Generating weights for the Weil representation attached to an even order cyclic quadratic module

TL;DR

This paper develops geometric methods to compute generating weights of vector valued modular forms associated with Weil representations of cyclic quadratic modules, revealing their asymptotic distribution as parameters grow.

Contribution

It introduces geometric techniques to determine generating weights for Weil representations of cyclic quadratic modules and analyzes their limiting behavior.

Findings

Generated weights for modules with cyclic quadratic groups of order 2p^r.

Found that generating weights tend to a simple distribution as p or r increase.

Provided explicit computations for prime p > 3.

Abstract

We develop geometric methods to study the generating weights of free modules of vector valued modular forms of half-integral weight, taking values in a complex representation of the metaplectic group. We then compute the generating weights for modular forms taking values in the Weil representation attached to cyclic quadratic modules of order 2p^r, where p is a prime greater than three. We also show that the generating weights approach a simple limiting distribution as p grows, or as r grows and p remains fixed.