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ON THE TATE-SHAFAREVICH GROUPS OVER DEGREE 3 NON-GALOIS EXTENSIONS
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  • Journal title : Honam Mathematical Journal
  • Volume 38, Issue 1,  2016, pp.85-93
  • Publisher : The Honam Mathematical Society
  • DOI : 10.5831/HMJ.2016.38.1.85
 Title & Authors
ON THE TATE-SHAFAREVICH GROUPS OVER DEGREE 3 NON-GALOIS EXTENSIONS
Yu, Hoseog;
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 Abstract
Let A be an abelian variety defined over a number field K and let L be a degree 3 non-Galois extension of K. Let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Assuming that III(A/L) is finite, we compute [III(A/K)][III()]/[III(A/L)], where [X] is the order of a finite abelian group X.
 Keywords
Tate-Shafarevich group;abelian varieties;restriction of scalars;
 Language
English
 Cited by
 References
1.
M. I. Bashmakov, The cohomology of abelian varieties over a number field, Russian Math. Surveys 27, no. 6 (1972), 25-70. crossref(new window)

2.
K. S. Brown, Cohomology of groups, Grad. Texts in Math. 87. Springer-Verlag 1982.

3.
J. W. S. Cassels, Arithmetic on curves of genus 1. VII. The dual exact sequence, J. Reine Angrew. Math. 216 (1964), 150-158.

4.
J. W. S. Cassels, Arithmetic on curves of genus 1. VIII. On the conjectures of Birch and Swinnerton-Dyer, J. Reine Angrew. Math. 217 (1965), 180-189.

5.
J. S. Milne, Arithmetic Duality Theorems, Perspectives in Math. vol. 1. Academic Press Inc. 1986.

6.
B. Poonen and M. Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. Math. 150 (1999), 1109-1149. crossref(new window)

7.
C. Riehm, The Corestriction of Algebraic Structures, Inven. Math. 11 (1970), 73-98. crossref(new window)

8.
J. Tate, Relations between $K_2$ and Galois cohomology, Inventiones Math. 36 (1976), 257-274. crossref(new window)

9.
J. Tate, WC-group over p-adic fields, In: Seminaire Bourbaki, 1957-58, expose 156.

10.
J. Tate, Duality theorem in Galois cohomology over number fields, Proc. Int. Cong. Math., Stockholm (1962), 288-295.

11.
A. Weil, Adeles and algebraic groups, Progrss in Math. 23. Birkhauser 1982.

12.
H. Yu, On Tate-Shafarevich groups over Galois extensions, Israel J. Math. 141 (2004), 211-220. crossref(new window)

13.
H. Yu, On Tate-Shafarevich groups over cyclic extensions, Honam Math. J. 32 (2010), 45-51. crossref(new window)