On the trace and norm maps from $\Gamma_0(\mathfrak{p})$ to $\operatorname{GL}_2(A)$

TL;DR

This paper explores the relationships between trace and norm maps of Drinfeld modular forms from a congruence subgroup to the full modular group, focusing on their coefficient arithmetic modulo a prime ideal.

Contribution

It establishes connections between the coefficients of modular forms and their trace and norm images, revealing new arithmetic properties modulo a prime ideal.

Findings

Connections between coefficients of forms and their trace/norm images

Arithmetic properties of coefficients modulo prime ideal

Relations between $u$-series coefficients and trace/norm forms

Abstract

Let $f$ be a Drinfeld modular form for $\Gamma_0(\mathfrak{p})$. From such a form, one can obtain two forms for the full modular group $\operatorname{GL}_2(A)$: by taking the trace or the norm from $\Gamma_0(\mathfrak{p})$ to $\operatorname{GL}_2(A)$. In this paper we show some connections between the arithmetic modulo $\mathfrak{p}$ of the coefficients of the $u$-series expansion of $f$ and those of a form closely related to its trace, and of the coefficients of $f$ and those of its norm.