Universal Weil module

TL;DR

This paper constructs a universal Weil module over a minimal ring, extending the classical Weil representation to a broader algebraic setting and enabling scalar extension to various coefficient rings.

Contribution

It identifies the minimal ring for Weil representation over general coefficient rings and constructs a universal Weil module over this ring.

Findings

Most problems are solvable over the base ring and extendable by scalar extension.

All Weil representations arise from a single universal module over the minimal ring.

The work suggests potential developments in integral theta correspondence.

Abstract

The classical construction of the Weil representation, with complex coefficients, has long been expected to work for more general coefficient rings. This paper exhibits the minimal ring $\mathcal{A}$ for which this is possible, the integral closure of $\mathbb{Z}[\frac{1}{p}]$ in a cyclotomic field, and carries out the construction of the Weil representation over $\mathcal{A}$-algebras. As a leitmotif all along the work, most of the problems can actually be solved over the base ring $\mathcal{A}$ and transferred to any $\mathcal{A}$-algebra by scalar extension. The most striking fact is that all these Weil representations arise as the scalar extension of a single one with coefficients in $\mathcal{A}$. In this sense, the Weil module obtained is universal. Building upon this construction, we speculate and make predictions about an integral theta correspondence.