Divisibility Arising From Addition: The Application of Modular Functions to Infinite Partition Congruence Families

TL;DR

This paper explores how modular functions can be used to prove partition congruences, addressing theoretical and computational challenges in the field of partition theory and modular forms.

Contribution

It provides a survey of the utility of modular functions in proving partition congruences and discusses unresolved problems in the area.

Findings

Modular functions are useful in proving partition congruences.

Challenges include genus of modular curves and convergence issues.

The paper highlights open problems in the field.

Abstract

The theory of partition congruences has been a fascinating and difficult subject for over a century now. In attempting to prove a given congruence family, multiple possible complications include the genus of the underlying modular curve, representation difficulties of the associated sequences of modular functions, and difficulties regarding the piecewise $\ell$-adic convergence of elements of the associated space of modular functions. However, our knowledge of the subject has developed substantially and continues to develop. In this very brief survey, we will discuss the utility of modular functions in proving partition congruences, both theoretical and computational, and many of the problems in the subject that are yet to be overcome.