Computation of ATR Darmon points on non-geometrically modular elliptic curves

Abstract

ATR points were introduced by Darmon as a conjectural construction of algebraic points on certain elliptic curves for which in general the Heegner point method is not available. So far the only numerical evidence, provided by Darmon--Logan and G\"artner, concerned curves arising as quotients of Shimura curves. In those special cases the ATR points can be obtained from the already existing Heegner points, thanks to results of Zhang and Darmon--Rotger--Zhao. In this paper we compute for the first time an algebraic ATR point on a curve which is not uniformizable by any Shimura curve, thus providing the first piece of numerical evidence that Darmon's construction works beyond geometric modularity. To this purpose we improve the method proposed by Darmon and Logan by removing the requirement that the real quadratic field be norm-euclidean, and accelerating the numerical integration of Hilbert modular forms.