Computation of ATR Darmon Points on Nongeometrically Modular Elliptic Curves

Abstract

ATR points were introduced by Darmon as a conjectural construction of algebraic points on certain elliptic curves for which the Heegner-point method is not in general available. So far, the only numerical evidence, provided by Darmon–Logan and Gärtner, 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 from Zhang and Darmon–Rotger–Zhao. In this paper, we compute for the first time an algebraic ATR point on a curve that 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 base field be norm-Euclidean and accelerating the numerical integration of Hilbert modular forms.