Efficient calculation of Stark-Heegner points via overconvergent modular symbols

Abstract

This note presents a qualitative improvement to the algorithm presented in [DG] for computing Stark-Heegner points attached to an elliptic curve and a real quadratic field. This algorithm computes the Stark-Heegner point with ap-adic accuracy ofM significant digits in time which is polynomial inM, the primep being treated as a constant, rather than theO(p M ) operations required for the more naive approach taken in [DG]. The key to this improvement lies in the theory of overconvergent modular symbols developed in [PS1] and [PS2].