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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
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)

Marc Masdeu
Affiliation: Columbia University, Department of Mathematics, Room 415, MC 4441, 2990 Broadway, New York, New York 10027

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.