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On Tate-Shafarevich groups
of abelian varieties

Author: Cristian D. Gonzalez-Avilés
Journal: Proc. Amer. Math. Soc. 128 (2000), 953-961
MSC (1991): Primary 11G40, 11G05
Published electronically: September 23, 1999
MathSciNet review: 1653469
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Abstract: Let $K/F$ be a finite Galois extension of number fields with Galois group $G$, let $A$ be an abelian variety defined over $F$, and let ${\Russian W}(A_{^{/\! K}})$ and ${\Russian W}(A_{^{/\! F}})$ denote, respectively, the Tate-Shafarevich groups of $A$ over $K$ and of $A$ over $F$. Assuming that these groups are finite, we derive, under certain restrictions on $A$ and $K/F$, a formula for the order of the subgroup of ${\Russian W}(A_{^{/\! K}})$ of $G$-invariant elements. As a corollary, we obtain a simple formula relating the orders of ${\Russian W}(A_{^{/\! K}})$, ${\Russian W}(A_{^{/\! F}})$ and ${\Russian W}(A_{^{\,/\! F}}^{\chi })$ when $K/F$ is a quadratic extension and $A^{\chi }$ is the twist of $A$ by the non-trivial character $\chi $ of $G$.

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  • [1] Atiyah, M. and Wall, C.T.C., Cohomology of groups, in: Algebraic Number Theory (J.W.S. Cassels and A. Fröhlich, Eds.), pp. 94-115, Academic Press, London, 1967. MR 36:2593.
  • [2] Cartan, H. and Eilenberg, S., Homological Algebra, Princeton University Press, Princeton, N.J., 1956. MR 17:1040e
  • [3] Chamfy, C., Modules semi-locaux, In: Cohomologie Galoisienne des Modules Finis (Séminaire de l'Inst. de Math. de Lille sous la direction de G. Poitou), Dunod, Paris, 1967.
  • [4] Gonzalez-Avilés, C.D., On the conjecture of Birch and Swinnerton-Dyer, Trans. Amer. Math. Soc. 349 (1997), 4181-4200. MR 98c:11062
  • [5] Hochschild, G.P. and Serre, J-P., Cohomology of group extensions, Trans. Amer. Math. Soc. 74 (1953), 110-134. MR 14:619b
  • [6] Kolyvagin, V.A., Finiteness of $E(\mathbb{Q})$ and ${\Russian W}(E,\mathbb{Q})$ for a class of Weil curves, Math. USSR, Izv. 32 (1989), 523-542. MR 89m:11056
  • [7] Mazur, K., Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183-266. MR 56:3020
  • [8] Milne, J.S., Arithmetic Duality Theorems, Academic Press, Orlando, FL, 1986. MR 88e:14028
  • [9] Milne, J.S., On the arithmetic of abelian varieties, Invent. Math. 17 (1972), 177-190. MR 48:8512
  • [10] O'Meara, O.T., Introduction to Quadratic Forms, Third Corrected Printing, Springer-Verlag, Berlin, 1973.
  • [11] Rubin, K., On Tate-Shafarevich groups and $L$-functions of elliptic curves with complex multiplication, Invent. Math. 89 (1987), 527-560. MR 89a:11065
  • [12] Schaefer, E.F., Class groups and Selmer groups, J. Number Theory 56 (1996), 79-114. MR 97e:11068
  • [13] Serre, J-P., Local Fields, Grad. Texts in Math. 67, Springer-Verlag, New York, 1979. MR 82e:12016
  • [14] Silverman, J., Advanced Topics in the Arithmetic of Elliptic Curves., Grad. Texts in Math. 151, Springer-Verlag, New York, 1994. MR 96b:11074
  • [15] Tate, J., WC-groups over $\mathfrak{p}$-adic fields, Séminaire Bourbaki, Exposé 156 (1957/58). MR 21:4162

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Additional Information

Cristian D. Gonzalez-Avilés
Affiliation: Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile

Received by editor(s): May 18, 1998
Published electronically: September 23, 1999
Additional Notes: The author was supported by Fondecyt grant 1981175.
Dedicated: To Ricardo Baeza with gratitude
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2000 American Mathematical Society

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