Automorphic functions for nilpotent extensions of curves over finite fields

TL;DR

This paper explores automorphic functions on nilpotent extensions of curves over finite fields, constructing Hecke algebras and eigenfunctions, and analyzing the structure and dimension of cuspidal function spaces.

Contribution

It introduces a new framework for automorphic functions on nilpotent curve extensions, including the construction of Hecke algebras and eigenfunctions, and provides dimension bounds for cuspidal spaces.

Findings

Cuspidal functions are preserved under a noncommutative Hecke algebra action.

A commutative subalgebra of Hecke operators is constructed for GL_2.

The space of cuspidal functions for length 2 extensions is finite-dimensional with explicit bounds.

Abstract

We define and study the subspace of cuspidal functions for $G$-bundles on a class of nilpotent extensions $C$ of curves over a finite field. We show that this subspace is preserved by the action of a certain noncommutative Hecke algebra $\mathcal{H}_{G,C}$. In the case $G=\rm{GL}_2$, we construct a commutative subalgebra in $\mathcal{H}_{G,C}$ of Hecke operators associated with simple divisors. In the case of length 2 extensions and of $G=\rm{GL}_2$, we prove that the space of cuspidal functions (for bundles with a fixed determinant) is finite-dimensional and provide bounds on its dimension. In this case we also construct some Hecke eigenfunctions using the relation to Higgs bundles over the corresponding reduced curve.