Computing the Cassels-Tate pairing on 3-isogeny Selmer groups via cubic norm equations
Accepted version
Peer-reviewed
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Authors
van Beek, Monique
Fisher, Tom
Abstract
We explain a method for computing the Cassels-Tate pairing on the 3-isogeny Selmer groups of an elliptic curve. This improves the upper bound on the rank of the elliptic curve coming from a descent by 3-isogeny, to that coming from a full 3-descent. One ingredient of our work is a new algorithm for solving cubic norm equations, that avoids the need for any S-unit computations. As an application, we show that the elliptic curves with torsion subgroup of order 3 and rank at least 13, found by Eroshkin, have rank exactly 13.
Description
Keywords
elliptic curves, Cassels-Tate pairing, descent, norm equations
Journal Title
ACTA ARITHMETICA
Conference Name
Journal ISSN
0065-1036
1730-6264
1730-6264
Volume Title
185
Publisher
Institute of Mathematics, Polish Academy of Sciences