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

Guitart, Xavier; Masdeu, Marc

doi:10.1090/S0025-5718-2014-02853-6

Elementary matrix decomposition and the computation of Darmon points with higher conductor
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by Xavier Guitart and Marc Masdeu;

Math. Comp. 84 (2015), 875-893

DOI: https://doi.org/10.1090/S0025-5718-2014-02853-6

Published electronically: July 17, 2014

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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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