On Tate-Shafarevich groups of abelian varieties

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 $G$.