Weil representations associated to finite quadratic modules

TL;DR

This paper provides an explicit, computationally accessible formula for the Weil representation matrices associated with finite quadratic modules, extending previous results to more general cases.

Contribution

It introduces a comprehensive explicit formula for the Weil representation matrices of finite quadratic modules, incorporating p-adic invariants and generalizing prior work.

Findings

Explicit formula involving p-adic invariants

Complements earlier results by Scheithauer

Easy to implement computationally

Abstract

To a finite quadratic module, that is, a finite abelian group D together with a non-singular quadratic form Q:D --> Q/Z, it is possible to associate a representation of either the modular group, SL(2,Z), or its metaplectic cover, Mp(2,Z), on C[D], the group algebra of D. This representation is usually called the Weil representation associated to the finite quadratic module. The main result of this paper is a general explicit formula for the matrix coefficients of this representation. The formula, which involves the p-adic invariants of the quadratic module, is given in a way which is easy to implement on a computer. The result presented completes an earlier result by Scheithauer for the Weil representation associated to a discriminant form of even signature.