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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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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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