Algorithms for Chow-Heegner points via iterated integrals

Abstract

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.

Efficient Computation of Rankin p-Adic L-Functions

“…(1) The above argument does not quite show that H := d −(1+t) (g [p] ) × h lies in M k (N, χ, Z p , 1 p+1 ) as for this one would need to replace "K" by "B" in (4). However, the author just assumed this was true, and this was not contradicted by our experiments; in particular, when one could relate the value of the Rankin p-adic L-function to the p-adic logarithm of a point on an elliptic curve, the relationship held to exactly the precision predicted by the algorithm.…”

Torsion properties of modified diagonal classes on triple products of modular curves

“…Darmon, Rotger, and Sols [7] have studied such points, in the broader context of Shimura curves over totally real fields, notably by computing their images under the complex Abel-Jacobi map in terms of iterated integrals. Methods have been developed by Darmon, Daub, Lichtenstein, Rotger, and Stein [5] to numerically calculate such points in the case of modular curves.…”

An effective Chabauty–Kim theorem

“…To complete the proof of Theorem 1.1, it remains to estimate the degree of the coefficients of a nice . We follow the construction of basis differentials in [DDLR15, § 4.2]. Let be a non-Weierstrass point of whose mod reduction is different from , and let be a non-constant function in .…”