Twisting eigensystems of Drinfeld Hecke eigenforms by characters

Abstract

We address some questions posed by Goss related to the modularity of Drinfeld modules of rank 1 defined over the field of rational functions in one variable with coefficients in a finite field. For each positive characteristic valued Dirichlet character, we introduce certain projection operators on spaces of Drinfeld modular forms with character of a given weight and type such that when applied to a Hecke eigenform return a Hecke eigenform whose eigensystem has been twisted by the given Dirichlet character. Unlike the classical case, however, the effect on Goss' $u$-expansions for these eigenforms --- and even on Petrov's $A$-expansions --- is more complicated than a simple twisting of the $u-$ (or $A-$) expansion coefficients by the given character. We also introduce Eisenstein series with character for irreducible levels $\mathfrak{p}$ and show that they and their Fricke transforms are Hecke eigenforms with a new type of $A$-expansion and $A$-expansion in the sense of Petrov, respectively. We prove congruences between certain cuspforms in Petrov's special family and the Eisenstein series and their Fricke transforms introduced here, and we show that in each weight there are as many linearly independent Eisenstein series with character as there are cusps for $\Gamma_1(\mathfrak{p})$.