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Algorithms for Chow-Heegner points via iterated integrals


Authors: Henri Darmon, Michael Daub, Sam Lichtenstein and Victor Rotger
Journal: Math. Comp. 84 (2015), 2505-2547
MSC (2010): Primary 11F67, 11G05, 11Y16, 14C15
DOI: https://doi.org/10.1090/S0025-5718-2015-02927-5
Published electronically: March 2, 2015
MathSciNet review: 3356037
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Abstract: Let $E_{/\mathbf Q}$ be an elliptic curve of conductor $N$ and let $f$ be the weight $2$ newform on $\Gamma _0(N)$ 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, $P_{T,f}\in E({\bar {\mathbf Q}})$, indexed by algebraic correspondences $T\subset X_0(N)\times X_0(N)$. 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 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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Additional Information

Henri Darmon
Affiliation: Department of Mathematics, McGill University, Montreal, Canada
MR Author ID: 271251
Email: darmon@math.mcgill.ca

Michael Daub
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California
Email: mwdaub@math.berkeley.edu

Sam Lichtenstein
Affiliation: University of California at Stanford, Stanford, California
Email: saml@math.stanford.edu

Victor Rotger
Affiliation: Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain
MR Author ID: 698263
Email: victor.rotger@upc.edu

Received by editor(s): December 23, 2011
Received by editor(s) in revised form: October 28, 2013, and December 23, 2013
Published electronically: March 2, 2015
Dedicated: With an appendix by William Stein
Article copyright: © Copyright 2015 American Mathematical Society