On mod $\mathfrak{p}$ congruences for Drinfeld modular forms of level $\mathfrak{p}\mathfrak{m}$

TL;DR

This paper investigates congruences between Drinfeld modular forms of level  and explores related conjectures, extending classical results to the function field setting.

Contribution

It introduces new results on mod  congruences for Drinfeld modular forms, generalizing classical congruence conjectures to the function field context.

Findings

Established analogues of classical congruence conjectures for Drinfeld modular forms.

Proved new theorems extending previous results to broader levels.

Connected classical and function field modular form theories.

Abstract

In~\cite{CS04}, Calegari and Stein studied the congruences between classical cusp forms $S_k(\Gamma_0(p))$ of prime level and made several conjectures about them. In~\cite{AB07} (resp., ~\cite{BP11}) the authors proved one of those conjectures (resp., their generalizations). In this article, we study the analogous conjecture and its generalizations for Drinfeld modular forms.