In search of a Langlands transform

Michael Harris, Institut de Mathématiques de Jussieu
Fine Hall 314

Class field theory expresses Galois groups of abelian extensions of a number field F in terms of harmonic analysis on the multiplicative group of locally compact topological ring, the adèle ring, attached to F. Among the most far-reaching predictions of the Langlands program is the existence of a vast generalization of class field theory, in the form of a correspondence between n-dimensional representations of the Galois group of F and automorphic representations, which arise in the harmonic analysis on a homogeneous space for GL(n) of the adèle ring of F. In some cases, the cohomology of Shimura varieties provides a procedure for transforming automorphic representations to Galois representations. I will outline the scope and limitations of this partial realization of the hypothetical Langlands correspondence. The project to use Shimura varieties to construct Galois representations is now nearly complete: I will also describe joint work with Lan, Taylor, and Thorne that constructs a new family of Galois representations that cannot be obtained by any known version of a Langlands transform.