A $p$-adic construction of ATR points on $\mathbb{Q}$-curves

Abstract

In this note we consider certain elliptic curves defined over real quadratic fields isogenous to their Galois conjugate. We give a construction of algebraic points on these curves defined over almost totally real number fields. The main ingredient is the system of Heegner points arising from Shimura curve uniformizations. In addition, we provide an explicit $p$-adic analytic formula which allows for the effective, algorithmic calculation of such points.