# ON THE TATE-SHAFAREVICH GROUPS OVER BIQUADRATIC EXTENSIONS

Yu, Hoseog

• Accepted : 2014.12.10
• Published : 2015.03.25
• 32 11

#### Abstract

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.

#### Keywords

Tate-Shafarevich group;abelian varieties;restriction of scalars

#### References

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