Vinakyak Vatsal (UBC)
Theta functions after Waldspurger and Mumford
We study the Fourier coefficients of modular forms of half-integer
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.