Arithmetic
of the oscillator representation
This unfinished manuscript of 1987 uses
Mumford's theory of theta functions to construct the bundle of Siegel
modular forms of
weight 1/2 algebraically, along with the action of Hecke operators and
of certain
homogeneous differential operators. Since the oscillator
representation is naturally defined
over the maximal abelian extension of Q,
one needs to replace the Siegel modular variety by
a certain infinite cover (parametrizing additive characters) to obtain
a theory rational over Q. It
was intended to include proofs of a q-expansion
principle in weight 1/2 and a study of the rationality of the theta
correspondence for dual
reductive pairs when one of the factors is compact at the real place,
but the
corresponding sections were never written. The article in its
unfinished state may still be of
some interest.