  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).
