Class Numbers, Congruent Numbers and Umbral Moonshine

TL;DR

This paper explores deep connections between sporadic simple groups, modular forms, and classical number theory problems, revealing new classifications and implications for the congruent number problem.

Contribution

It establishes a link between umbral moonshine and quadratic class numbers, classifies modules for Mathieu groups, and connects these to the ancient congruent number problem.

Findings

Connection between umbral moonshine and quadratic class numbers

Classification of modules for Mathieu groups

Implications for the congruent number problem

Abstract

In earlier work we initiated a program to study relationships between finite groups and arithmetic geometric invariants of modular curves in a systematic way. In the present work we continue this program, with a focus on the two smallest sporadic simple Mathieu groups. To do this we first elucidate a connection between a special case of umbral moonshine and the imaginary quadratic class numbers. Then we use this connection to classify a distinguished set of modules for the smallest sporadic Mathieu group. Finally we establish consequences of the classification for the congruent number problem of antiquity.