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A report on Wiles' Cambridge lectures
Authors:
K. Rubin and A. Silverberg
Journal:
Bull. Amer. Math. Soc. 31 (1994), 15-38
MSC:
Primary 11F80; Secondary 11D41, 11G05
MathSciNet review:
1256978
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Abstract: In lectures at the Newton Institute in June of 1993, Andrew Wiles announced a proof of a large part of the Taniyama-Shimura Conjecture and, as a consequence, Fermat's Last Theorem. This report for nonexperts discusses the mathematics involved in Wiles' lectures, including the necessary background and the mathematical history.
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- B. Birch and W. Kuyk, eds., Modular functions of one variable. IV, Lecture Notes in Math., vol. 476, Springer-Verlag, New York, 1975, pp. 74-144. MR 0376533 (51:12708)
- [2]
- J. Buhler, R. Crandall, R. Ernvall, and T. Metsänkylä, Irregular primes and cyclotomic invariants to four million, Math. Comp. 61 (1993), 151-153. MR 1197511 (93k:11014)
- [3]
- J. W. S. Cassels and A. Frohlich, Algebraic number theory, Academic Press, London, 1967. MR 0215665 (35:6500)
- [4]
- P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507-530. MR 0379379 (52:284)
- [5]
- L. E. Dickson, History of the theory of numbers (Vol. II), Chelsea Publ. Co., New York, 1971.
- [6]
- H. M. Edwards, Fermat's Last Theorem. A genetic introduction to algebraic number theory, Springer-Verlag, New York, 1977. MR 616635 (83b:12001a)
- [7]
- M. Eichler, Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion, Arch. Math. (Basel) 5 (1954), 355-366. MR 0063406 (16:116d)
- [8]
- G. Faltings, p-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), 255-299. MR 924705 (89g:14008)
- [9]
- -, Crystalline cohomology and p-adic Galois representations, Algebraic Analysis, Geometry and Number Theory, Proceedings of the JAMI Inaugural Conference (J. I. Igusa, ed.), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25-80. MR 1463696 (98k:14025)
- [10]
- M. Flach, A finiteness theorem for the symmetric square of an elliptic curve, Invent. Math. 109 (1992), 307-327. MR 1172693 (93g:11066)
- [11]
- G. Frey, Links between solutions of
and elliptic curves, Number Theory, Ulm 1987, Proceedings, Lecture Notes in Math., vol. 1380, Springer-Verlag, New York, 1989, pp. 31-62. MR 1009792 (90g:11069)
- [12]
- S. Gelbart, Automorphic forms on adele groups, Ann. of Math. Stud., vol. 83, Princeton Univ. Press, Princeton, NJ, 1975. MR 0379375 (52:280)
- [13]
- B. Gross, Kolyvagin's work on modular elliptic curves, L-functions and Arithmetic, London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 235-256. MR 1110395 (93c:11039)
- [14]
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Fourth ed., Oxford Univ. Press, London, 1971.
- [15]
- Y. Hellegouarch, Étude des points d'ordre fini des variétés de dimension un définies sur un anneau principal, J. Reine Angew. Math. 244 (1970), 20-36. MR 0274451 (43:216)
- [16]
- -, Points d'ordre fini des variétés abéliennes de dimension un, Colloque de Théorie des Nombres (Univ. Bordeaux, Bordeaux, 1969), Bull. Soc. Math. France, Mém. 25, Soc. Math. France, Paris, 1971, pp. 107-112.
- [17]
- -, Points d'ordre fini sur les courbes elliptiques, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A540-A543.
- [18]
- -, Points d'ordre
sur les courbes elliptiques, Acta. Arith. 26 (1974/75), 253-263. MR 0379507 (52:412)
- [19]
- V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift (Vol. II) (P. Cartier et al., eds.), Birkhäuser, Boston, 1990, pp. 435-483. MR 1106906 (92g:11109)
- [20]
- R. Langlands, Base change for
, Ann. of Math. Stud., vol. 96, Princeton Univ. Press, Princeton, NJ, 1980. MR 574808 (82a:10032)
- [21]
- B. Mazur, Deforming Galois representations, Galois groups over Q (Y. Ihara, K. Ribet, and J.-P. Serre, eds.), Math. Sci. Res. Inst. Publ., vol. 16, Springer-Verlag, New York, 1989, pp. 385-437. MR 1012172 (90k:11057)
- [22]
- -, Number theory as gadfly, Amer. Math. Monthly 98 (1991), 593-610. MR 1121312 (92f:11077)
- [23]
- B. Mazur and J. Tilouine, Représentations galoisiennes, différentielles de Kähler et "conjectures principales", Inst. Hautes Études Sci. Publ. Math. 71 (1990), 65-103. MR 1079644 (92e:11060)
- [24]
- J. Oesterlé, Nouvelles approches du "théorème" de Fermat, Séminaire Bourbaki no. 694 (1987-1988), Astérisque 161/162 (1988) 165-186. MR 992208 (90g:11038)
- [25]
- -, On a variation of Mazur's deformation functor, Compositio Math. 87 (1993), 269-286. MR 1227448 (94h:11054)
- [26]
- P. Ribenboim, 13 lectures on Fermat's Last Theorem, Springer-Verlag, New York, 1979. MR 551363 (81f:10023)
- [27]
- K. Ribet, On modular representations of
arising from modular forms, Invent. Math. 100 (1990), 431-476. MR 1047143 (91g:11066)
- [28]
- -, Report on mod
representations of , Motives (U. Jannsen, S. Kleiman, and J-P. Serre, eds.), Proc. Sympos. Pure Math., vol. 55 (Part 2), Amer. Math. Soc., Providence, RI, 1994 (to appear).
- [29]
- K. Rubin, The main conjecture. (Appendix to Cyclotomic fields I and II, S. Lang), Graduate Texts in Math., vol. 121, Springer-Verlag, New York, 1990, pp. 397-419. MR 1029028 (91c:11001)
- [30]
- -, Kolyvagin's system of Gauss sums, Arithmetic Algebraic Geometry (G. van der Geer, F. Oort, and J. Steenbrink, eds.), Progr. Math., vol. 89, Birkhäuser, Boston, 1991, pp. 309-324.
- [31]
- -, The "main conjectures" of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), 25-68. MR 1079839 (92f:11151)
- [32]
- J.-P. Serre, Sur les représentations modulaires de degré 2 de
, Duke Math. J. 54 (1987), 179-230. MR 885783 (88g:11022)
- [33]
- G. Shimura, Correspondances modulaires et les fonctions
de courbes algébriques, J. Math. Soc. Japan 10 (1958), 1-28. MR 0095173 (20:1679)
- [34]
- -, Construction of class fields and zeta functions of algebraic curves, Ann. of Math. 85 (1967), 58-159. MR 0204426 (34:4268)
- [35]
- -, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, Princeton, NJ, 1971. MR 0314766 (47:3318)
- [36]
- -, On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields, Nagoya Math. J. 43 (1971), 199-208. MR 0296050 (45:5111)
- [37]
- -, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), 523-544. MR 0318162 (47:6709)
- [38]
- -, Yutaka Taniyama and his time. Very personal recollections, Bull. London Math. Soc. 21 (1989), 186-196. MR 976064 (90b:01090)
- [39]
- J. Silverman, The arithmetic of elliptic curves, Graduate Texts in Math., vol. 106, Springer-Verlag, New York, 1986. MR 817210 (87g:11070)
- [40]
- F. Thaine, On the ideal class groups of real abelian number fields, Ann. of Math. (2) 128 (1988), 1-18. MR 951505 (89m:11099)
- [41]
- J. Tunnell, Artin's conjecture for representations of octahedral type, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 173-175. MR 621884 (82j:12015)
- [42]
- A. Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149-156. MR 0207658 (34:7473)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0273-0979-1994-00512-9
PII:
S 0273-0979(1994)00512-9
Article copyright:
© Copyright 1994 American Mathematical Society
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