Almost Totally Complex Points on Elliptic Curves
نویسنده
چکیده
Let F/F0 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 F0. 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 numerical evidence for the validity of our conjectures.
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