p-adic congruences and modular forms of half integer weight (Q1061774)

This paper investigates p-adic interpolation of modular forms of half integer weight, and gives evidence that under certain circumstances p- adic congruences between forms of integer weight descend via the Shimura lifting to congruences between corresponding forms of half integer weight. First, an example is given: the Ramanujan congruence descends to weight 13/2; this gives a mod 691 factorization into a product of Bernoulli numbers for the algebraic part of the central critical value of the twisted L-function of the discriminant form. Next, it is shown that it makes sense to speak of half integer weight forms of p-adic weight. Then the generalized class number Eisenstein series of half integer weight are interpolated p-adically; this motivates certain conjectures concerning congruences between central critical values of Hecke L-series corresponding to elliptic curves. These conjectures are then shown to be compatible with the Katz-Manin-Vishik interpolation of two-variable Eisenstein series. Finally, numerical evidence in support of the conjectures is given in the case of two families of Hecke series corresponding to certain elliptic curves with complex multiplication by third and fourth roots of unity.