ALMOST TOTALLY COMPLEX POINTS ON ELLIPTIC CURVES

作者:Guitart Xavier*; Rotger Victor; Zhao Yu
来源:Transactions of the American Mathematical Society, 2014, 366(5): 2773-2802.

摘要

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%26apos;s ATR method to higher-dimensional modular abelian varieties, from which they inherit the following features: they are algebraic provided Darmon%26apos;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.

  • 出版日期2014-5