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

By C. Breuil, Brian Conrad, Fred Diamond, R. Taylor. Journal of The American Mathematical Society, 2001

987 citations recorded by the source corpus.

We complete the proof that every elliptic curve over the rational numbers is modular.

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9b312aef35930092e379fd4abbc49e276809e32e

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In Algebra and Number Theory

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Citations in the corpus

Low-lying zeros of families of elliptic curves (2004, background): “ing to the Shimura-Taniyama conjecture (proved by Wiles et al [W], [TW], [ BCDT ]) there exists a weight two primitive cuspidal newform f(z) on 0(N) such that L(s,E) = L(s,f).”
Rankin--Eisenstein classes and explicit reciprocity laws (2015, background): “One knows that both E and ρ∗ are modular, associated to modular forms f of weight 2 ([10]) and g of weight 1 ([21]).”
Iwasawa Main Conjecture for Supersingular Elliptic Curves (2014, methodology): “By Taniyama-Shimura conjecture proved by Wiles [26] and Breuil-Conrad-Diamond-Taylor [1] we”
Fibonacci numbers at most one away from a perfect power (2008, background): “Since then the proof of the Modularity Theorem has been completed in a series of papers the last of which is [5].”
SIEGEL MODULAR FORMS OF GENUS 2 ATTACHED TO ELLIPTIC CURVES (2006, neutral): “N T ] 1 6 Se p 20 06”
Effective equidistribution of eigenvalues of Hecke operators (2009, positive): “Indeed, thanks to the famous Shimura–Taniyama– Weil conjecture, now proved by the work of Wiles, Taylor–Wiles and Breuil–Conrad–Diamond–Taylor (see [2]), in the special case k = 2 and d = 1, estimating s(N,k)d is equivalent to counting the number of isogeny classes of elliptic curves defined over Q ”
On the weights of mod p Hilbert modular forms (2006, methodology): “3.4) to adopt the conventions of [4 ]a nd [20], rather than those of [ 2 ]; thus rather than working with the usual contravariant equivalence of categories, we work with a covariant version of it, so that our formulae for generic fibres will differ by duality and a twist from those following the con”
Ranks of elliptic curves (2002, background): “The ShimuraTaniyama Conjecture, recently proved by Breuil, Conrad, Diamond, and Taylor [5] by extending work of Wiles [76], implies the following long-standing conjecture of Hasse and Weil.”
Unearthing the visions of a master: harmonic Maass forms and number theory (2008, result): “By the work of Breuil, Conrad, Diamond, Taylor, and Wiles [46, 81, 198, 209], we have the following deep result which confirmed the Shimura-Taniyama Conjecture.”
THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES OF MODULAR FORMS (2004, methodology): “We should stress the importance of making this set as small and explicit as possible; indeed the refinements in [22,13,5] which completed the proof of the Shimura–Taniyama–Weil conjecture can be viewed as work in this direction for weight two modular forms.”
The Modular Approach to Diophantine Equations (2007, background): “The proof was completed in a series of papers by Diamond [15], Conrad, Diamond and Taylor [11], and finally Breuil, Conrad, Diamond and Taylor [6].”
Arboreal Galois representations (2007, background): “Moreover, the Fontaine-Mazur conjecture (Fontaine and Mazur, 1995), which suggests that the representations from algebraic geometry should be exactly those that are finitely ramified and potentially semistable at p, has been successively chipped away, most notably by Wiles (Wiles, 1995) (completed b”
Explicit approaches to modular abelian varieties (2000, background): “1 The BSD conjecture By [10] we now know that every elliptic curve over Q is a quotient of the curve X0(N), whose complex points are the isomorphism classes of pairs consisting of a (generalized) elliptic curve and a cyclic subgroup of order N .”
On the p-adic L-function of a modular form at a supersingular prime (2003, positive): “SinceE is modular (see [ 29], [4]), we define the p-adic L-function of E to be thep-adic L-function of the corresponding modular form.”