Computing the Cassels–Tate pairing on the 3-Selmer group of an elliptic curve-Reference-Cited by-同舟云学术

Computing the Cassels–Tate pairing on the 3-Selmer group of an elliptic curve

Published:2014-09-09 Issue:07 Volume:10 Page:1881-1907
ISSN:1793-0421
Container-title:International Journal of Number Theory
language:en
Short-container-title:Int. J. Number Theory

Author:

Fisher Tom¹,Newton Rachel²

Affiliation:

1. Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK

2. Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands

Abstract

We extend the method of Cassels for computing the Cassels–Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank of an elliptic curve obtained by 3-descent.

Publisher

World Scientific Pub Co Pte Lt

Subject

Algebra and Number Theory

Link

https://www.worldscientific.com/doi/pdf/10.1142/S1793042114500602

Reference34 articles.

1. Jacobians of Genus One Curves

2. On the Jacobians of plane cubics

3. Complex Abelian Varieties

4. The Magma Algebra System I: The User Language

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