Algorithms for Chow-Heegner points via iterated integrals (original) (raw)
2015, Mathematics of Computation
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, the article [DRS] describes the construction of so-called Chow-Heegner points P T,f ∈ E(Q) indexed by algebraic correspondences T ⊂ X0(N )×X0(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 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.
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