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

Abstract

We extend the algorithm of (DG02) and (DP06) for computing p-adic Darmon points on elliptic curves to the case of composite conductor. We also extend the algorithm of (DL03) for computing ATR Darmon points to treat curves of nontrivial conductor. Both cases involve an algorithmic decomposition into elementary matrices in congruence subgroups 1(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.