Almost totally complex points on elliptic curves

Guitart, Xavier; Rotger, Victor; Zhao, Yu

doi:10.1090/S0002-9947-2013-05981-8

Almost totally complex points on elliptic curves
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by Xavier Guitart, Victor Rotger and Yu Zhao PDF

Trans. Amer. Math. Soc. 366 (2014), 2773-2802 Request permission

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 for 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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