicians held in Amsterdam in 1954 K Kodaira and J P Serre were awarded
123. properties for the L functions associated to compatible systems of rational adic representations a typical example being that of the Sato Tate conjecture In Chap II Serre associates to every algebraic number field k a projective family Sm of commutative algebraic groups defined over Q From the point of view of motives these groups are just the commutative motivic Galois groups For each modulus m of k he constructs an exact sequence of commutative algebraic groups 1 gt Tn Sm Cm gt 1 in which Cy is a finite group and Tm is a torus The characters of Sm are essentially the Grdssencharakteren of type Ao in the sense of Weil of conductor dividing m They appear in the theory of complex multiplication In Chap III the concept of a locally algebraic abelian adic representation is defined The main result is that such Galois representations come from linear rep resentations in the algebraic sense of the family Sm When the number field k is obtained by the composition of quadratic fields it is shown that every semisim ple abelian rational adic representation is locally algebraic The proof is based upon transcendence results of C L Siegel and S Lang Serre observes that the re sult should also be true for any algebraic number field this was proved later by M Waldschmidt 1986 as a consequence of a stronger transcendence result In Chap IV the results of the previous chapters are applied to the adic rep
124. r we have Gal k E k GL2 F 7 for almost all The proof is based upon local results relative to the ac tion of the tame inertia group on the points of finite order of the elliptic curves This action can be expressed in terms of products of fundamental characters and the main point is that these exponents have a uniform bound namely the ramification index of the local field This boundedness plays a role similar to that of the local algebraicity which had been used in McGill Serre conjectures that similar bounds are valid for higher dimensional cohomology this has been proved recently as a by product of Fontaine s theory He also raised several questions concerning the effectiveness of the results The paper includes many numerical examples in which all the prime numbers for which Gal k E k 4 GL2 Fe are computed In the summary of the course Serre also mentions that if A is an abelian surface such that End A is an order of a quaternion field D defined over Q a so called fake elliptic curve then the group p Gal Q Q where p Lec is open in D A This was proved later by M Ohta 1974 following Serre s guidelines 13 3 Modular Forms and adic Representations Many arithmetical functions can be recovered from the Fourier coefficients of modular functions or modular forms In an early contribution at the S minaire Delange Pisot Poitou S138 CE 80 1969 one finds the remarkable conjecture that cer
125. ritten with the help of H Bass In the first chapter Serre shows that it is possible to recover a group G which acts on a tree X from the quo tient graph or fundamental domain G X and the stabilizers of the vertices and of the edges If G X is a segment then G may be identified with an amalgam of two groups and moreover every amalgam of two groups can be obtained in this way The study of relations between amalgams and fixed points show that groups such as SL3 Z and Sp Z are not amalgams since one can show that they always have fixed points when they act on trees see S163 1974 the method extends to all G Z where G is any reductive group scheme over Z which is simple of rank gt 2 In the second chapter the results are applied to the study of the groups SL2 k where k is a local field The group SL2 k acts on the Bruhat Tits tree associated to the space k The vertices of this tree are the classes of lattices of k In this way Serre recovers a theorem due to Y Ihara by which every torsion free discrete subgroup of SL2 Qp is free According to MathSciNet this book is now Serre s most cited publication 14 2 1 A question raised by Grothendieck concerning linear representations of group schemes was answered by Serre in S136 81 1968 Suppose that C is a coalgebra over a Dedekind ring A which is flat If Com denotes the abelian cate gory of comodules over C which are of finite type as A modules one may consider the Gr
126. rum of the automorphism group G Aut A of a regular tree of valency q 1 which is a locally compact group with respect to the topology of simple convergence Then ug is the restriction to Q of the Plancherel measure of G Several interesting consequences are derived from the equidistribution theorem For instance it 1s shown that the maximum of the dimension of the Q simple factors of the Jacobian Jo N of the modular curve Xo N tends to infinity as N ov In particular there are only finitely many inte gers N gt 1 such that Jo V is isogenous over Q to a product of elliptic curves as was already stated in S237 159 1993 11 2 Varieties Over Finite Fields Let q p with p a prime number and e gt 1 and let F be a finite field with q elements The numbers Ngr r gt 1 of rational points over F of non singular projective varieties defined over F are encapsulated in their zeta function One of the major achievements of A Grothendieck and his school was to provide the tools for the proof of the Weil conjectures 1949 relative to the nature of these functions cohomological interpretation rationality functional equation and the so called Riemann hypothesis Serre had a profound influence on the process An account of the landmark paper by P Deligne 1974 on the proof of the Riemann hypothesis for the zeta function of a non singular variety defined over a finite field can be found in S166 1974 The paper 117 197
127. s Aujourd hui il faut ajouter les ordinateurs Cela reste peu on reux dans la mesure o les math mati ciens ont rarement besoin de ressources de calcul tr s importantes A la diff rence par exemple de la physique des particules dont les besoins en calcul sont a la mesure des tr s grands quipements n cessaires au recueil des donn es les math maticiens ne mobilisent pas de grands centres de calcul En pratique l informatique change les conditions mat rielles du travail des math maticiens on passe beaucoup de temps devant son ordinateur Il a diff rents us ages Tout d abord le nombre des math maticiens a consid rablement augment A mes d buts il y a55 ou 60 ans le nombre des math maticiens productifs tait de quelques milliers dans le monde entier l quivalent de la population d un village A Vheure actuelle ce nombre est d au moins 100000 une ville Cet accroissement a des cons quences pour la mani re de se contacter et de s informer L ordinateur et Internet acc l rent les changes C est d autant plus pr cieux que les math mati ciens ne sont pas ralentis comme d autres par le travail exp rimental nous pouvons communiquer et travailler tr s rapidement Je prends un exemple Un math maticien a trouv une d monstration mais il lui manque un lemme de nature technique Au moyen d un moteur de recherche comme Google il rep re des coll gues qui ont travaill sur la q
128. s d buts Est ce que l on fait des math matiques aujourd hui comme on les faisait il y a cinquante ans Bien s r on fait des math matiques aujourd hui comme il y a cinquante ans videmment on comprend davantage de choses l arsenal de nos m thodes a aug ment Il y a un progr s continu Ou parfois un progr s par coups certaines branches restent stagnantes pendant une d cade ou deux puis brusquement se r veillent quand quelqu un introduit une id e nouvelle Si l on voulait dater les math matiques modernes un terme bien dangereux il faudrait sans doute remonter aux environs de 1800 avec Gauss Et en remontant plus loin si vous rencontriez Euclide qu auriez vous vous dire Euclide me semble tre plut t quelqu un qui a mis en ordre les math matiques de son poque Il a jou un r le analogue celui de Bourbaki il y a cinquante ans Ce n est pas par hasard que Bourbaki a choisi d intituler ses ouvrages des l ments de Math matique c est par r f rence aux l ments d Euclide Notez aussi que Math matique est crit au singulier Bourbaki nous enseigne qu il n y a pas plusieurs math matiques distinctes mais une seule math matique Et il nous l enseigne sa fa on habituelle pas par de grands discours mais par l omission d une lettre la fin d un mot Pour en revenir a Euclide je ne pense pas qu il ait produit des contributions r ellement originales
129. s II and III are devoted to the study of Galois cohomology in the com mutative and the non commutative cases most of the results of Chap II were due to Tate and an important part of those of Chap III were due to Borel Serre 9 1 1 The Bourbaki report S105 GE 60 1964 summarizes M Lazard s seminal paper 1964 on p adic Lie groups One of Lazard s main results is that a profi 50 P Bayer nite group is an analytic p adic group if and only if it has an open subgroup H which is a pro p group and is such that H H C HP if p 2 or H H C H if p 2 If G is a compact p adic Lie group such that cd G n lt ov then G is a Poincar pro p group of dimension n and the character x x det Ad x is the dualizing character of G Here Ad x denotes the adjoint automorphism of Lie G defined by x The group G has finite cohomological dimension if and only if it is torsion free the proof combines Lazard s results with the theorem of Serre men tioned below 9 1 2 The paper S114 GE 66 1965 proves that if G is a profinite group p torsion free then for every open subgroup U of G we have the equality cd U cd G between their respective cohomological p dimensions The proof is rather intricate In it Serre makes use of Steenrod powers a tool which he had acquired during his topological days cf S44 1953 As a corollary every torsion free pro p group which contains a free open subgroup is free Serre asked whether
130. s formula for the set X of all totally ramified extensions of K of given degree n contained in K namely 2 ge n LexXn where g is the norm of the wild component of the discriminant of L K Al though the formula could in principle be deduced from earlier results of Krasner Serre proves it independently in two elegant and different ways The first proof is de rived from the volume of the set of Eisenstein polynomials The second uses the p adic analogue of Weil s integration formula applied to the multiplicative group D of a division algebra D of center K such that D K n 8 p adic Analysis Let V be an algebraic variety over a finite field k of characteristic p One of Weil s conjectures is that the zeta function Zy t is a rational function of t This was proved in 1960 by B Dwork His method involved writing Zy t as an alternating product of p adic Fredholm determinants This motivated Serre to study the spectral the ory of completely continuous operators acting on p adic Banach spaces S99 CE 55 1962 The paper which is self contained provides an excellent introduction to p adic analysis Given a completely continuous endomorphism u defined on a Banach space E over a local field the Fredholm determinant det 1 tu is a power series in f which has an infinite radius of convergence and thus defines an entire function of t The Fredholm resolvent P t u det 1 tu 1 tu of u is an entire func
131. s known that 7 is lacunary for r 2 4 6 8 10 14 26 Serre proves that if r is even the above list is complete By one of the theorems proved in his Chebotarev paper see above this is equiva lent to showing that n is of CM type only if r 2 4 6 8 10 14 or 26 The proof consists of showing that the complex multiplication if it exists comes from either QCG or Q V 3 13 3 8 The paper S216 GE 143 1987 entitled Sur les repr sentations modulaires de degr 2 de Gal Q Q contains one of Serre s outstanding contributions In this profound paper dedicated to Y I Manin Serre formulates some very precise con jectures on Galois representations which extend those made twelve years before in Bordeaux S169 104 1975 We shall only mention two of these conjectures conjectures 3 2 39 and 3 2 49 known nowadays as Serre s modularity conjec tures or simply Serre s modularity conjecture Let p Gal Q Q GL F be a continuous irreducible representation of odd determinant 3 2 39 There exists a cusp form f with coefficients in F which is an eigenfunc tion of the Hecke operators whose associated representation ppf is isomor phic to the original representation p 3 2 49 The smallest possible type of the form f of 3 2 39 is equal to N p k p p where the level N p is the Artin conductor of p it reflects the ram ification at the primes p the character p is x I k det p
132. s residue field which is assumed to be perfect Given an abelian variety A defined over K the authors start from the existence of the N ron model A of A with respect to v which is a group scheme of finite type over Spec O Serre and Tate define the concept of potential good reduction of A which generalized that of good reduction They prove that A has good reduction at v if and only if the Tate module 7 A is unramified at v where denotes a prime which differs from the characteristic of ky This criterion is partially due to A P Ogg in the case of elliptic curves and partially to I Shafarevich In their proof the struc ture of the connected component A of the special fiber A of Ay appears it is an extension of an abelian variety B by a linear group L and L is a product of a torus S by a unipotent group U The abelian variety A has good reduction if and only if L 1 it has potential good reduction if and only if L U and it has semistable reduction if and only if L S A second theorem says that A has potential good reduction if and only if the image of the inertia group v for the adic represen tation pe Gal K K Aut Ts is finite An appropriate use of the characters of Artin and Swan then allows the definition of the conductor of A The semistable reduction theorem conjectured by Serre in 1964 and proved later by Grothendieck in SGA 7 would allow the definition of the conductor for every abelian variety The sem
133. s the ring of integers in the quadratic field of discriminant d being E F q 1 a d a 4q and under the assumption that d is the discriminant of an imaginary quadratic field Polarizations on these abelian varieties correspond to positive def inite hermitian forms on R modules Thus in the cases where there is no inde composable positive definite hermitian module of discriminant 1 one obtains the non existence of curves whose Jacobian is of that type And conversely if such a hermitian module exists one obtains a principally polarized abelian variety if fur thermore its dimension is 2 this abelian variety is a Jacobian and one gets a curve whose number of points is g 1 2a a similar but less precise result holds for genus 3 one finds a curve with either g 1 3a or g 1 3a points Partic ular results on the classification of these modules in dimensions 2 and 3 due to D W Hoffmann 1991 and a procedure for gluing isogenies are used to deter mine the existence or non existence of certain polarized abelian varieties useful in their turn to show that for all finite fields F there exists a genus 3 curve over Fg such that its number of rational points is within 3 of the Serre Weil upper or lower bound 11 3 2 In a letter published in S230 155 1991 Serre answered a problem posed by M Tsfasman at Luminy on the maximal number of points of a hypersur 60 P Bayer face defined over a finite field On
134. spaces whose fibers are products of elliptic curves with com plex multiplication Serre S107 63 1964 constructed examples of non singular projective varieties defined over an algebraic number field K which are Galois con jugate but have non isomorphic fundamental groups In particular although they have the same Betti numbers they are not homeomorphic 13 2 3 The paper S118 GE 70 1966 is about the adic Lie groups and the adic Lie algebras associated to elliptic curves defined over an algebraic number field k and without complex multiplication The central result is that gg is as large as possible namely it is equal to End V when the ground field is Q In the proof Serre uses a wide range of resources Lie algebras and adic Lie groups Hasse Witt invariants of elliptic curves pro algebraic groups the existence of canonical liftings of ordinary curves in characteristic p the Lie subalgebras of the ramification groups Chebotarev s density theorem as well as class field theory Hodge Tate theory and so on Serre also observes that if a conjecture of Tate on Galois actions on the Tate modules is true then the determination of the Lie algebra g can be 66 P Bayer carried out for any algebraic number field Tate s conjecture was proved almost two decades later by G Faltings 1983 in a celebrated paper in which he also proved two more conjectures one due to L J Mordell and the other due to I Shafar
135. tain congruences satisfied by the Ramanujan t function can be explained by the existence for each prime of a 2 dimensional adic representation pe Gal Q Q gt Aut Ve unramified away from and such that Tr oe F t p det oe F p for each Frobenius element Fp at any prime p Assume this conjecture which was proved a few months later by Deligne see below The adic representa tion pe leaves a lattice of Ve stable and thus may be viewed as a representation in GL2 Zy When varying the different primes the above representations pg 68 P Bayer make up a compatible system of rational adic representations of Q in the sense of McGill and the images of pg are almost always the largest possible The primes for which this does not happen are called the exceptional primes and they are fi nite in number More specifically in the case of the t function the exceptional primes are 2 3 5 7 23 691 this was proved later by Swinnerton Dyer For exam ple t p 1 p mod 691 is the congruence discovered by Ramanujan As a consequence the value of t p mod cannot be deduced from any congruence on p if is a non exceptional prime 13 3 1 Serre s conjecture on the existence of adic representations associated to modular forms was soon proved by Deligne 1971 This result has been essential for the study of modular forms modulo p for that of p adic modular forms as well as for the work of H
136. tely decomposable is bounded The second question has a negative answer in characteristic p gt 0 The examples are constructed either by means of modular curves or as coverings of curves of genus 2 or 3 The highest genus obtained is 1297 11 1 5 Let p be a prime number In the paper S254 CE 170 1997 Serre deter mines the asymptotic distribution of the eigenvalues of the Hecke operators T on spaces of modular forms when the weight or the level varies More precisely let Tp denote the Hecke operator associated to p acting on the space S N k of cusp forms of weight k for the congruence group l o N with gcd N p 1 and let l T p D By Deligne s theorem on the Ramanujan Petersson conjecture the eigenvalues of the operator l belong to the interval Q 2 2 Let us con sider sequences of pairs of integers N k such that ky is even k N as oo and p does not divide N The main theorem proved in the paper states that the family x x N k of eigenvalues of the L N k is equidistributed 58 P Bayer in the interval Q with respect to a measure up which is given by an explicit for mula similar but not identical to the Sato Tate measure In fact in the paper we find measures yg which are defined for every q gt 1 and have the property that limg gt oo Mg Loo Where uo is the Sato Tate measure In order to give an interpre tation of ug Serre identifies Q with a subset of the spect
137. th Soc Providence RI 2004 Wen Ching Winnie Li On negative eigenvalues of regular graphs C R Acad Sci Paris S r I Math 333 10 907 912 With comments by J P Serre Kristin Lauter Geometric methods for improving the upper bounds on the number of ra tional points on algebraic curves over finite fields J Algebraic Geom 10 1 19 36 2001 With an appendix by J P Serre Jean Pierre Serre In Wolf Prize in Mathematics Vol 2 523 551 World Sci Publ Co 2002 K Lauter The maximum or minimum number of rational points on genus three curves over finite fields Compositio Math 134 1 87 111 With an appendix by J P Serre On a theorem of Jordan Math Medley 29 3 18 Reprinted in Bull Amer Math Soc N S 40 4 429 440 2003 List of Publications for Jean Pierre Serre 95 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 2003 Cohomological invariants Witt invariants and trace forms Notes by Skip Garibaldi In Co homological invariants in Galois cohomology 1 100 Univ Lecture Ser Vol 28 Amer Math Soc Providence RI 2004 Discours prononc en s ance publique le 30 septembre 2003 en hommage a Armand Borel 1923 2003 Gaz Math 102 25 28 with J Arthur E Bombieri K Chandrasekharan F Hirzebruch G Prasad T A Springer and J Tits Armand Borel 1923 2003 Notices Amer Math Soc 51 5
138. the discrete analogue of this statement is true i e whether every torsion free group G which contains a free subgroup of finite index is free This was proved a few years later by J Stallings 1968 and R Swan 1969 9 1 3 More than thirty years later Serre dedicated S255 173 1998 to John Tate The paper deals with the Euler characteristic of profinite groups Given a profi nite group G of finite cohomological p dimension and a discrete G module A which is a vector space of finite dimension over the finite field F the Euler characteristic e G A 1 dim H G A is defined under the assumption that dim H i G A lt for all i Let Greg be the subset of G made up by the regular elements Serre proves that there exists a distribution ug over Greg with values in Q such that e G A PA UG Where pA Greg gt Zp denotes the Brauer character of the G module A This distribution can be described explicitly in several cases e g when G is a p torsion free p adic Lie group thanks to Lazard s theory 9 2 Galois Cohomology Let G Gal K k be the Galois group of a field ex tension and suppose that A is a discrete G module The abelian Galois cohomol ogy groups H4 Gal K k A are usually denoted by H1 K k A or simply by H41 k A when K ks is a separable closure of k Abelian Galois cohomology with special emphasis on the results of Tate was the content of the course S104 59 1963 and of the sec
139. the historian Fernand Braudel As for probability theory it is useful for its applications both to mathematics and to practical questions From a purely mathematical point of view it is a branch of measure theory Can one really describe it as a new way in which to represent the world Surely not in mathematics Have computers changed the manner in which mathematics is conducted It used to be said that mathematical research was cheap paper and pencils that was all we needed Nowadays you have to add computers It is not very expensive since mathematicians rarely need a lot of processing power This is different from say particle physics where a lot of equipment is required In practice computers have changed the material conditions of mathematicians work we spend a lot of time in front of our computer It has several different uses First of all there are now considerably more mathematicians When I started out some 55 or 60 years ago there were only a few thousand productive mathematicians in the whole world the equivalent of a village Now this number has grown to at least 100 000 a city This growth has consequences for the way mathematicians contact each other and gain information The computer and Internet have acceler ated exchanges This is especially important for us since we are not slowed down as others by experimental work we can communicate and work very rapidly Let me give you an example If a mathematic
140. the English translation of CG M Kneser 1965 proved Conjecture II when k is a p adic field and G Harder 1965 did the same when k is a totally imaginary algebraic number field and G does not have any factor of type Eg this restriction was removed more than 20 years later by V I Chernousov 1989 More generally for any algebraic number field k and any semisimple simply connected linear algebraic group G the natural mapping H k G gt I Loreat l ky G is bijective Hasse s principle in agreement with a conjecture of Kneser A detailed presentation of this fact can be found in a book by V Platonov and A Rapinchuk 1991 9 3 1 The Galois cohomology of semisimple linear groups was taken up again by Serre in the course S234 153 1991 One of his objectives was to discuss the cohomological invariants of H l k G i e see below the relations which connect the non abelian cohomology set H k G and certain Galois cohomology groups H i k C where C is commutative e g C Z 22 More precisely let us consider a smooth linear algebraic group G defined over a field k an integer i gt 0 and a finite Galois module C over ko whose order is coprime to the characteristic By definition a cohomological invariant of type H C is a morphism of the functor k gt H k G into the functor k gt H k C defined over the category of field extensions k of ko Suppose that the characteristic of k is not 2
141. the properties of the Artin representation a notion due to Weil in his paper L avenir des math matiques 1947 The third part of CL is about group cohomology It includes the cohomological interpretation of the Brauer group Br k of a field k and class formations a la Artin Tate Local class field theory takes up the fourth part of the book The reciprocity isomorphism is obtained from the class formation associated to the original local field it is made explicit by means of a computation of norm residue symbols based on a theorem of B Dwork 1958 48 P Bayer One also finds in CL the first definitions of non abelian Galois cohomology Given a Galois extension K k and an algebraic group G defined over k the ele ments of the set H Gal K k G K describe the classes of principal homoge neous G spaces over k which have a rational point in K Easy arguments show that H Gal K k G K 1 when G is one of the following algebraic groups additive G4 multiplicative Gm general linear GL and symplectic Sp 7 2 1 Another exposition of local class field theory can be found in the lecture S127 75 1967 it differs from the one given in CL by the use of Lubin Tate theory of formal groups which allows a neat proof of the existence theorem 7 3 A Local Mass Formula Let K denote a local field with finite residue field k of g elements and let K be a separable closure of K In S186 GE 115 1978 one finds a mas
142. the s plane and a very simple functional equation 13 5 2 The subject of adic representations had already been considered by Serre in S177 112 1977 in his address to the Kyoto Symposium on Algebraic Num ber Theory In this paper which is rich in problems and conjectures we find the statement of the conjecture of Shimura Taniyama Weil according to which any elliptic curve over Q of conductor N is a quotient of the modular curve Xo N 13 5 3 The paper S229 154 1991 is a short introduction to the theory of mo tives Along these lines we also highlight the paper S239 GE 160 1993 which corresponds to a text Serre wrote for Bourbaki in 1968 The paper deals with alge braic envelopes of linear groups and their relationship with different types of alge bras coalgebras and bialgebras Its last section contains an account of the dictionary between compact real Lie groups and complex reductive algebraic groups 13 5 4 To finish this section we shall briefly summarize the paper entitled Pro pri t s conjecturales des groupes de Galois motiviques et des repr sentations t adiques S243 161 1994 Serre formulates a series of conjectures regard ing adic representations which generalize many of his previous results We denote by M the category of pure motives over a subfield k of C which we suppose to be of finite type over Q The motivic Galois group Gy is related to the absolute Ga lois group of k by means of an e
143. theorems of Faltings Frobenius tori McGill theory properties of inertia groups at the places which divide as well as group theoretic information regarding the subgroups of GLy F supplied by theorems of V Nori 1985 1987 The proofs of the above results have not been published in a formal way but one can find an account of them in Serre s letters to K Ribet 133 1981 and CE 138 1986 to D Bertrand GE 134 1984 and to M F Vign ras 137 1986 74 P Bayer 13 5 Motives A first lecture on zeta and L functions in the setting of the theory of schemes of finite type over Spec Z was given by Serre in S112 GE 64 1965 One finds in it a generalization of Chebotarev s density theorem to schemes of arbi trary dimension 13 5 1 In his lecture in the S minaire Delange Pisot Poitou GE 87 1969 70 Serre introduces several definitions and formulates several conjectures about the local factors gamma factors included of the zeta function of a smooth projective variety over a number field The local factors at the primes of good reduction do not raise any problem The interesting cases are a the primes with bad reduction b the archimedean primes In both cases Serre gives definitions based in case a on the action of the local Galois group on the adic cohomology and in case b on the Hodge type of the real cohomology The main conjecture is that such a zeta function has an analytic continuation to
144. tion of t with values in End Given an element a K one shows that the endomorphism 1 au is invertible if and only if det 1 au 0 If this is the case then the relation det 1 au 1 au P a u P a u 1 au is satisfied If a K is a zero of order h of the function det 1 fu then the space E uniquely decomposes into a direct sum of two closed subspaces N F Jean Pierre Serre An Overview of His Work 49 which are invariant under u The endomorphism 1 au is nilpotent on N and invertible on F the dimension of N is h just as in F Riesz theory over C Serre proves that given an exact sequence of Banach spaces and continuous lin ear mappings 0 Eo Ss E are En 0 and given completely continu ous endomorphisms u of E such that dj ou uj 0 di for O lt i lt n then j det 1 tu 1 this is useful for understanding some of Dwork s com putations 8 0 1 In S113 65 1965 the compact p adic analytic manifolds are classified Given a field K locally compact for the topology defined by a discrete valuation any compact analytic manifold X defined over K of dimension n at each of its points is isomorphic to a disjoint finite sum of copies of the ball A where A denotes the valuation ring of K Two sums r A and r A are isomorphic if and only ifr r mod g 1 where q denotes the number of elements of the residue field of A The class of r m
145. tive the congruence kernel C is a finite cyclic group whose order is at most equal to the number of roots of unity in k if k is not totally imaginary one has C 1 the problem has a positive answer If 5 1 the problem has a quite negative answer C is an infinite group The proof is very interesting In the case S 1 it uses number theory while in the case S gt 1 it uses topology We shall now provide some of the details of this proof In the case 5 gt 2 Serre shows that C is contained in the centre of GX and then makes use of a theory of C Moore in order to determine it and in partic ular to show that it is finite and cyclic The finiteness of C has some important consequences For instance given an S arithmetic subgroup N C SL 0 a field Jean Pierre Serre An Overview of His Work 55 K of characteristic zero and a linear representation p N G K there exists a subgroup N C N of finite index such that the restriction of p to N is algebraic This implies that p is semisimple Moreover for every k N module V of finite rank over K we have H N V 0 In particular when taking for V the adjoint representation one sees that N is rigid If 5 1 Serre shows that for most S arithmetic subgroups N the group N is infinite this is enough to show that the S congruence problem has a negative an swer There are three cases char k p gt 0 k Q and k an imaginary quadratic field In each case there is
146. tiviques et des repr sentations adiques In Motives Proc Sympos Pure Math 55 377 400 Amer Math Soc Providence RI with E Bayer Fluckiger Torsions quadratiques et bases normales autoduales Amer J Math 116 1 1 64 Sur la semi simplicit des produits tensoriels de repr sentations de groupes Invent Math 116 1 3 5 13 530 Cohomologie galoisienne progr s et probl mes In S minaire Bourbaki 1993 94 Expos 783 229 257 Ast risque No 227 R sum des cours de 1993 1994 Annuaire du Coll ge de France 91 98 1995 Travaux de Wiles et Taylor I In S minaire Bourbaki 1994 95 Expos 803 319 332 Ast risque No 237 1996 Two letters on quaternions and modular forms mod p Israel J Math 95 281 299 With introduction appendix and references by R Livn Exemples de plongements des groupes PSL2 F dans des groupes de Lie simples Invent Math 124 1 3 525 562 94 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 List of Publications for Jean Pierre Serre 1997 Deux lettres sur la cohomologie non ab lienne In Geometric Galois actions 1 London Math Soc Lecture Note Ser Vol 242 175 182 Cambridge Univ Press Cambridge Semisimplicity and tensor products of group representations converse theorems J Algebra 194 2 496 520 With an appendix by Walter Feit Robert Steinberg
147. ts of the product of such elliptic curves is as large as possible for almost all the primes Jean Pierre Serre An Overview of His Work 61 12 Field Theory The paper S195 123 1980 reproduces a letter of Serre answering a question raised by J D Gray about Klein s lectures on the icosahedron The icosahedral group G As acts on a curve X of genus zero by extending the field of definition k of X and G to an algebraic closure one obtains an embedding of G in the projec tive linear group PGL 2 Moreover the field k must contain 4 5 The quotient X G is isomorphic to P If z is a k point of X G its lifting to X generates a Galois extension k of k whose Galois group is a subgroup of G Serre explains that the main question posed by Hermite and Klein turns out to be whether one obtains all Galois extensions of k with Galois group G in this way He then shows that the an swer to this question is almost yes Suppose that k k is a Galois extension with Galois group G Serre uses a descent method and works with twisted curves Xx The curves Xx are controlled by a quaternion algebra Hy To go from Xy to Hy Serre follows two procedures either using the non trivial element of H G Z 2Z which corresponds to the binary icosahedral group or considering the trace form Tr z in a quintic extension k k defining k k Then he shows that k k comes from a covering X X G if and only if Xy has a rational point over k if and on
148. ues Luminy 1989 Ast risque 198 200 11 351 353 Les petits cousins In Miscellanea mathematica 277 291 Springer Verlag Berlin Erratum Letters of Ren Baire to mile Borel In Cahiers du S minaire d Histoire des Math matiques Vol 12 p 513 Univ Paris VI Paris Rev tements de courbes alg briques In S minaire Bourbaki 1991 92 Expos 749 167 182 Ast risque No 206 R sum des cours de 1990 1991 Annuaire du Coll ge de France 111 121 1992 Topics in Galois theory Research Notes in Mathematics Vol 1 Jones and Bartlett Pub lishers Boston MA Lecture notes by H Darmon With a foreword by Darmon and the author R sum des cours de 1991 1992 Annuaire du Coll ge de France 105 113 1993 with T Ekedahl Exemples de courbes alg briques jacobienne compl tement d compos able C R Acad Sci Paris S r I Math 317 5 509 513 Smith Minkowski et l Acad mie des Sciences Gaz Math 56 3 9 Gebres L Enseign Math 2 39 1 2 33 85 R sum des cours de 1992 1993 Annuaire du Coll ge de France 109 110 1994 A letter as an appendix to the square root parameterization paper of Abhyankar In Alge braic geometry and its applications West Lafayette IN 1990 85 88 Springer Verlag New York U Jannsen S Kleiman and J P Serre editors Motives Proc Symp Pure Math 55 Part 1 and 2 Amer Math Soc Providence RI Propri t s conjecturales des groupes de Galois mo
149. uestion et leur envoie un e mail De cette mani re il a toutes les chances de trouver en quelques jours ou m me en quelques heures la personne qui a effectivement d montr le lemme dont il a besoin Bien entendu ceci ne concerne que des probl mes auxiliaires des points de d tail pour lesquels on d sire renvoyer des r f rences existantes plut t que de refaire soi m me les d monstrations Sur des questions vraiment difficiles mon math maticien aurait peu de chances de trouver quelqu un qui puisse lui venir en aide L ordinateur et Internet sont donc des outils d acc l ration de notre travail Ils permettent aussi de rendre les manuscrits accessibles dans le monde entier sans attendre leur parution dans un journal C est tr s pratique Notez que cette acc l ra tion a aussi des inconv nients Le courrier lectronique produit des correspondances informelles que l on conserve moins volontiers que le papier On jette rarement des lettres alors que l on efface ou l on perd facilement les emails quand on change d ordinateur par exemple On a publi r cemment en version bilingue fran ais sur une page et anglais sur la page d en face ma correspondance avec A Grothendieck entre 1955 et 1987 cela n aurait pas t possible si elle avait t lectronique Jean Pierre Serre My First Fifty Years at the Coll ge de France 23 of today The time of mathematicians is the longue dur e of my late colleague
150. ume that I is torsion free Then H41 T M x Hm l Is M for every l module M and for every integer q the dualizing module Is H Xs Z being free over Z Moreover cd T m and the group H1 T Z T is equal to 0 if g m and is equal to Zs if q m The proofs can be found in the two papers by Borel Serre S159 1973 S172 1976 which are reproduced in A Borel uvres Collected Papers no 98 and no 105 see also the survey S190 120 1979 11 Arithmetic of Algebraic Varieties 11 1 Modular Curves The three publications S142 1970 S170 1975 and S183 1977 correspond to lectures delivered by Serre in the S minaire Bourbaki They were very popular in the seventies as introductory texts for the study of the arithmetic of modular curves Jean Pierre Serre An Overview of His Work 57 11 1 1 The first lecture S142 1970 deals with a theorem of Y Manin 1969 according to which given a number field K an elliptic curve E defined over K and a prime number p the order of the p component of the torsion group Etr K is bounded by an integer depending only on K and p The proof relies on a previous result by V A Demjanenko and Manin on the finiteness of the number of rational points of certain algebraic curves this is applied afterwards to the modular curve Xo N where N N p K 11 1 2 The second lecture S170 1975 was written jointly with B Mazur Its purpose is to present results of
151. un espace euclidien par des fibres compactes C R Acad Sci Paris 230 2258 2260 Cohomologie des extensions de groupes C R Acad Sci Paris 231 643 646 Homologie singuli re des espaces fibr s I La suite spectrale C R Acad Sci Paris 231 1408 1410 H Holden R Piene eds The Abel Prize 83 DOI 10 1007 978 3 642 01373 7_14 Springer Verlag Berlin Heidelberg 2010 84 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 List of Publications for Jean Pierre Serre 1951 Homologie singuli re des espaces fibr s II Les espaces de lacets C R Acad Sci Paris 232 31 33 Homologie singuli re des espaces fibr s II Applications homotopiques C R Acad Sci Paris 232 142 144 Groupes d homotopie In S minaire Bourbaki 1950 1951 Expos 44 6 pp Utilisation des nouvelles op rations de Steenrod dans la th orie des espaces fibr s d apr s A Borel et J P Serre In S minaire Bourbaki 1951 1952 Expos 54 10 pp Applications de la th orie g n rale divers probl mes globaux In S minaire H Cartan E N S 1951 1952 Expos 20 26 pp Homologie singuli re des espaces fibr s Applications Ann of Math 2 54 425 505 with A Borel D termination des p puissances r duites de Steenrod dans la cohomologie des groupes classiques Applications C R Acad Sci Par
152. velopment of your discipline since the time when you were starting out Is mathematics conducted nowadays as it was 50 years ago Of course you do mathematics today like 50 years ago Clearly more things are understood the range of our methods has increased There is continuous progress Or sometimes leaps forward some branches remain stagnant for a decade or two and then suddenly there s a reawakening as someone introduces a new idea If you want to put a date on modern mathematics a very dangerous term you would have to go back to about 1800 and Gauss Going back further if you were to meet Euclid what would you say to him Euclid seems to me like someone who just put the mathematics of his era into or der He played a role similar to Bourbaki s 50 years ago It is no coincidence that Bourbaki decided to give its treatise the title l ments de Math matique This is a reference to Euclid s l ments Note that Math matique is written in the singular Bourbaki tells us that rather than several different mathematics there is one single mathematics And he tells us in his usual way not by a long discourse but by the omission of one letter from the end of one word Coming back to Euclid I don t think that he came up with genuinely original contributions Archimedes would be much more interesting to talk to He was the great mathematician of antiquity He did extraordinary things both in mathematics and physics
153. ver Tor G is equivalent to H K G 0 for every extension K of k For A a non abelian finite simple group 80 P Bayer Serre reproduces a table by Griess Ryba 1999 giving the pairs A G for which G is of exceptional type and A embeds projectively in G In order to see that the table is in fact complete the classification of finite simple groups is used 14 3 2 Part of the material of the paper S280 2007 arose from a series of three lectures at the Ecole Polytechnique F d rale de Lausanne in May 2005 Given a reductive group G over a field k and a prime different from char k and A a finite subgroup of G k the purpose of the paper is to give an upper bound for ve A that is the adic valuation of the order of A in terms of invariants of G k and Serre provides two types of such bounds which he calls S bounds and M bounds in recognition of previous work by I Schur 1905 and H Minkowski 1887 The Minkowski bound M n applies to the situation G GL and k Q and is optimal in the sense that for every n and for every there exists a finite subgroup A of GL Q for which ve A M n By making use of the at the time newly created theory of characters due to Frobenius Schur extended Minkowskr s results to an arbitrary number field k he defined a number M n such that veg A lt M n for any finite subgroup of GL2 C such that Tr g belongs to k for any g A As in the cas
154. w to use finite fields for problems concerning infinite fields Arithmetic Geometry Cryp tography and Coding Theory 183 194 Contemp Math 487 AMS Providence RI Born Degrees education Positions Visiting positions Memberships A Curriculum Vitae for Jean Pierre Serre September 15 1926 in Bages France Ecole Normale Sup rieure Paris 1945 1948 Agr g des sciences math matiques 1948 Docteur s sciences Sorbonne 1951 Attach puis charg de recherches CNRS 1948 1953 Maitre de recherches CNRS 1953 1954 Maitre de conf rences Facult des Sciences de Nancy 1954 1956 Professeur Coll ge de France 1956 1994 Professeur honoraire Coll ge de France 1994 Harvard University 1957 1964 1974 1976 1979 1981 1983 1985 1988 1990 1992 1994 1995 1996 2003 2005 2007 Institute for Advanced Study Princeton 1955 1957 1959 1961 1963 1967 1970 1972 1978 1983 1999 I H E S Bures sur Y vette 1963 1964 G ttingen Universit t 1970 McGill University 1967 2006 Mexico University 1956 Princeton University 1952 1999 Singapore University 1985 American Academy of Arts and Sciences 1960 Acad mie des Sciences de Paris correspondant 1973 titulaire 1977 London Mathematical Society Honorary Member 1973 Fellow of the Royal Society 1974 Royal Netherlands Academy of Arts and Sciences 1978 National Academy of Sciences USA 1979 97 98 A
155. wards and prizes Honorary degrees Curriculum Vitae for Jean Pierre Serre Royal Swedish Academy of Sciences 1981 American Philosophical Society 1998 Russian Academy of Sciences 2003 Norwegian Academy of Science and Letters 2009 Fields Medal 1954 Prix Peccot Vimont 1955 Prix Francoeur 1957 Prix Gaston Julia 1970 M daille Emile Picard 1971 Balzan Prize 1985 M daille d or du C N R S 1987 AMS Steele Prize 1995 Wolf Prize 2000 Abel Prize 2003 Cambridge 1978 Stockholm 1980 Glasgow 1983 Athens 1996 Harvard 1998 Durham 2000 London 2001 Oslo 2002 Oxford 2003 Bucharest 2004 Barcelona 2004 Madrid 2006 McGill 2008
156. work 76 P Bayer k G modules of finite dimension then their tensor product Vi V2 is a semisimple k G module Serre proves that this statement remains true in characteristic p gt 0 provided that p is large enough More precisely if V 1 lt i lt m are semisimple k G modules and p gt dim V 1 then the k G module V Vm is also semisimple The bound on p is best possible as the case G SL2 k shows In order to prove this Serre first considers the case in which G is the group of points of a simply connected quasi simple algebraic group and the representations Vi and Vz are algebraic irreducible and of restricted type In this case the proof relies on arguments on dominant weights due to J C Jantzen 1993 The general case is reduced to the previous one by using a saturation process due to V Nori which Serre had already used in his study of the adic representations associated with abelian varieties see Sect 13 4 above The study of these topics is continued in S252 171 1997 where one finds converse theorems such as if V V is semisimple and dim V is not divisible by char k then V is semisimple Here the proofs use only linear or multilinear algebra they are valid in any tensor category 14 1 2 In the Bourbaki report S273 2004 SEM 2008 Serre extends the notion of complete reducibility that is semisimplicity to subgroups F not only of GL but of any reductive group G
157. xact sequence 1 G9 gt Gy Gal k k 1 Given a motive E over k let M E be the smallest Tannakian subcategory of M which contains E Suppose that the standard conjectures and Hodge conjecture are true Under these assumptions and in an optimistic vein Serre formulates a series of conjectures and questions aimed at the description of Grothendieck s motivic paradise We stress the following ones 12 The motivic Galois group Gy is pro reductive 27 The motivic Galois group G yz is characterized by its tensor invariants Jean Pierre Serre An Overview of His Work 75 39 The group G m E Q 18 the closure in the Zariski topology of the image of the adic representation pg Gal k k gt G M E Q associated to E 49 The connected pro reductive group G9 decomposes as O C D where C is a pro torus equal to the identity component of the centre of en and D is a pro semisimple group equal to the derived group of G9 59 If S CG then S is the projective limit of the tori Tm defined in his McGill book 69 Every homomorphism G9 PGL has a lifting to G9 gt GL 7 Which connected reductive groups are realized as Gm g Are G2 and Eg possible 82 The group Ge Im pe E is open in G m E Qe Let pE pe E Gal k k gt Gee C Gu A Suppose that G m is connected Then F is a maximal motive if and only if Im pz is open in the group Gy Ey AS where A is

2003 Jean-Pierre Serre

Contents

Download Pdf Manuals

Related Search

Related Contents