A Reading List for the Boston University Conference on FERMAT’S LAST THEOREM August 9-18, 1995

The main goal of our conference is to understand the breakthrough papers of Wiles and of Taylor and Wiles in which the Taniyama-Shimura-Weil conjecture is proved for semistable elliptic curves.

[W] Wiles, A.: Modular elliptic curves and Fermat’s Last Theorem. Annals of Math. 141 (1995), 443-551.

[TW] Taylor, R. L., Wiles, A.: Ring theoretic properties of certain Hecke algebras. Annals of Math. 141 (1995), 553-572.

In the following pages, we list standard references for some of the background required for understanding these papers. Most of this material will be discussed at the conference, but it may be useful to review some of it in advance.

SURVEY ARTICLES.

Shortly after Wiles first announced a proof of Fermat’s Last Theorem at his Cambridge University lectures in June of 1993, Karl Rubin and Alice Silverberg wrote a very useful survey article [RS] outlining the main ideas. Though the announcement of a proof of Fermat’s Last Theorem was premature, Wiles’ original methods did suffice for proving that there are infinitely many j-invariants of modular elliptic curves. This was a remarkable result in its own right. The proof was outlined in Henri Darmon’s article [D] which, unfortunately, has not yet appeared in published form. Since the ideas described in [RS] and [D] are very close to those in Wiles’ final proof [W] of the Semistable Taniyama-Shimura-Weil Conjecture (from which Fermat’s Last Theorem follows), these articles are very much worth reading.

[RS] Rubin, K., Silverberg, A.: A report on Wiles’ Cambridge lectures. Bulletin of the AMS 31 (1994), 15-38.

[D] Darmon, H.: The Shimura-Taniyama conjecture (d’après Wiles). Russian Math Surveys. To appear.

At least two excellent survey articles have been written since Wiles announced his new proof in October of 1994. Unfortunately, neither has yet appeared in final published form.

[DDT] Darmon, H., Diamond, F., Taylor, R. L.: Fermat’s Last Theorem. In Current Developments in Mathematics, 1995, International Press. To appear. (A preliminary soft cover version of this volume is available at this time from International Press for $12 by writing to Julie Lynch in the mathematics department at Harvard University or by sending email to julie@math.harvard.edu. The final hard cover version of these Proceedings should appear by January of 1995 and be available for approximately $30 from International Press.)

[Ri1] Ribet, K.: Galois representations and modular forms. Bull. AMS 32 (1995), 375-402. (To appear in October, 1995).

TANIYAMA-SHIMURA-WEIL ⟹ FERMAT. For a pleasant exposition of what the Taniyama-Shimura-Weil Conjecture asserts and how it implies Fermat’s Last Theorem, any one (or all) of the following papers is recommended. The first three also contain brief introductions to Wiles’ work.

[C] Cox, D.: Introduction to Fermat’s Last Theorem. American Math. Monthly 101 (1994), 3-14.

[G] Gouvea, F .Q.: “A Marvelous Proof”. American Math. Monthly 101 (1994),

203-222.

[HR] Hayes, B., Ribet, K.: Fermat’s Last Theorem and modern arithmetic. American Scientist 82 (2) (1994), 144-156.

[Mz1] Mazur, B.: Number theory as gadfly. American Math. Monthly 98 (1991), 593-610.

[Oe] Oesterlé, J.: Nouvelles approches du “theorème” de Fermat. Asterisque 161-162 (1988), 165-186.

[Ri2] Ribet, K.: From the Taniyama-Shimura conjecture to Fermat’s Last Theorem, Annales de la Faculté des Sciences de l’Université de Toulouse 11 (1990), 116-139. (A preprint version of this paper is available by anonymous ftp from math.berkeley.edu in the directory pub/Preprints/Ken Ribet, where it is the file toulouse.tex.)

BACKGROUND READING

Elliptic curves. The 1974 survey article by Tate [T] is still the best place to find a quick introduction to the arithmetic theory of elliptic curves. For some of the details behind Tate’s article, see Silverman’s book [Si]. Knapp’s book [Kn] also describes the Taniyama-Shimura-Weil Conjecture as well as the Eichler-Shimura theory for weight 2 modular forms.

[Kn] Knapp, A. W.: Elliptic curves. Mathematical notes 40. Princeton University Press, Princeton, N.J., 1992.

[Si] Silverman, J. H.: Arithmetic of elliptic curves. Graduate texts in mathematics 106. Springer-Verlag, New York, 1985.

[T] Tate, J.: The arithmetic of elliptic curves. Invent. Math. 23 (1974), 179-206.

Modular forms. The Atkin-Lehner theory of newforms is described in chapter VIII of Lang’s book [La1]. A detailed account of the abelian varieties and galois representations attached to weight two newforms, is contained in chapter 6 of Shimura’s book [Sh]. For a simplified description of the Eichler-Shimura theory, see [Kn] above.

[La1] Lang, S.: Introduction to modular forms. Springer-Verlag, New York, 1976.

[Sh] Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Princeton University Press, Princeton, 1971.

Galois representations. Chapter 1 of Serre’s book [Se1] is a good introduction to Galois representations—localization, frobenii, ramification, and the Cebotarev Density Theorem.

[Se1] Serre, J.-P.: Abelian ℓ-adic Representations and Elliptic Curves. W. A. Benjamin, Inc., New York, Amsterdam, 1968.

Finite flat group schemes. Oort’s book [O] is a good place to start. Shatz’s article [Sha] also icludes an exposition of Raynaud’s fundamental paper [R].

[O] Oort, F.: Commutative Group Schemes. Lecture Notes in Mathematics 15, Springer-Verlag, Berlin, New York, 1966.

[R] Raynaud, M.: Schémas en groupes de type (p, …, p). Bull. Soc. Math. France 102 (1974), 241-280.

[Sha] Shatz, S.: Group schemes, formal groups, and p-divisible groups. In Arithmetic geometry, G. Cornell, J. H. Silverman, editors. Springer-Verlag, New York, 1986.

Galois cohomology. Serre’s classic text on local fields [Se2] includes a readable and self-contained account (chapters VII, VIII) of the definition and basic properties of group cohomology. For a more utilitarian introduction, see Appendix B of Silverman’s book [Si] (above). The basic properties of Galois cohomology, including the duality theory of Tate and Poitou may be found in chapter 1 of Milne’s book [Mi].

[Mi] Milne, J.: Arithmetic Duality Theorems. Perspectives in mathematics. Academic Press, San Diego, 1986.

[Se2] Serre, J.-P.: Local fields. Graduate texts in mathematics 67. Springer-Verlag, New York, 1979. (Chapters VII and VIII).

Deforming Galois representations. Mazur’s theory of deformations of Galois representations is an essential prerequisite for understanding Wiles’ work. We are fortunate to have Mazur’s beautifully written article [Mz2]. Ramakrishna’s paper [Ram] is also quite well written and contains some important improvements required by Wiles.

[Mz2] Mazur, B.: Deforming Galois representations. In Galois groups over Q: proceedings of a workshop held March 23-27, 1987, Y. Ihara, K. Ribet, J.-P. Serre, editors. Mathematical Sciences Research Institute publications 16. Springer-Verlag, New York,1989, pp. 385-437.

[Ram] Ramakrishna, R.: On a variation of Mazur’s deformation functor. Compositio Math. 87 (1993), 269-286.

Commutative algebra. Wiles uses some standard results from commutative algebra that may not be entirely familiar to most number theorists, e.g. the theories of Gorenstein rings and local complete intersection rings (see [M] and the appendix of [MzR]). Wiles also makes use of the theory of Fitting ideals, the basics of which are compactly described in Chapter 10, §13 of Lang’s Algebra book [La2].

[La2] Lang, S.: Algebra. Second Edition. Addison-Wesley, Menlo Park, California, 1984.

[M] Matsumura, H.: Commutative ring theory, Cambridge University Press, Cambridge (U.K.), 1986.

[MzR] Mazur, B., Roberts, L.: Local Euler characteristics. Invent. Math. 9 (1970), 201-234.

Generalized Selmer groups. For an account of classical Selmer groups (for elliptic curves) see Chapter X of Silverman’s book [Si]. Generalized Selmer groups were first defined by Ralph Greenberg in [Gr] by imposing “local conditions”. Wiles uses them to describe the tangent spaces associated to his deformation theories—a key step in his proof.

[Gr] Greenberg, R.: Iwasawa theory for for p-adic representations. In Algebraic number theory, in honor of K. Iwasawa. Adv. Studies in Pure Math. 17. Academic Press, San Diego, 1989, pp. 97-137.

Automorphic forms and the Langlands program. For an introduction to automorphic representation theory for GL(2) and the dictionary between the languages of classical modular forms and automorphic representations, Gelbart’s book [Ge1] is the place to look. His survey article [Ge2] describes the Taniyama-Shimura-Weil Conjecture in the language of automorphic representation theory and interprets it in the broader context of the Langlands program. For general discussions of the Langlands program, see [Ge3],

[GeA] and [L].

[Ge1] Gelbart, S.: Automorphic forms on adele groups. Advances in Mathematics 21, No. 3 (1976), 235-292.

[Ge2] Gelbart, S.: Elliptic curves and automorphic representations. Advances in Mathematics 21, No. 3 (1976), 235-292.

[Ge3] Gelbart, S.: An Elementary Introduction to the Langlands Program. Bull. of the AMS. 10, No. 2 (1984), 177-219.

[GeA] Gelbart, S., Arthur, J.: Lectures on Automorphic L-functions. In L-functions and Arithmetic, Proceedings of the Durham Symposium, July, 1989, J. Coates and M. Taylor (eds), Cambridge University Press, 1991, pp. 1-59.

[L] Langlands, R. P.: Representation Theory: its rise and role in Number Theory. In Proceedings of the Gibbs Symposium, Yale University, May 15-17, 1989, American Mathematical Society, 1990, pp. 181-210.

Connections between Fermat’s Last Theorem and elliptic curves. The contruction of special elliptic curves from solutions of the Fermat equation was first described by Hellegouarch [H]. Gerhart Frey was the first to suggest that these elliptic curves might violate the Taniyama-Shimura-Weil Conjecture (see [F]).

[F] Frey, G.: Links between solutions of A − B = C and elliptic curves. In Number Theory, proceedings of the Journees arithmetiques, held in Ulm, 1987, H.P. Schlickewei, E. Wirsing, editors. Lecture notes in mathematics 1380. Springer-Verlag, Berlin, New York, 1989.

[H] Hellegouarch, Y.: Points d’ordre 2p^{h} sur les courbes elliptiques. Acta. Arith. 26

(1974/75), 253-263. Serre’s Conjectures. Serre’s original paper [Se3] is a masterpiece. For a progress report, see Ribet’s paper [Ri3].

[Se3] Serre, J.-P.: Sur les représentations modulaires de degré 2 de Gal(\overline{Q}/Q), Duke Math. J. 54 (1987), 179-230.

[Ri3] Ribet, K.: Report on mod ℓ Representations of Gal(\overline{Q}/Q). Proc. Symp. in Pure Math. 55 (2) (1994), 639-676.