On some constancy of Hecke eigensystems for Drinfeld cuspforms of level $\Gamma_1(\mathfrak{n}\wp^r)$

TL;DR

This paper proves a constancy property of Hecke eigensystems for Drinfeld cuspforms across levels differing by powers of a prime, showing eigensystems of finite slope are level-independent.

Contribution

It establishes that Hecke eigensystems of finite slope in Drinfeld cuspforms are invariant under increasing the level by powers of a prime, revealing a new level-constancy phenomenon.

Findings

Hecke eigensystems of finite slope appear at all levels if they appear at the base level.

The eigensystems are invariant under level raising by prime powers.

The result applies to Drinfeld cuspforms over polynomial rings over finite fields.

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 $k\geq 2$ and $r\geq 1$ 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 show 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 $S_k(\Gamma_1(\mathfrak{n}\wp))$.