Motives and the Langlands program

Introduction

This proposal belongs to the field of arithmetic geometry and lies at the interface of number theory, algebraic geometry, and the theory of automorphic forms. More precisely, I aim to use algebraic cycles in the form of motivic sheaves to address fundamental questions about the Langlands program for function fields. This aims at motivic Langlands parametrizations.

Langlands Program Overview

The Langlands program predicts a decomposition of the space of automorphic forms by motivic Langlands parameters. A breakthrough of V. Lafforgue for function fields, with generalizations by Xue, Zhu, Drinfeld, and Gaitsgory, led to tremendous progress in the construction of l-adic Langlands parameters.

Recent Developments

The recent work of Fargues–Scholze for non-Archimedean local fields uses similar techniques in another context. This should be viewed as the l-adic realization of the conjectured motivic parametrization.

Advances in Motivic Sheaves

Independently, recent advances for motivic sheaves, as envisioned by Grothendieck and realized by Voevodsky, Ayoub, and Cisinski–Déglise, provide powerful techniques to handle algebraic cycles. Despite the immense progress in both areas, hardly any advances have been made specifying the motivic nature of Langlands' original prediction.

Project Goals

My project aims to develop new tools to make motivic sheaves amenable for applications in the Langlands program for function fields.

Key Constructions

In my ongoing joint work with Scholbach, we have successfully implemented key constructions such as:

1. IC-Chow groups of shtuka spaces
2. The motivic Satake equivalence (Math. Ann.)

This was achieved by bypassing the use of standard conjectures on algebraic cycles.

Geometry of Infinite Dimensional Varieties

As in my joint work with Haines, this in particular requires studying the geometry of infinite dimensional varieties such as the affine Grassmannian.

Future Directions

Based on these results, I will attack the longstanding open problem on the relation of automorphic forms and motives in the function field case.