Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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ON TATE-SHAFAREVICH GROUPS OVER CYCLIC EXTENSIONS

  • Yu, Ho-Seog (Department of Applied Mathematics, Sejong University)
  • Received : 2009.12.03
  • Accepted : 2010.01.15
  • Published : 2010.03.25
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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 = <${\sigma}$> 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+${\cdots}$+$x^{n-1}$ and let $A^x$ be the twist of $A^{n-1}$ defined by $f^{-1}{\circ}f^{\sigma}$ = M(x) where $f:A^{n-1}{\rightarrow}A^x$ is an isomorphism defined over L. In this paper we compute [III(A/K)][III($A^x$/K)]/[III(A/L)] in terms of cohomology, where [X] is the order of an finite abelian group X.

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