A $p$-adic Approach to the Weil Representation of Discriminant Forms Arising from Even Lattices

TL;DR

This paper introduces a $p$-adic method to explicitly compute the Weil representation of even lattices without relying on theta functions, revealing new insights into its structure and roots of unity factors.

Contribution

It provides a direct $p$-adic evaluation approach to the Weil representation, decomposing it into $p$-parts and analyzing the metaplectic cover properties.

Findings

Decomposition of Weil representation into $p$-parts.

Explicit $p$-adic formulas for the Weil representation actions.

Identification of roots of unity factors in the operators.

Abstract

Suppose that $M$ is an even lattice with dual $M^{*}$ and level $N$. Then the group $Mp_{2}(\mathbb{Z})$, which is the unique non-trivial double cover of $SL_{2}(\mathbb{Z})$, admits a representation $\rho_{M}$, called the Weil representation, on the space $\mathbb{C}[M^{*}/M]$. The main aim of this paper is to show how the formulae for the $\rho_{M}$-action of a general element of $Mp_{2}(\mathbb{Z})$ can be obtained by a direct evaluation which does not depend on ``external objects'' such as theta functions. We decompose the Weil representation $\rho_{M}$ into $p$-parts, in which each $p$-part can be seen as subspace of the Schwartz functions on the $p$-adic vector space $M_{\mathbb{Q}_{p}}$. Then we consider the Weil representation of $Mp_{2}(\mathbb{Q}_{p})$ on the space of Schwartz functions on $M_{\mathbb{Q}_{p}}$, and see that restricting to $Mp_{2}(\mathbb{Z})$ just gives the $p$-part of $\rho_{M}$ again. The operators attained by the Weil representation are not always those appearing in the formulae from 1964, but are rather their multiples by certain roots of unity. For this, one has to find which pair of elements, lying over a matrix in $SL_{2}(\mathbb{Q}_{p})$, belong to the metaplectic double cover. Some other properties are also investigated.