Arithmetic quotients of the Bruhat-Tits building for projective general linear group in positive characteristic

TL;DR

This paper investigates a special subspace of automorphic forms for GL_d over function fields, describing its constituents and constructing integral modular symbols using Bruhat-Tits building geometry, with finiteness results.

Contribution

It characterizes the constituents of the subspace with Steinberg local component and constructs integral modular symbols in this setting, providing finiteness and bounds.

Findings

Describes automorphic constituents with multiplicity one.

Constructs integral modular symbols using Bruhat-Tits building.

Establishes finiteness and bounds on the quotient space.

Abstract

Let $d \ge 1$. We study a subspace of the space of automorphic forms of $\mathrm{GL}_d$ over a global field of positive characteristic (or, a function field of a curve over a finite field). We fix a place $\infty$ of $F$, and we consider the subspace $\mathcal{A}_{\mathrm{St}}$ consisting of automorphic forms such that the local component at $\infty$ of the associated automorphic representation is the Steinberg representation (to be made precise in the text). We have two results. One theorem (Theorem 16) describes the constituents of $\mathcal{A}_{\mathrm{St}}$ as automorphic representation and gives a multiplicity one type statement. For the other theorem (Theorem 12), we construct, using the geometry of the Bruhat-Tits building, an analogue of modular symbols in $\mathcal{A}_{\mathrm{St}}$ integrally (that is, in the space of $\mathbb{Z}$-valued automorphic forms). We show that the quotient is finite and give a bound on the exponent of this quotient.