**
****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.

The transfer to tex introduced numerous typographical errors, many of which undoubtedly remain.