Visualising Sha[2] in Abelian surfaces

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 $\Sh(E_1/\mathbb{Q} )$.