Seminar series
Date
Mon, 28 Oct 2024
16:00
16:00
Location
C3
Speaker
Dmitri Whitmore
Organisation
University of Cambridge
The (global) Langlands programme is a vast generalization of classical reciprocity laws. Roughly, it predicts a correspondence between:
1) modular forms (and their generalizations, automorphic forms)
2) representations of the Galois group of a number field.
While many constructions of Galois representations from automorphic forms exist, the converse direction is often harder to establish. The main tools to do so are modularity lifting theorems and are proved via the Taylor-Wiles method, originating from Wiles' proof of Fermat's Last Theorem.
I will introduce these ideas and their applications, focusing particularly on the problem of modularity of elliptic curves. I will then briefly discuss a generalization of the Taylor-Wiles method developed in my thesis which led to new modularity theorems in the setting of quadratic extensions of totally real fields by building of work of Boxer-Calegari-Gee-Pilloni.