On some constancy of Hecke eigensystems for Drinfeld cuspforms of finite slope

Abstract

Let $p$ be a rational prime, let $q>1$ be a $p$-power integer, let ${\mathbb{F}}_q$ be the field of $q$ elements and let $A={\mathbb{F}}_q[t]$ be the polynomial ring over ${\mathbb{F}}_q$. Let $\mathfrak{n}\in A$ be a nonzero element and let $\wp \in A$ be a monic irreducible polynomial of positive degree. Let $k\ge 2$ and $r\ge 1$ be integers. Let $S_k({\Gamma }_1(\mathfrak{n}{\wp }^r))$ be the space of Drinfeld cuspforms of level ${\Gamma }_1(\mathfrak{n}{\wp }^r)$ and weight $k$. In this paper, we prove that the multiplicity of a Hecke eigensystem of finite $\wp$-slope in $S_k({\Gamma }_1(\mathfrak{n}{\wp }^r))$ is equal to $q^{(r-1)\mathrm{deg}(\wp )}$ times that in $S_k({\Gamma }_1(\mathfrak{n}\wp ))$. In particular, this shows that a Hecke eigensystem of finite $\wp$-slope appears in $S_k({\Gamma }_1(\mathfrak{n}{\wp }^r))$ if and only if it appears in $S_k({\Gamma }_1(\mathfrak{n}\wp ))$.

13 pages; title changed, Theorem 1.1 slightly improved, Remarks 5.5 and 5.6 added