When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.

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Modular elliptic curves and Fermat’s Last Theorem

Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?)

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

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Modular curves and the Eisenstein ideal

Modular Forms, Elliptic Curves, and Modular Curves.- Modular Curves as Riemann Surfaces.- Dimension Formulas.- Eisenstein Series.- Hecke Operators.- Jacobians and Abelian Varieties.- Modular Curves as Algebraic Curves.- The Eichler-Shimura Relation and L-functions.- Galois Representations.

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A First Course in Modular Forms

where G is the Galois group GaI ( I ) /Q) and F is a finite field of characteristic I > 3. Suppose that p is modular of level N, i.e., that it arises from a weight-2 newform of level dividing N and trivial "Nebentypus character." Then p is an odd representation: the matrix p(c) (where c is a complex conjugation in G) has eigenvalues + 1, 1. Since + 1 and 1 are distinct in F, p is absolutely irreducible and has a model over every subfield of F containing the set trace(p ). We assume that F has been chosen so that it is generated by this set. Assume that p is a prime which exactly divides N (we write p I[ N), and restrict p to a decomposition group of G for the prime p. View the restriction as a representation p p of Gal(t)p/Qp). We say that p is finite at p ([34] or [35], w if there is a finite flat F-vector space scheme H over Zp for which the action of Gal( t )p/Qp) on the F-vector space H(1)p) gives pp. ( I f / ~ p,/9 is finite at p if and only if pp is unramified.) It is clear that p is finite at p whenever p is already modular of level N/p. In w of [34] (cf. [35]), Serre conjectured the converse, i.e., that if p is finite, then p is modular of level N/p. Soon thereafter, Mazur 1-20] proved this conjecture in the case

Kenneth A. Ribet

Papers

On modular representations of\((\bar Q/Q)\) arising from modular forms

Modular elliptic curves and fermat's last theorem

Values of Abelian L-functions at Negative Integers over Totally Real Fields.

Galois representations attached to eigenforms with nebentypus

Galois Action on Division Points of Abelian Varieties with Real Multiplications