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On the index of the Heegner subgroup of elliptic curve  Carlos CastanoBernard
;  Date: 
3 Sep 2007  Abstract:  Let E be an elliptic curve of conductor N and rank one over Q. So there is a
nonconstant morphism X+0(N) > E defined over Q, where X+0(N) = X0(N)/wN and
wN is the Fricke involution of the modular curve X+0(N). Under this morphism
the traces of the Heegner points of X+0(N) map to rational points on E. In this
paper we study the index I of the subgroup generated by all these traces on
E(Q). We propose and also discuss a conjecture that says that if N is prime and
I > 1, then either the number of connected components of the real locus
X+0(N)(R) is greater than 1 or (less likely) the order S of the TateSafarevich
group is nontrivial. This conjecture is backed by computations performed on
each E that satisfies the above hypothesis in the range N < 129999. This paper
was prepared for the proceedings of the Conference on Algorithmic Number
Theory, Turku, May 811, 2007. We tried to make the paper as self contained as
possible.  Source:  arXiv, 0709.0132  Services:  Forum  Review  PDF  Favorites 


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