ON TATE-SHAFAREVICH GROUPS OVER CYCLIC EXTENSIONS

• Journal title : Honam Mathematical Journal
• Volume 32, Issue 1,  2010, pp.45-51
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2010.32.1.045
Title & Authors
ON TATE-SHAFAREVICH GROUPS OVER CYCLIC EXTENSIONS
Yu, Ho-Seog;

Abstract
Let A be an abelian variety defined over a number field K and let L be a cyclic extension of K with Galois group G = <$\small{{\sigma}}$> of order n. Let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and of A over L. Assume III(A/L) is finite. Let M(x) be a companion matrix of 1+x+$\small{{\cdots}}$+$\small{x^{n-1}}$ and let $\small{A^x}$ be the twist of $\small{A^{n-1}}$ defined by $\small{f^{-1}{\circ}f^{\sigma}}$ = M(x) where $\small{f:A^{n-1}{\rightarrow}A^x}$ is an isomorphism defined over L. In this paper we compute [III(A/K)][III($\small{A^x}$/K)]/[III(A/L)] in terms of cohomology, where [X] is the order of an finite abelian group X.
Keywords
Tate-Shafarevich group;corestriction map;transgression map;Kronecker product;
Language
English
Cited by
1.
ON THE TATE-SHAFAREVICH GROUPS OVER BIQUADRATIC EXTENSIONS,;

호남수학학술지, 2015. vol.37. 1, pp.1-6
1.
ON THE TATE-SHAFAREVICH GROUPS OVER DEGREE 3 NON-GALOIS EXTENSIONS, Honam Mathematical Journal, 2016, 38, 1, 85
2.
ON THE TATE-SHAFAREVICH GROUPS OVER BIQUADRATIC EXTENSIONS, Honam Mathematical Journal, 2015, 37, 1, 1
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