# Galois theory, discriminants and torsion subgroup of elliptic curves

@article{GarcaSelfa2010GaloisTD,
title={Galois theory, discriminants and torsion subgroup of elliptic curves},
author={Irene Garc{\'i}a-Selfa and Enrique Gonz{\'a}lez-Jim{\'e}nez and Jos{\'e} M. Tornero},
journal={Journal of Pure and Applied Algebra},
year={2010},
volume={214},
pages={1340-1346}
}
• Published 10 November 2008
• Mathematics
• Journal of Pure and Applied Algebra
7 Citations
On the ubiquity of trivial torsion on elliptic curves
• Mathematics
• 2010
The purpose of this paper is to give a down-to-earth proof of the well-known fact that a randomly chosen elliptic curve over the rationals is most likely to have trivial torsion.
Arithmetic Algebraic Geometry
• Mathematics
• 2015
[3] , Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero, Math. Finiteness results for modular curves of genus at least 2, Amer.
Torsion of rational elliptic curves over quadratic fields
• Mathematics
• 2014
Let $$E$$E be an elliptic curve defined over $${\mathbb {Q}}$$Q. We study the relationship between the torsion subgroup $$E({\mathbb {Q}})_{{{\mathrm{tors}}}}$$E(Q)tors and the torsion subgroup

## References

SHOWING 1-10 OF 49 REFERENCES
A complete diophantine characterization of the rational torsion of an elliptic curve
• Mathematics
• 2007
We give a complete characterization for the rational torsion of an elliptic curve in terms of the (non-)existence of integral solutions of a system of diophantine equations.
Computing the Rational Torsion of an Elliptic Curve Using Tate Normal Form
• Mathematics
• 2002
It is a classical result (apparently due to Tate) that all elliptic curves with a torsion point of order n(4⩽n⩽10, or n=12) lie in a one-parameter family. However, this fact does not appear to have
Simplest Cubic Number Fields
• Mathematics
• 2012
In this paper we intend to show that certain integers do not occur as the norms of principal ideals in a family of cubic fields studied by Cohn, Shanks, and Ennola. These results will simplify the
Abelian L-adic representation and elliptic curves
This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent
Integral Points on Elliptic Curves Defined by Simplest Cubic Fields
The complete list of integral points on elliptic curves of the form y2 = f(X) is computed and it is proved that this list is exhaustive by using the methods of Tzanakis and de Weger, together with bounds on linear forms in elliptic logarithms due to S. David.
Torsion subgroups of elliptic curves in short Weierstrass form
• Mathematics
• 2005
In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves E/Q in short Weierstrass form, subject to certain inequalities for their
The simplest cubic fields
Abstract. The cyclic cubic fields generated by x3 = ax2 + (a + 3)x + 1 are studied in detail. The regulators are relatively small and are known at once. The class numbers are al2 2 ways of the form A
Diophantine analysis and torsion on elliptic curves
In a recent paper of Bennett and the author, it was shown that the elliptic curve defined by y2 = x3 + Ax + B, where A and B are integers, has no rational points of finite order if A is sufficiently
Algorithms for Modular Elliptic Curves
This book presents a thorough treatment of many algorithms concerning the arithmetic of elliptic curves with remarks on computer implementation and an extensive set of tables giving the results of the author's implementations of the algorithms.