Abelian varieties and Weil representations

Abstract

The main goal of this article is to construct and study a family of Weil representations over an arbitrary locally noetherian scheme without restriction on characteristic. The key point is to recast the classical theory in the scheme-theoretic setting. As in work of Mumford, Moret-Bailly and others, a Heisenberg group (scheme) and its representation can be naturally constructed from a pair of an abelian scheme and a nondegenerate line bundle, replacing the role of a symplectic vector space. Once enough is understood about the Heisenberg group and its representations (e.g., the analogue of the Stone–von Neumann theorem), it is not difficult to produce the Weil representation of a metaplectic group (functor) from them. As an interesting consequence (when the base scheme is $Spec{\overline{\mathbb{F}}}_p$), we obtain the new notion of mod $p$ Weil representations of $p$-adic metaplectic groups on ${\overline{\mathbb{F}}}_p$-vector spaces. The mod $p$ Weil representations admit an alternative construction starting from a $p$-divisible group with a symplectic pairing.

We have been motivated by a few possible applications, including a conjectural mod $p$ theta correspondence for $p$-adic reductive groups and a geometric approach to the (classical) theta correspondence.