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

TL;DR

This paper constructs examples of non-simple, non-principally polarised abelian surfaces over rationals with Tate-Shafarevich groups of non-square order, exploring their properties and invariance under isogeny.

Contribution

It provides explicit examples of abelian surfaces with non-square Tate-Shafarevich group orders, extending understanding beyond the principally polarised case.

Findings

Constructed abelian surfaces with Tate-Shafarevich group order k times a square for specific k.

Demonstrated invariance of the BSD conjecture under isogeny for these surfaces.

Extended the known cases of Tate-Shafarevich group orders beyond the square or twice a square.

Abstract

Let A be an abelian surface over a fixed number field. If A is principally polarised, then it is known that the order of the Tate-Shafarevich group of A must, if finite, be a square or twice a square. The situation for A not principally polarised remains unclear. For each k in {1,2,3,5,6,7,10,13} we construct a non-simple non-principally polarised abelian surface over the rationals whose Tate-Shafarevich group has order k times a square. To obtain this result, we explore the invariance under isogeny of the Birch and Swinnerton-Dyer conjecture.