Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Visualising Sha[2] in Abelian Surfaces   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).

8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
26
The categories of this paper are: bayes-an