Algebra and Number Theory Seminar: Base change for modular forms

I'll talk about the base change lifting from holomorphic modular forms to Hilbert modular forms for totally real fields F. A new proof of the existence of this base change lifting is contained in joint work with Laurent Clozel and Jack Thorne.

The base change lifting is a simple example of Langlands functoriality, corresponding on the Galois side to restriction to the absolute Galois group of F. When F is a solvable extension of Q, its existence was proved by Langlands using the twisted trace formula (earlier work by Doi and Naganuma covered the case where F is quadratic). Dieulefait used modularity lifting theorems and a delicate construction of chains of congruences between modular forms to prove the existence of the base change lifting without a solvability assumption. Our new proof replaces (at least some of) this chain of congruences with a `p-adic analytic continuation of functoriality' step, adapted from my work with Thorne on symmetric power functoriality.