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

TL;DR

This paper extends algorithms for computing Darmon points on elliptic curves to cases with composite conductors and nontrivial conductors, providing new computational methods and supporting conjectures on their rationality.

Contribution

It introduces extended algorithms for Darmon points on elliptic curves with composite and nontrivial conductors, involving elementary matrix decompositions in congruence subgroups.

Findings

Extended algorithms successfully compute Darmon points for new cases.

Provides computational evidence supporting rationality conjectures.

Enhances understanding of Darmon points in complex conductor scenarios.

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}(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.