# ON THE TATE-SHAFAREVICH GROUPS OVER DEGREE 3 NON-GALOIS EXTENSIONS

• Yu, Hoseog (Department of Mathematics, Sejong University)
• Accepted : 2016.01.25
• Published : 2016.03.25
• 94 13

#### 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($A_{\varphi}/K$)]/[III(A/L)], where [X] is the order of a finite abelian group X.

#### Keywords

Tate-Shafarevich group;abelian varieties;restriction of scalars

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

#### References

1. M. I. Bashmakov, The cohomology of abelian varieties over a number field, Russian Math. Surveys 27, no. 6 (1972), 25-70. https://doi.org/10.1070/RM1972v027n06ABEH001392
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. https://doi.org/10.2307/121064
7. C. Riehm, The Corestriction of Algebraic Structures, Inven. Math. 11 (1970), 73-98. https://doi.org/10.1007/BF01389807
8. J. Tate, Relations between $K_2$ and Galois cohomology, Inventiones Math. 36 (1976), 257-274. https://doi.org/10.1007/BF01390012
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. https://doi.org/10.1007/BF02772219
13. 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