On Tate-Shafarevich groups of abelian varieties
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- by Cristian D. Gonzalez-Avilés PDF
- Proc. Amer. Math. Soc. 128 (2000), 953-961 Request permission
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 $\Shcha (A_{^{/ K}})$ and $\Shcha (A_{^{/ F}})$ 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 $\Shcha (A_{^{/ K}})$ of $G$-invariant elements. As a corollary, we obtain a simple formula relating the orders of $\Shcha (A_{^{/ K}})$, $\Shcha (A_{^{/ F}})$ and $\Shcha (A_{^{ / F}}^{\chi })$ when $K/F$ is a quadratic extension and $A^{\chi }$ is the twist of $A$ by the non-trivial character $\chi$ of $G$.References
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Additional Information
- Cristian D. Gonzalez-Avilés
- Affiliation: Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile
- Email: cgonzale@abello.dic.uchile.cl
- Received by editor(s): May 18, 1998
- Published electronically: September 23, 1999
- Additional Notes: The author was supported by Fondecyt grant 1981175.
- Communicated by: David E. Rohrlich
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 953-961
- MSC (1991): Primary 11G40, 11G05
- DOI: https://doi.org/10.1090/S0002-9939-99-05244-2
- MathSciNet review: 1653469
Dedicated: To Ricardo Baeza with gratitude