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

@inproceedings{Wiles1995ModularEC,
title={Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?)},
author={Andrew Wiles},
year={1995}
}
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…
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## References

SHOWING 1-10 OF 108 REFERENCES
Arithmetic moduli of elliptic curves
• Mathematics
• 1985
This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova"
A result on modular forms in characteristic p
d The action of the derivation e = q ~ on the q-expansions of modular forms in characteristic p is one of the fundamental tools in the Serre/Swinnerton-Dyer theory of mod p modular forms. In this
Iwasawa theory for the symmetric square of an elliptic curve.
• Mathematics
• 1987
Up until the present time, most work in Iwasawa theory has dealt with either the cyclotomic theory or descent theory on abelian varieties. We began work on the material in this paper several years
Multiplicities of p-finite mod p Galois representations in Jo(Np)
Let M > 1 be an integer. Let Jo(M) be the Jacobian Pic~ of the modular curve Xo(M)q. Let TM be the .subring of End(Jo(M)) generated by the Hecke operators Tn with n > 1. Suppose that p is a maximal
Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer
1. Quick Review of Elliptic Curves 2 2. Elliptic Curves over C 4 3. Elliptic Curves over Local Fields 6 4. Elliptic Curves over Number Fields 12 5. Elliptic Curves with Complex Multiplication 15 6.
Ring-Theoretic Properties of Certain Hecke Algebras
• Mathematics
• 1995
The purpose of this article is to provide a key ingredient of [W2] by establishing that certain minimal Hecke algebras considered there are complete intersections. As is recorded in [W2], a method
Iwasawa Theory for $p$-adic Representations
Several years ago Mazur and Wiles proved a fundamental conjecture of Iwasawa which gives a precise link between the critical values of the Riemann zeta function (and, more generally, Dirichlet
General Selmer Groups and Critical Values of Hecke L-functions
Let K be an imaginary quadratic eld and let O be the ring of integers of K. Let E be an elliptic curve deened over Q with complex multiplication by O. Let be the Grr ossencharacter attached to the
BASE CHANGE FOR GL(2)
R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, ?) attached to a cuspdidal automorphic
On modular representations of $$(\bar Q/Q)$$ arising from 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