On the trace and norm maps from ${\Gamma }_0(\mathfrak{p})$ to ${\mathrm{GL}}_2(A)$

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 ${\mathrm{GL}}_2(A)$: by taking the trace or the norm from ${\Gamma }_0(\mathfrak{p})$ to ${\mathrm{GL}}_2(A)$. 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.