Almost totally complex points on elliptic curves

Abstract

Let $F/F_0$ be a quadratic extension of totally real number fields, and let $E$ be an elliptic curve over $F$ which is isogenous to its Galois conjugate over $F_0$. A quadratic extension $M/F$ is said to be almost totally complex (ATC) if all archimedean places of $F$ but one extend to a complex place of $M$. The main goal of this note is to provide a new construction of a supply of Darmon-like points on $E$, which are conjecturally defined over certain ring class fields of $M$. These points are constructed by means of an extension of Darmon's ATR method to higher dimensional modular abelian varieties, from which they inherit the following features: they are algebraic provided Darmon's conjectures on ATR points hold true, and they are explicitly computable, as we illustrate with a detailed example that provides certain numerical evidence for the validity of our conjectures.

24 pages

References in corpus (1)

• Darmon's points and quaternionic Shimura varieties

Cited by in corpus (1)

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