Generalizations of Shimura varieties

This project aims to make progress on two central problem in the Langlands program.

The first is the Artin conjecture, a special case of which predicts that the L-function attached to any two-dimensional irreducible complex representation of Galois groups Gal(L/Q) has analytic continuation to the whole complex plane.

In the Langlands program, this conjecture is refined to predict that the L-function is equal to the L-function of an automorphic form.

The conjecture can be split into the so called even and odd cases, corresponding algebraic Maass form (the even case) and weight one modular forms (the odd case). The odd case is solved, but the even case remains one of the most famous open problems in the field.

We aim to approach the even case through a conjecture of Peter Scholze, rooted in p-adic geometry, by applying ideas from the p-adic local Langlands correspondence for GL(2,Qp).

Our second goal is the study of geometric structures arising in the recent reformulation of the local Langlands correspondence by Scholze and Laurent Fargues.

We will study Ran spaces and affine Grassmannians, which are highly infinite-dimensional objects in p-adic geometry whose sheaf theory should cast a new light on representation theory of p-adic groups.