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

2012/06/08 by Keil, Stefan
#11G07 (Primary) 11G35 (Secondary) #11G10 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1206.1822

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.

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