Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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TATE-SHAFAREVICH GROUPS OVER THE COMMUTATIVE DIAGRAM OF 8 ABELIAN VARIETIES

  • Hoseog Yu (Department of Mathematics and Statistics, Sejong University)
  • Received : 2023.04.03
  • Accepted : 2023.03.17
  • Published : 2023.09.14
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Abstract

Suppose that there are 8 abelian varieties defined over a number field K which satisfy a commutative diagram. We show that if we know that three out of four short exact sequences satisfy the rate formula of Tate-Shafarevich groups, then the unknown short exact sequence satisfies the rate formula of Tate-Shafarevich groups, too.

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References

  1. J. S. Milne, On the arithmetic of abelian varieties, Inventiones Math. 17 (1972), 177-190.  https://doi.org/10.1007/BF01425446
  2. J. S. Milne, Arithmetic Duality Theorems, Perspectives in Math. Vol. 1. Academic Press Inc., 1986. 
  3. 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
  4. H. Yu, On the rate of Tate-Shafarevich groups over cyclic extensions of order p2, Honam Math. J. 36 (2014), 417-424. https://doi.org/10.5831/HMJ.2014.36.2.417