Examples of non-simple abelian surfaces over the rationals with non-square order Tate-Shafarevich group (Q404322)

In this paper the author constructs examples of non-simple abelian surfaces \(B/\mathbb{Q}\) such that the Tate-Shafarevich group of \(B\) is finite and has order \(k\) times a square, where \(k\in\{1,2,3,5,6,7,10,13\}\). The starting point is a pair of elliptic curves \(E,E'\) such that \(A=E\times E'\) has a Galois invariant subgroup \(G\) of order a squarefree multiple of \(k\) and such that \(B=A/G\) is not principally polarized.NEWLINENEWLINEIf the Tate-Shafarevich group of \(A\) is finite then a formula due to Cassels and Tate relates the quotient \(\#\Sha(A)/\#\Sha(B)\) with the Tamagawa numbers, periods, the regulators and the size of the torsion subgroups of the abelian surfaces \(A\) and \(B\). (This formula of Cassels and Tate implies that if the Birch and Swinnerton-Dyer conjecture holds for an abelian variety then it holds for any variety isogenous to it.)NEWLINENEWLINEFor each of the value of \(k\) the author finds an example where this quotient is non-square. Most of the paper deals with actually determining each of the entries in this formula.