| For -D a fundamental discriminant and p a prime, we investigate the surjectivity of the reduction map from elliptic curves with CM by O-D to supersingular elliptic curves over Fp whenever p does not split in O-D . Under GRH for Dirichlet L-functions and the L-functions of weight 2 newforms, we are able to show an effectively computable bound D p such that the reduction map is surjective for every D > Dp with p nonsplit. Our investigation takes a detour through a study of quaternion algebras and quadratic forms. In particular, in showing our result, we obtain as a side effect the following result. For each positive definite quadratic form Q whose associated theta series is in Kohnen's plus space of weight 3/2 and level 4p, M+3/2 (4p), we show an effectively computable bound DQ, dependent upon GRH such that Q represents every D for which D > DQ and p does not split in O-D . Moreover, we give an explicit algorithm to compute D Q (respectively Dp), and for small p we explicitly compute DQ (resp. Dp). For a further restricted set of p, we moreover obtain a computationally feasible bound, allowing us to give a full list of fundamental discriminants -D for which the map is not surjective. To determine the full list we develop a specialized algorithm to compute which D < Dp are represented more efficiently whenever all of the elliptic curves are defined over Fp . Additionally, we obtain as an additional side effect a new proof and an explicit algorithm, conditional upon GRH, for the Ramanujan-Petersson conjecture for weight 3/2 cusp forms of level 4N in Kohnen's plus space with N odd and squarefree. |
