Computationally Feasible Bounds for Representations of Integers by Ternary Quadratic Forms and CM Lifts of Supersingular Elliptic Curves By

iAbstract 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 Dp 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 boundDQ, 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 DQ (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 th