Traces of Hecke operators on Drinfeld modular forms for $\mathrm{GL}_2(\mathbb{F}_q[T])$

TL;DR

This paper derives explicit formulas and algorithms for traces of Hecke operators on Drinfeld modular forms over function fields, improving bounds and analyzing form decompositions.

Contribution

It provides closed-form trace formulas for low-degree primes, algorithms for higher degrees, and confirms conjectures on form decomposition under certain eigenvalue multiplicity conditions.

Findings

Closed-form trace formulas for primes of degree ≤ 2

Algorithms for primes of higher degree

Improved Ramanujan bound and form decomposition results

Abstract

In this paper, we study traces of Hecke operators on Drinfeld modular forms of level 1 in the case $A = \mathbb{F}_q[T]$. We deduce closed-form expressions for traces of Hecke operators corresponding to primes of degree at most 2 and provide algorithms for primes of higher degree. We improve the Ramanujan bound and deduce the decomposition of cusp forms of level $\Gamma_0(\mathfrak{p})$ into oldforms and newforms, as conjectured by Bandini-Valentino, under the hypothesis that each Hecke eigenvalue has multiplicity less than $p$.