Local units, elliptic units, Heegner points and elliptic curves

1987 ◽  
Vol 88 (2) ◽  
pp. 405-422 ◽  
Author(s):  
Karl Rubin
Keyword(s):  
Elliptic Curves ◽  
Heegner Points ◽  
Elliptic Units

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 On the units generated by Weierstrass forms

2014 ◽  
Vol 17 (A) ◽  
pp. 303-313
Author(s):  
Ömer Küçüksakallı
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Curves Over Finite Fields ◽  
Elliptic Units ◽  
Fast Arithmetic

AbstractThere is an algorithm of Schoof for finding divisors of class numbers of real cyclotomic fields of prime conductor. In this paper we introduce an improvement of the elliptic analogue of this algorithm by using a subgroup of elliptic units given by Weierstrass forms. These elliptic units which can be expressed in terms of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}x$-coordinates of points on elliptic curves enable us to use the fast arithmetic of elliptic curves over finite fields.

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.

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