scholarly journals Heegner points and p-adic L-functions for elliptic curves over certain totally real fields

2011 ◽  
pp. 867-945 ◽  
Author(s):  
Chung Pang Mok
Keyword(s):  
Elliptic Curves ◽  
Heegner Points ◽  
Totally Real ◽  
Totally Real Fields

Introduction and Statement of Main Results

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Abelian Varieties ◽  
Main Idea ◽  
Shimura Curves ◽  
Modular Curves ◽  
Heegner Points ◽  
Cm Points ◽  
Totally Real ◽  
Totally Real Fields ◽  
Derivatives Of ◽  
Original Formula

This chapter states the main result of this book regarding Shimura curves and abelian varieties as well as the main idea of the proof of a complete Gross–Zagier formula on quaternionic Shimura curves over totally real fields. It begins with a discussion of the original formula proved by Benedict Gross and Don Zagier, which relates the Néeron–Tate heights of Heegner points on X⁰(N) to the central derivatives of some Rankin–Selberg L-functions under the Heegner condition. In particular, it considers the Gross–Zagier formula on modular curves and abelian varieties parametrized by Shimura curves. It then decribes CM points and the Waldspurger formula before concluding with an outline of our proof, along with the notation and terminology.

scholarly journals Heegner Points and the Rank of Elliptic Curves over Large Extensions of Global Fields

2008 ◽  
Vol 60 (3) ◽  
pp. 481-490 ◽  
Author(s):  
Florian Breuer ◽  
Bo-Hae Im
Keyword(s):  
Elliptic Curves ◽  
Absolute Galois Group ◽  
Heegner Points ◽  
Global Function ◽  
Infinite Rank ◽  
Weil Group ◽  
Class Fields ◽  
Separable Closure ◽  
Totally Real ◽  
Real Number Field

AbstractLetkbe a global field,a separable closure ofk, andGkthe absolute Galois groupofoverk. For everyσ ∈ Gk, letbe the fixed subfield ofunderσ. LetE/kbe an elliptic curve overk. It is known that the Mordell–Weil grouphas infinite rank. We present a new proof of this fact in the following two cases. First, when k is a global function field of odd characteristic andEis parametrized by a Drinfeld modular curve, and secondly whenkis a totally real number field andE/kis parametrized by a Shimura curve. In both cases our approach uses the non-triviality of a sequence of Heegner points onEdefined over ring class fields.

scholarly journals Modularity of nearly ordinary 2-adic residually dihedral Galois representations

2014 ◽  
Vol 150 (8) ◽  
pp. 1235-1346 ◽  
Author(s):  
Patrick B. Allen
Keyword(s):  
Elliptic Curves ◽  
Galois Representations ◽  
Real Field ◽  
Two Dimensional ◽  
Hida Families ◽  
Totally Real ◽  
Totally Real Fields ◽  
Totally Real Field

AbstractWe prove modularity of some two-dimensional,$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$-adic Galois representations over a totally real field that are nearly ordinary at all places above$2$and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the$2$-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above $2$.

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