nLab modularity theorem




The modularity theorem states that given an elliptic curve EE over the rational numbers with a conductor N EN_E, there is a cusp form fS 2(Γ 0(N E))f \in S_2(\Gamma_0(N_E)) such that a 1(f)=1a_1(f) = 1 and a p(f)=a p(E)a_p(f) = a_p(E) for all primes pp.

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.


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)


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.