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.