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.References
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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
- © 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
Dedicated: With an appendix by William Stein