Abstract. Let E /Q be an elliptic curve of conductor N and let f be the weight 2 newform on Γ 0 (N ) associated to it by modularity. Building on an idea of S. Zhang, an article by Darmon, Rotger, and Sols describes the construction of so-called Chow-Heegner points,It also gives an analytic formula, depending only on the image of T in cohomology under the complex cycle class map, for calculating P T,f numerically via Chen's theory of iterated integrals. The present work describes an algorithm based on this formula for computing the Chow-Heegner points to arbitrarily high complex accuracy, carries out the computation for all elliptic curves of rank 1 and conductor N < 100 when the cycles T arise from Hecke correspondences, and discusses several important variants of the basic construction.

Abstract. Suppose that f (resp. g) is a modular form of integral (resp. halfintegral) weight with coefficients in the ring of integers O K of a number field K. For any ideal m ⊂ O K , we bound the first prime p for which f | T p (resp. g | T p 2 ) is zero (mod m). Applications include the solution to a question of Ono (2001) concerning partitions.