Visualising Sha[2] in Abelian Surfaces

Abstract

Given an elliptic curve E1 over a number field 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 E2. We show that by comparing the 2-Selmer groups of E1, E2 and A, we can obtain information about Sha(E1/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 E1 x E2. One of the constructions can be applied iteratively to obtain information about Sha(E1/K)[2^n]. We give an example where we use this iterated application to exhibit an element of order 4 in Sha(E1/Q).