Cuspidal cohomology of stacks of shtukas

TL;DR

This paper develops a cohomological framework for stacks of shtukas over function fields, generalizing automorphic forms, and proves the finiteness and Hecke-finiteness of cuspidal cohomology.

Contribution

It introduces a constant term morphism and defines cuspidal cohomology for stacks of shtukas, extending classical automorphic form concepts to a geometric setting.

Findings

Cuspidal cohomology is finite-dimensional.

Cuspidal cohomology equals Hecke-finite cohomology.

Constructs a constant term morphism for shtukas.

Abstract

Let $G$ be a connected split reductive group over a finite field ${\mathbb F}_q$ and $X$ a smooth projective geometrically connected curve over ${\mathbb F}_q$. The $\ell$-adic cohomology of stacks of $G$-shtukas is a generalization of the space of automorphic forms with compact support over the function field of $X$. In this paper, we construct a constant term morphism on the cohomology of stacks of shtukas which is a generalization of the constant term morphism for automorphic forms. We also define the cuspidal cohomology which generalizes the space of cuspidal automorphic forms. Then we show that the cuspidal cohomology has finite dimension and that it is equal to the (rationally) Hecke-finite cohomology defined by V. Lafforgue.