Heegner Point Computations Via Numerical p-Adic Integration

Abstract

Building on ideas of Pollack and Stevens, we present an efficient algorithm for integrating rigid analytic functions against measures obtained from automorphic forms on definite quaternion algebras. We then apply these methods, in conjunction with the Jacquet-Langlands correspondence and the Cerednik-Drinfeld theorem, to the computation of p-adic periods and Heegner points on elliptic curves defined over ℚ and \({\mathbb{Q}}(\sqrt{5})\) which are uniformized by Shimura curves.