### On Tate-Shafarevich groups of abelian varieties, by Cristian D. Gonzalez-Aviles

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
${\cyr W}(A_{^{/\! K}})$ and ${\cyr W}(A_{^{/\! F}})$ 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 ${\cyr W}(A_{^{/\! K}})$
of $G$-invariant elements. As a corollary, we obtain a simple formula
relating the orders of ${\cyr W}(A_{^{/\! K}})$, ${\cyr W}(A_{^{/\! F}})$ and
${\cyr W}(A_{^{\,/\! F}}^{\chi})$ when $K/F$ is a quadratic extension and
$A^{\chi}$ is the twist of $A$ by the non-trivial character $\chi$ of
$G$.

Cristian D. Gonzalez-Aviles <cgonzale@abello.dic.uchile.cl>