Hecke operators on differential modular forms mod p

TL;DR

This paper characterizes primitive differential series mod p that are eigenvectors of Hecke operators and correspond to differential modular forms, establishing a natural correspondence with classical modular forms mod p.

Contribution

It provides a complete description of primitive differential series mod p of order 1 as eigenvectors of Hecke operators, linking them to classical modular forms.

Findings

Primitive differential series mod p of order 1 are eigenvectors of all Hecke operators.

A natural one-to-one correspondence between these series and classical modular forms mod p.

The set of differential series is fully characterized in terms of Fourier expansions.

Abstract

A description is given of all primitive differential series mod p of order 1 which are eigenvectors of all the Hecke operators and which are differential Fourier expansions of differential modular forms of arbitrary order and given weight; this set of differential series is shown to be in a natural one-to-one correspondence with the set of series mod p (of order 0) which are eigenvectors of all the Hecke operators and which are Fourier expansions of (classical) modular forms of appropriate weight.