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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Algorithms for Chow-Heegner points via iterated integrals
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by Henri Darmon, Michael Daub, Sam Lichtenstein and Victor Rotger PDF
Math. Comp. 84 (2015), 2505-2547 Request permission

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
  • © Copyright 2015 American Mathematical Society
  • 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
  • MathSciNet review: 3356037