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

JOURNAL OF NUMBER THEORY

卷:144

Article

Keil, Stefan

关键词: Abelian surface;    Tate-Shafarevich group;    Cassels-Tate equation;   

DOI  :  10.1016/j.jnt.2014.04.018

来源: Elsevier

【 摘 要 】

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. For each k is an element of {1,2,3,5,6,7,10,13} we construct a non-simple non-principally polarised abelian surface B/Q having Tate-Shafarevich group of order k times a square. To obtain this result, we explore the invariance under isogeny of the Birch and Swinnerton-Dyer conjecture. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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