Class Numbers, Cyclic Simple Groups and Arithmetic

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.