호남수학학술지 (Honam Mathematical Journal)

  • 제32권1호
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  • Pages.45-51
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  • 2010
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  • 1225-293X(pISSN)
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  • 2288-6176(eISSN)
호남수학회 (The Honam Mathematical Society)
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ON TATE-SHAFAREVICH GROUPS OVER CYCLIC EXTENSIONS

  • Yu, Ho-Seog (Department of Applied Mathematics, Sejong University)
  • 투고 : 2009.12.03
  • 심사 : 2010.01.15
  • 발행 : 2010.03.25
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초록

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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참고문헌

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피인용 문헌

  1. ON THE TATE-SHAFAREVICH GROUPS OVER DEGREE 3 NON-GALOIS EXTENSIONS vol.38, pp.1, 2016, https://doi.org/10.5831/HMJ.2016.38.1.85
  2. ON THE TATE-SHAFAREVICH GROUPS OVER BIQUADRATIC EXTENSIONS vol.37, pp.1, 2015, https://doi.org/10.5831/HMJ.2015.37.1.1