A construction of $\mathfrak v$-adic modular forms

TL;DR

This paper constructs $rak v$-adic modular forms related to Drinfeld modules, extending recent progress in the $rak v$-adic theory and revealing new structures in the Hecke action.

Contribution

It introduces a new construction of $rak v$-adic cusp forms inspired by Petrov's $A$-expansions, advancing the understanding of $rak v$-adic modular forms for Drinfeld modules.

Findings

Construction of $rak v$-adic cusp forms from Petrov's $A$-expansions.

Demonstration of a decomposition of the Hecke action on these forms.

Extension of $rak v$-adic theory to new classes of modular forms.

Abstract

The classical theory of $p$-adic (elliptic) modular forms arose in the 1970's from the work of J.-P.\ Serre \cite{se1} who took $p$-adic limits of the $q$-expansions of these forms. It was soon expanded by N.\ Katz \cite{ka1} with a more functorial approach. Since then the theory has grown in a variety of directions. In the late 1970's, the theory of modular forms associated to Drinfeld modules was born in analogy with elliptic modular forms \cite{go1}, \cite{go2}. The associated expansions at $\infty$ are quite complicated and no obvious limits at finite primes ${\mathfrak v}$ were apparent. Recently, however, there has been progress in the $\mathfrak v$-adic theory, \cite{vi1}. Also recently, A.\ Petrov \cite{pe1}, building on previous work of \cite{lo1}, showed that there is an intermediate expansion at $\infty$ called the "$A$-expansion," and he constructed families of cusp forms with such expansions. It is our purpose in this note to show that Petrov's results also lead to interesting ${\mathfrak v}$-adic cusp forms \`a la Serre. Moreover the existence of these forms allows us to readily conclude a mysterious decomposition of the associated Hecke action.