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

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.