The Shimura-Taniyama-Weil modularity conjecture asserts that all elliptic curves over Q arise as images of quotients of the Poincare upper half plane by congruence subgroups of the modular group SL2(Z). Wiles proved Fermat’s Last Theorem by establishing the modularity of semistable elliptic curves over Q. Subsequent work of Breuil-Conrad-Diamond-Taylor established the modularity of elliptic curves over Q in full generality. My work with J-P. Wintenberger gave a proof of the generalized Shimura-Taniyama-Weil conjecture which asserts that all “odd, rank 2 motives over Q” are modular. This is a corollary of our proof of Serre’s modularity conjecture.
Very little is known when one looks at the same question over finite extensions of Q. I will talk about the recent beautiful work of Ana Caraiani and James Newton which proves modularity of all elliptic curves over Q(i). An input into their proof is a result, proved in joint work with Patrick Allen and Jack Thorne, that proves the analog of Serre’s conjecture for mod 3 representations that arise from elliptic curves over Q(i).
My talk will give a general introduction to this circle of ideas centred around the modularity conjecture for motives and Galois representations over number fields. We know only fragments of what is conjectured, but what little we know is already quite remarkable!
