Algorithms for Chow-Heegner points via iterated integrals

Henri Darmon, Michael Daub, Sam Lichtenstein, Victor Rotger
2015 Mathematics of Computation  
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, P T,f ∈ E(Q), indexed by algebraic correspondences T ⊂ X 0 (N ) × X 0 (N ). 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
more » ... ly 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.
doi:10.1090/s0025-5718-2015-02927-5 fatcat:ffuy436fdfbnvbql3dnr3vxvfq