Partition Eisenstein series and semi-modular forms

TL;DR

This paper introduces semi-modular forms, a new class of functions invariant under specific subgroups of $GL_2( Z)$, and constructs Eisenstein-like series over integer partitions to generate these forms, expanding the classical modular form framework.

Contribution

It defines semi-modular forms invariant under certain subgroups and constructs Eisenstein-like series over partitions to produce these forms, broadening the scope of modular form theory.

Findings

Identification of semi-modular forms invariant under special subgroups

Construction of Eisenstein-like series over integer partitions

Extension of classical modular forms to new function classes

Abstract

We identify a class of "semi-modular" forms invariant on special subgroups of $GL_2(\mathbb Z)$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an Eisenstein-like series summed over integer partitions, and use it to construct families of semi-modular forms.