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).听 Wiles proved Fermat鈥檚 Last Theorem by establishing the modularity of semistable elliptic curves over Q.听Subsequent work of Breuil-Conrad-Diamond-Taylor established the modularity of elliptic听 curves over Q in full generality. My work with J-P. Wintenberger gave a proof of 听the generalized Shimura-Taniyama-Weil conjecture which asserts听that all 鈥渙dd, rank 2 motives over Q鈥 are modular.听This is听 a corollary of our proof of Serre鈥檚 modularity conjecture.
Very little听is known when one looks at the same question over finite extensions of Q. I will talk about the recent beautiful 听work of听 Ana Caraiani and James Newton which proves modularity of all elliptic听 curves over Q(i). An input听into their proof is a result, proved in joint work with Patrick Allen and Jack Thorne, that proves the analog听 of Serre鈥檚 conjecture for mod 3 representations that arise from elliptic听 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听 representations over number fields. We know only fragments of what is听 conjectured, but what little we know is already quite remarkable!
