Chow–Heegner Points on CM Elliptic Curves and Values of p-adic L-functions

2012 ◽  
Vol 2014 (3) ◽  
pp. 745-793 ◽  
Author(s):  
Massimo Bertolini ◽  
Henri Darmon ◽  
Kartik Prasanna
Keyword(s):  
Elliptic Curves ◽  
Heegner Points

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 Heegner points and the arithmetic of elliptic curves over ring class extensions

2012 ◽  
Vol 132 (8) ◽  
pp. 1707-1719 ◽  
Author(s):  
Robert Bradshaw ◽  
William Stein
Keyword(s):  
Elliptic Curves ◽  
Heegner Points

scholarly journals ON THE -ADIC VARIATION OF HEEGNER POINTS

2019 ◽  
Vol 19 (6) ◽  
pp. 2127-2164 ◽  
Author(s):  
Francesc Castella
Keyword(s):  
Elliptic Curves ◽  
Selmer Groups ◽  
Direct Relation ◽  
Heegner Points ◽  
Reciprocity Law ◽  
Heegner Cycles ◽  
Hida Families ◽  
Explicit Reciprocity Law

In this paper, we prove an ‘explicit reciprocity law’ relating Howard’s system of big Heegner points to a two-variable $p$-adic $L$-function (constructed here) interpolating the $p$-adic Rankin $L$-series of Bertolini–Darmon–Prasanna in Hida families. As applications, we obtain a direct relation between classical Heegner cycles and the higher weight specializations of big Heegner points, refining earlier work of the author, and prove the vanishing of Selmer groups of CM elliptic curves twisted by 2-dimensional Artin representations in cases predicted by the equivariant Birch and Swinnerton-Dyer conjecture.

Sign in / Sign up

Export Citation Format

Share Document