On the excursion algebra

TL;DR

The paper develops a structural understanding of the excursion algebra ${\ Exc}(X,{\mathsf G})$ for a scheme $X$ over a finite field and a reductive group ${\mathsf G}$. It shows that ${\ Exc}(X,{\mathsf G})$ is classical, decomposes into reduced factors, and is finitely generated over local Hecke algebras, with a canonical ell-independent rational structure; for ${\mathsf G}=GL_n$ the global Hecke algebra surjects onto ${\ Exc}(X,{\mathsf G})$. A central technical tool is a contraction mechanism, implemented via an ${\mathbb A}^1$-action, that contracts Weil/sheaf data and the stack of local systems to a geometrically semisimple locus, enabling a reduction to a simpler, semi-simple picture. The work builds a robust functorial framework linking ${\rm QCoh}({\mathbb A}^1)\text{-comod}$ with filtered ${\mathbb Z}$-modules, and extends these ideas to categories of Weil and Weil-restrictive local systems, establishing t-exact contractions and semi-simplicity results that underpin the key properties of the excursion algebra. The results pave the way for applications to function-field Ramanujan–Petersson-type phenomena and the broader Langlands program over function fields, including independence-of-ell aspects.

Abstract

The excursion algebra associated to a scheme X over a finite field and a reductive group G is the algebra of global functions on the stack of arithmetic G-local systems on X. When X is a curve, this algebra acts on the space of automorphic functions. In this paper we establish some basic properties of this algebra.

Theorems & Definitions (106)

• Theorem 1.1.7
• Remark 1.1.8
• Remark 1.2.5
• Proposition 1.2.7
• Corollary 1.2.9
• Proposition 1.4.3
• proof
• Remark 2.1.5
• Proposition 2.1.9
• proof