Elementary Matrix Decomposition and The Computation of Darmon Points with Higher Conductor

2012/09/20 by Xavier Guitart, Guitart, Xavier, Marc Masdeu +1
Mathematics · #11G05 #11G07 #11S80 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G05 #msc:11G07 #msc:11S80

paper · pdf · doi:10.48550/arxiv.1209.4614

16 pages, 6 tables

arxiv created 2012/09/20 · arxiv updated 2012/09/21

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 Γ(N) for ideals 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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