Modular Weil representation and compatibility of cuspidals with congruences

TL;DR

This paper extends the Weil representation theory to ll-modular coefficients over non-archimedean local fields, generalizing classical results and establishing compatibility of cuspidal representations with congruences.

Contribution

It introduces ll-modular Weil representations, generalizes the Stone-von Neumann theorem, and studies the irreducibility and congruence compatibility of theta lifts in this setting.

Findings

Generalized Weil representation to ll-modular coefficients.

Proved irreducibility of cuspidal theta lifts under certain conditions.

Established compatibility of cuspidals with congruences via integral theta lifts.

Abstract

Let $F$ be a non-archimedean local field of characteristic different from $2$ and of residual characteristic $p$. We generalise the theory of the Weil representation over $F$ with complex coefficients to $\ell$-modular representations \textit{i.e.} when the complex coefficients are replaced by a coefficient field $R$ of characteristic $\ell \neq p$. We obtain along the way a generalisation of the Stone-von Neumann theorem to the $\ell$-modular setting, together with the Weil representation with coefficients in $R$ on the $R$-metaplectic group. Surprisingly enough, the latter $R$-metaplectic group happens to be split over the symplectic group if $\ell = 2$. The theory also makes sense when $F$ is a finite field of odd characteristic. We also establish the irreducibility of the theta lift in the cuspidal case as long as $\ell$ does not divide the pro-orders of the groups at stake and we provide a compatibility to congruences in this setting via an integral version of the theta lift.