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
), we obtain the new
notion of mod
Weil
representations of
-adic
metaplectic groups on
-vector
spaces. The mod
Weil representations admit an alternative construction starting from a
-divisible
group with a symplectic pairing.
We have been motivated by a few possible applications, including a conjectural mod
theta correspondence
for
-adic
reductive groups and a geometric approach to the (classical) theta correspondence.
Keywords
abelian varieties, Weil representations, Heisenberg groups
