On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises

Abstract

IntroductionIn this paper, building on work of Wiles [Wi] and of Taylor and Wiles [TW], we will prove the following two theorems (see §2.2). Theorem A. If E /Q is an elliptic curve, then E is modular. Theorem B. If ρ : Gal(Q/Q) → GL 2 (F 5 ) is an irreducible continuous representation with cyclotomic determinant, then ρ is modular.We will first remind the reader of the content of these results and then briefly outline the method of proof.If N is a positive integer, then we let Γ 1 (N ) denote the subgroup of SL 2 (Z) consisting of matrices that modulo N are of the form 1 * 0 1 .The quotient of the upper half plane by Γ 1 (N ), acting by fractional linear transformations, is the complex manifold associated to an affine algebraic curve Y 1 (N ) /C . This curve has a natural model Y 1 (N ) /Q , which for N > 3 is a fine moduli scheme for elliptic curves with a point of exact order N . We will let X 1 (N ) denote the smooth projective curve which contains Y 1 (N ) as a dense Zariski open subset.Recall that a cusp form of weight k ≥ 1 and level N ≥ 1 is a holomorphic function f on the upper half complex plane H such that• for all matrices Key words and phrases. Elliptic curve, Galois representation, modularity. The first author was supported by the CNRS. The second author was partially supported by a grant from the NSF. The third author was partially supported by a grant from the NSF and an AMS Centennial Fellowship, and was working at Rutgers University during much of the research. The fourth author was partially supported by a grant from the NSF and by the Miller Institute for Basic Science. The space S k (N ) of cusp forms of weight k and level N is a finite-dimensional complex vector space. If f ∈ S k (N ), then it has an expansionand we define the L-series of f to befor anywith c ≡ 0 mod N and d ≡ p mod N . The operators T p for p | N can be simultaneously diagonalised on the space S k (N ) and a simultaneous eigenvector is called an eigenform. If f is an eigenform, then the corresponding eigenvalues, a p (f ), are algebraic integers and we haveLet λ be a place of the algebraic closure of Q in C above a rational prime and let Q λ denote the algebraic closure of Q thought of as a Q algebra via λ. If f ∈ S k (N ) is an eigenform, then there is a unique continuous irreducible representation [DS]. Moreover ρ is odd in the sense that det ρ of complex conjugation is −1. Also, ρ f,λ is potentially semi-stable at in the sense of Fontaine. We can choose a conjugate of ρ f,λ which is valued in GL 2 (O Q λ ), and reducing modulo the maximal ideal and semi-simplifying yields a continuous representationwhich, up to isomorphism, does not depend on the choice of conjugate of ρ f,λ . Now suppose that ρ : G Q → GL 2 (Q ) is a continuous representation which is unramified outside finitely many primes and for which the restriction of ρ to a decomposition group at is potentially semi-stable in the sense of Fontaine. To ρ| Gal(Q /Q ) we can associate both a pair of Hodge-Tate numbers and a Weil-Deligne representation of the Weil g...

Finite descent obstructions and rational points on curves

“…In [Kolyvagin 1988;Kolyvagin and Logachëv 1989], it is proved that A(‫)ޑ‬ and X(‫,ޑ‬ A) are both finite. Corollary 6.2 then implies that A(‫ށ‬ ‫ޑ‬ ) f-ab ޑ‬ Wiles [1995], Taylor and Wiles [1995], and Breuil, Conrad, Diamond and Taylor [Breuil et al 2001] have proved that E is modular, and so the first assertion applies. Now let X be a principal homogeneous space for the abelian variety A.…”

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“…Their result was later extended by Breuil, Conrad, Diamond and Taylor [Breuil et al 2001] to all elliptic curves over ‫.ޑ‬ This correspondence allows facts about elliptic curves to be proven using modular forms, and vice versa. (See [Koblitz 1993] for more background on the theory of elliptic curves and modular forms.…”

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“…In [Kolyvagin 1988;Kolyvagin and Logachëv 1989], it is proved that A(‫)ޑ‬ and X(‫,ޑ‬ A) are both finite. Corollary 6.2 then implies that A(‫ށ‬ ‫ޑ‬ ) f-ab ޑ‬ Wiles [1995], Taylor and Wiles [1995], and Breuil, Conrad, Diamond and Taylor [Breuil et al 2001] have proved that E is modular, and so the first assertion applies. Now let X be a principal homogeneous space for the abelian variety A.…”