Geodesic
Computing the Cassels–Tate pairing on 3-isogeny Selmer groups via cubic norm equations
Monique van Beek 1   ; Tom Fisher 2
1 Canons Village 1 46 Global edu-ro 145beon-gil Daejeong-eup, Seogwipo-si Jeju-do, 63644, Republic of Korea
2 DPMMS, Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB, UK
Acta Arithmetica, Tome 185 (2018) no. 4, pp. 367-396 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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 $13$.
DOI : 10.4064/aa171108-11-4
Keywords: explain method computing cassels tate pairing isogeny selmer groups elliptic curve improves upper bound rank elliptic curve coming descent isogeny coming full descent ingredient work algorithm solving cubic norm equations avoids s unit computations application elliptic curves torsion subgroup order rank least found eroshkin have rank exactly

Monique van Beek  1   ; Tom Fisher  2

1 Canons Village 1 46 Global edu-ro 145beon-gil Daejeong-eup, Seogwipo-si Jeju-do, 63644, Republic of Korea
2 DPMMS, Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB, UK 
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     title = {Computing the {Cassels{\textendash}Tate} pairing on 3-isogeny {Selmer} groups via cubic norm equations},
     journal = {Acta Arithmetica},
     pages = {367--396},
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Monique van Beek; Tom Fisher. Computing the Cassels–Tate pairing on 3-isogeny Selmer groups via cubic norm equations. Acta Arithmetica, Tome 185 (2018) no. 4, pp. 367-396. doi: 10.4064/aa171108-11-4

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