호남수학학술지 (Honam Mathematical Journal)

  • 제37권1호
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  • Pages.1-6
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  • 2015
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  • 1225-293X(pISSN)
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  • 2288-6176(eISSN)
호남수학회 (The Honam Mathematical Society)
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ON THE TATE-SHAFAREVICH GROUPS OVER BIQUADRATIC EXTENSIONS

  • Yu, Hoseog (Department of Mathematics, Sejong University)
  • 투고 : 2014.11.06
  • 심사 : 2014.12.10
  • 발행 : 2015.03.25
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초록

Let A be an abelian variety defined over a number field K. Let L be a biquadratic extension of K with Galois group G and let III (A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Assuming III(A/L) is finite, we compute [III(A/K)]/[III(A/L)] where [X] is the order of a finite abelian group X.

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참고문헌

  1. K. S. Brown, Cohomology of groups, Grad. Texts in Math. 87. Springer-Verlag 1982.
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  3. J. S. Milne, Arithmetic Duality Theorems, Perspectives in Math. vol. 1. Academic Press Inc. 1986.
  4. J. Tate, Relations between $K_2$ and Galois cohomology, Inventiones Math. 36 (1976), 257-274. https://doi.org/10.1007/BF01390012
  5. J. Tate, Duality theorem in Galois cohomology over number fields, Proc. Int. Cong. Math., Stockholm (1962), 288-295.
  6. A. Weil, Adeles and algebraic groups, Progrss in Math. 23. Birkhauser 1982.
  7. H. Yu, On Tate-Shafarevich groups over Galois extensions, Israel J. Math. 141 (2004), 211-220. https://doi.org/10.1007/BF02772219
  8. H. Yu, On Tate-Shafarevich groups over cyclic extensions, Honam Math. J. 32 (2010), 45-51. https://doi.org/10.5831/HMJ.2010.32.1.045