Current Developments in Mathematics

Volume 2018

Motives and $L$-functions

Pages: 57 – 123



Frank Calegari (University of Chicago, Illinois, U.S.A.)


We survey some conjectures and recent developments in the Langlands program, especially on the conjectures linking motives and Galois representations with automorphic forms and $L$-functions. We then give an idiosyncratic discussion of various recent modifications of the Taylor–Wiles method due in part to the author in collaboration with David Geraghty, and then give some applications. The final goal is to give some hints about ideas in recent joint work with George Boxer, Toby Gee, and Vincent Pilloni. This is an expanded version of the author’s talk in the 2018 Current Developments in Mathematics conference. It will be presented in three parts: exposition, development, and capitulation.

The author was supported in part by NSF Grant DMS-1701703.

Published 17 December 2019