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Visualising Sha[2] in Abelian surfaces

Author: Nils Bruin
Translated by:
Journal: Math. Comp. 73 (2004), 1459-1476
MSC (2000): Primary 11G05; Secondary 14G05, 14K15
Published electronically: January 8, 2004
MathSciNet review: 2047096
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Abstract: Given an elliptic curve $E_1$ over a number field $K$ and an element $s$ in its $2$-Selmer group, we give two different ways to construct infinitely many Abelian surfaces $A$ such that the homogeneous space representing $s$ occurs as a fibre of $A$ over another elliptic curve $E_2$. We show that by comparing the $2$-Selmer groups of $E_1$, $E_2$ and $A$, we can obtain information about $\Sh (E_1/K)[2]$ and we give examples where we use this to obtain a sharp bound on the Mordell-Weil rank of an elliptic curve.

As a tool, we give a precise description of the $m$-Selmer group of an Abelian surface $A$ that is $m$-isogenous to a product of elliptic curves $E_1\times E_2$.

One of the constructions can be applied iteratively to obtain information about $\Sh(E_1/K)[2^n]$. We give an example where we use this iterated application to exhibit an element of order $4$ in $\Sh(E_1/\mathbb{Q} )$.

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Additional Information

Nils Bruin
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

Keywords: Visualisation, Shafarevich-Tate group, elliptic curve, two-descent, Mordell-Weil group
Received by editor(s): February 2, 2002
Received by editor(s) in revised form: September 13, 2002
Published electronically: January 8, 2004
Additional Notes: The research in this paper was funded by the Pacific Institute for the Mathematical Sciences, Simon Fraser University, the University of British Columbia, and the School of Mathematics of the University of Sydney
Article copyright: © Copyright 2004 American Mathematical Society

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