Theta operators, Goss polynomials, and v-adic modular forms

TL;DR

This paper studies hyperderivatives of Drinfeld modular forms, expressing them via Goss polynomials, and shows that v-adic modular forms are stable under hyperdifferentiation, preserving v-integrality after multiplication by Carlitz factorial.

Contribution

It provides explicit formulas for hyperderivatives in terms of Goss polynomials and proves the stability of v-adic modular forms under hyperdifferentiation.

Findings

Hyperderivatives expressed via Goss polynomials.

v-adic modular forms are preserved under hyperdifferentiation.

Hyperdifferentiation preserves v-integrality after multiplication by Carlitz factorial.

Abstract

We investigate hyperderivatives of Drinfeld modular forms and determine formulas for these derivatives in terms of Goss polynomials for the kernel of the Carlitz exponential. As a consequence we prove that v-adic modular forms in the sense of Serre, as defined by Goss and Vincent, are preserved under hyperdifferentiation. Moreover, upon multiplication by a Carlitz factorial, hyperdifferentiation preserves v-integrality.