In his celebrated work completed in 1995, Wiles, in part with Taylor, proved that every semistable elliptic curve over $\mathbb{Q}$ is modular, in the sense that its $L$-function is also that of a modular form. Their methods were subsequently extended by Breuil, Conrad, Taylor and myself to prove the modularity of all elliptic curves over $\mathbb{Q}$. The Modularity Theorem can be viewed as a special case of Langlands reciprocity conjectures, which continue to see exciting advances stemming from Wiles’ work in combination with further innovations. In the first half of the talk, I’ll give an overview of Wiles’ method and subsequent developments.

In addition to its most famous consequence, namely Fermat’s Last Theorem, modularity also underpins all major progress on the Birch–Swinnerton-Dyer Conjecture. Like the Modularity Theorem, the Birch–Swinnerton-Dyer Conjecture can also be viewed as an instance of a vast family of conjectures, in this case relating arithmetic invariants to special values of $L$-functions. In the second half of the talk, I’ll explain how the proof of the Modularity Theorem is itself related, by work of Hida, to another instance of these conjectures, namely for adjoint $L$-functions.