Algorithms for Chow–Heegner points via iterated integrals

Abstract. Let E/Q be an elliptic curve of conductor N and let f be the weight 2 newform on Γ0(N) associated to it by modularity. Building on an idea of S. Zhang, an article by Darmon, Rotger, and Sols describes the con-struction of so-called Chow-Heegner points, PT,f ∈ E(Q̄), indexed by algebraic correspondences T ⊂ X0(N) × X0(N). It also gives an analytic formula, de-pending only on the image of T in cohomology under the complex cycle class map, for calculating PT,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. 1