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