nLab modularity theorem

Statement

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.

Generalizations

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.

Related concepts

• elliptic curve, modular curve, modular form

• Fermat's last theorem

• Ribet's theorem

• Wiles' proof of Fermat's last theorem

References

• 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

• Nuno Freitas, Bao V. Le Hung, Samir Siksek Elliptic Curves Over Real Quadratic Fields Are Modular, Invent. Math. 201 (2015), no. 1, 159–206. (arXiv:1310.7088)

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:

• Wikipedia, Modularity theorem