Vinakyak Vatsal (UBC)
Theta functions after Waldspurger and Mumford
We study the Fourier coefficients of modular forms of half-integer
weight by
realizing them as theta constants arising from Mumford's theory of
algebraic theta functions associated to
polarized abelian varieties. In the case of interest, the abelian
varieties are three-folds isogenous to the product
of elliptic curves, and the polarization is deduced from a positive
definite quadratic form in three variables.
These theta functions give a geometric realization of the automorphic
theta correspondence associated to the ternary quadratic form implicit
in the polarization, as studied byWaldspurger. Thus our construction
of geometric theta functions gives a geometric incarnation of
Waldspurger's correspondence, which in turn gives a mechanism for
studying the values modulo p of quadratic twists of L-functions.