Dimension variation of Gouv\^{e}a-Mazur type for Drinfeld cuspforms of level $\Gamma_1(t)$

TL;DR

This paper proves a function field analogue of the Gouva-Mazur conjecture, demonstrating that the dimension of slope lpha ext{ } generalized eigenspaces for Drinfeld cuspforms remains constant under certain weight congruences.

Contribution

It establishes a new invariance property of slope dimensions for Drinfeld cuspforms, extending the Gouva-Mazur conjecture to the function field setting.

Findings

Dimension of slope lpha ext{ } eigenspaces is invariant under specific weight congruences.

Proves a function field analogue of the Gouva-Mazur conjecture.

Shows stability of eigenspace dimensions for weights differing by multiples of p^m.

Abstract

Let $p$ be a rational prime and $q>1$ a $p$-power. Let $S_k(\Gamma_1(t))$ be the space of Drinfeld cuspforms of level $\Gamma_1(t)$ and weight $k$ for $\mathbb{F}_q[t]$. For any non-negative rational number $\alpha$, we denote by $d(k,\alpha)$ the dimension of the slope $\alpha$ generalized eigenspace for the $U$-operator acting on $S_k(\Gamma_1(t))$. In this paper, we prove a function field analogue of the Gouv\^{e}a-Mazur conjecture for this setting. Namely, we show that for any $\alpha\leq m$ and $k_1,k_2>\alpha+1$, if $k_1\equiv k_2 \bmod p^m$, then $d(k_1,\alpha)=d(k_2,\alpha)$.