Class Numbers, Cyclic Simple Groups and Arithmetic

TL;DR

This paper explores the connections between finite cyclic groups, class numbers of imaginary quadratic fields, and the arithmetic geometry of modular curves, introducing optimal modules in the context of mock Jacobi forms.

Contribution

It introduces the concept of optimal modules for finite groups within holomorphic mock Jacobi forms and classifies them for cyclic groups of prime order in a specific weight and index setting.

Findings

Classification of optimal modules for cyclic groups of prime order.

Identification of the role of class numbers of imaginary quadratic fields.

Connection between group classification and arithmetic geometry of modular curves.

Abstract

Here we initiate a program to study relationships between finite groups and arithmetic-geometric invariants in a systematic way. To do this we first introduce a notion of optimal module for a finite group in the setting of holomorphic mock Jacobi forms. Then we classify optimal modules for the cyclic groups of prime order, in the special case of weight two and index one, where class numbers of imaginary quadratic fields play an important role. Finally we exhibit a connection between the classification we establish and the arithmetic geometry of imaginary quadratic twists of modular curves of prime level.