Abstract:
Let O be a maximal order in the quaternion algebra Bp over Q ramfi ed at p and ∞. The paper is about the computational problem: Construct a supersingular elliptic curve E over Fp such that End(E) ≅ O. We present an algorithm that solves this problem by taking gcds of the reductions modulo p of Hilbert class polynomials. New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice OT of O, the order O is eff ectively characterized by the three successive minima and two other short vectors of OT . The desired conditions turn out to hold whenever the j-invariant j(E), of the elliptic curve with End(E) = O, lies in Fp. We can then prove that our algorithm terminates with running time O(p1+ε) under the aforementioned conditions. As a further application we present an algorithm to simultaneously match all maximal order types with their associated j-invariants. Our algorithm has running time O(p2:5+ε) operations and is more efficient than Cerviño's algorithm for the same problem.