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Mathematics of Computation

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Elementary matrix decomposition and the computation of Darmon points with higher conductor


Authors: Xavier Guitart and Marc Masdeu
Journal: Math. Comp. 84 (2015), 875-893
MSC (2010): Primary 11G40; Secondary 11F41, 11Y99
DOI: https://doi.org/10.1090/S0025-5718-2014-02853-6
Published electronically: July 17, 2014
MathSciNet review: 3290967
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Abstract: We extend the algorithm of Darmon–Green and Darmon–Pollack for computing $p$-adic Darmon points on elliptic curves to the case of composite conductor. We also extend the algorithm of Darmon–Logan for computing ATR Darmon points to treat curves of nontrivial conductor. Both cases involve an algorithmic decomposition into elementary matrices in congruence subgroups $\Gamma _1({\mathfrak N})$ for ideals ${\mathfrak N}$ in certain rings of $S$-integers. We use these extensions to provide additional evidence in support of the conjectures on the rationality of Darmon points.


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Additional Information

Xavier Guitart
Affiliation: Universitat Politècnica de Catalunya, Departament de Matematica Aplicada II, C/Jordi Girona, 1-3, 08034 Barcelona (Spain)
MR Author ID: 887813
Email: xevi.guitart@gmail.com

Marc Masdeu
Affiliation: Columbia University, Department of Mathematics, Room 415, MC 4441, 2990 Broadway, New York, New York 10027
MR Author ID: 772165
Email: masdeu@math.columbia.edu

Received by editor(s): May 14, 2013
Received by editor(s) in revised form: June 5, 2013
Published electronically: July 17, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.