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ON TATE-SHAFAREVICH GROUPS OVER CYCLIC EXTENSIONS
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  • 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;
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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 = <> 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++ and let be the twist of defined by = M(x) where is an isomorphism defined over L. In this paper we compute [III(A/K)][III(/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 crossref(new window)
1.
ON THE TATE-SHAFAREVICH GROUPS OVER DEGREE 3 NON-GALOIS EXTENSIONS, Honam Mathematical Journal, 2016, 38, 1, 85  crossref(new windwow)
2.
ON THE TATE-SHAFAREVICH GROUPS OVER BIQUADRATIC EXTENSIONS, Honam Mathematical Journal, 2015, 37, 1, 1  crossref(new windwow)
 References
1.
K. S. Brown, Cohomology of groups, Grad. Texts in Math. 87. Springer-Verlag 1982.

2.
K. Cartan and S. Eilenberg, Homological algebra, Princeton University Press 1956.

3.
C. D. Gonzalez-Aviles, On Tate-Shafarevich groups of abelian varieties, Proc. Amer. Math. Soc. 128 (2000), 953-961. crossref(new window)

4.
G. Hochschild and J-P. Serre, Cohomology of Group Extension, Trans. Amer. Math. Soc. 74 (1953), 110-134 crossref(new window)

5.
J. S. Milne, On the arithmetic of abelian varieties, Inventiones Math. 17 (1972), 177-190. crossref(new window)

6.
J. S. Milne, Arithmetic Duality Theorems, Perspectives in Math. vol. 1. Academic Press Inc. 1986.

7.
Hwasln Park, Idempotent relations and the conjecture of Birch and Swinnerton-Dyer, In: Algebra and Topology 1990 (Taejon, 1990), 97-125.

8.
Carl Riehm, The Corestriction of Algebraic Structures, Inven. Math. 11 (1970), 73-98. crossref(new window)

9.
L. Solomon, Similarity of the companion matrix and its transpose, Linear Algebra Appl. 302/303 (1999), 555-561. crossref(new window)

10.
J. Tate, Relations between $K_2$ and Galois cohomology, Inventiones Math. 36 (1976), 257-274. crossref(new window)

11.
H. Yu, On Tate-Shafarevich groups over Galois extensions, Israel Journal of Math. 141 (2004), 211-220. crossref(new window)