transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
The modularity theorem states that given an elliptic curve $E$ over the rational numbers with a conductor $N_E$, there is a cusp form $f \in S_2(\Gamma_0(N_E))$ such that $a_1(f) = 1$ and $a_p(f) = a_p(E)$ for all primes $p$.
The “semistable” case of the modularity theorem was proven in Wiles 1995, Taylor & Wiles 1995. The full theorem was proved by Breuil, Conrad, Diamond & Taylor 2001.
The modularity theorem was extended to the case of elliptic curves over real quadratic fields by Freitas-Le Hung-Siksek in FLS15, and work in progress by Caraiani-Newton extends this to the case of elliptic curves over imaginary quadratic fields.
More generally, the term “modularity” is often applied to mean the question of whether a Galois representation comes from a modular form or one of its higher-dimensional generalizations such as Hilbert modular forms or Siegel modular forms.
Andrew Wiles, Modular Elliptic Curves and Fermat’s Last Theorem, Annals of Mathematics Second Series, 141 3 (1995) 443-551 $[$doi:10.2307/2118559$]$
Richard Taylor, Andrew Wiles, Ring-Theoretic Properties of Certain Hecke Algebras, Annals of Mathematics Second Series 141 3 (1995) 553-572 $[$doi:10.2307/2118560$]$
Christophe Breuil, Brian Conrad, Fred Diamond, Richard Taylor, On the modularity of elliptic curves over $\mathbf{Q}$: Wild 3-adic exercises, Journal of the American Mathematical Society 14 4 (2001) 843–939 $[$doi:10.1090/S0894-0347-01-00370-8$]$
A generalization to elliptic curves over real quadratic fields is in
Review:
Gerd Faltings, The Proof of Fermat’s Last Theorem by R. Taylor and A. Wiles, Notices of the AMS (1995) $[$pdf$]$
Fred Diamond, Jerry Shurman, A First Course in Modular Forms, GTM 228, Springer (2005) $[$doi:10.1007/978-0-387-27226-9, ISBN-13: 978-0387232294$]$
R. van Dobben de Bruyn, The Modularity Theorem, (pdf)
See also:
Last revised on November 30, 2022 at 20:18:05. See the history of this page for a list of all contributions to it.