Dimension variation of Gouvêa-Mazur type for Drinfeld cuspforms of level ${\Gamma }_1(t)$

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 Gouvêa-Mazur conjecture for this setting. Namely, we show that for any $\alpha \le m$ and $k_1,k_2>\alpha +1$, if $k_1\equiv k_2\mathrm{mod}{p}^m$, then $d(k_1,\alpha )=d(k_2,\alpha )$.

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