ON THE RATIO OF TATE-SHAFAREVICH GROUPS OVER CYCLIC EXTENSIONS OF ORDER p2

• Journal title : Honam Mathematical Journal
• Volume 36, Issue 2,  2014, pp.417-424
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2014.36.2.417
Title & Authors
ON THE RATIO OF TATE-SHAFAREVICH GROUPS OVER CYCLIC EXTENSIONS OF ORDER p2
Yu, Hoseog;

Abstract
Let A be an abelian variety defined over a number field K and p be a prime. Define $\small{{\varphi}_i=(x^{p^i}-1)/(x^{p^{i-1}}-1)}$. Let $\small{A_{{\varphi}i}}$ be the abelian variety defined over K associated to the polynomial $\small{{\varphi}i}$ and let Ш($\small{A_{{\varphi}i}}$) denote the Tate-Shafarevich groups of $\small{A_{{\varphi}i}}$ over K. In this paper assuming Ш(A/F) is finite, we compute [Ш($\small{A_{{\varphi}1}}$)][Ш($\small{A_{{\varphi}2}}$)]/[Ш($\small{A_{{\varphi}1{\varphi}2}}$)] in terms of K-rational points of $\small{A_{{\varphi}i}}$, $\small{A_{{\varphi}1{\varphi}2}}$ and their dual varieties, where [X] is the order of a finite abelian group X.
Keywords
Tate-Shafarevich group;abelian varieties;cyclic extension;
Language
English
Cited by
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