On Tate-Shafarevich groups over Galois extensions
Let \(A\) be an abelian variety defined over a number field \(K\), let \(L\supset K\) be a finite Galois extension with Galois group \(G\), let \(\chi\) be a non-trivial character of \(G\) and let \(A^{\chi}\) be the twist of \(A\) by \(\chi\). The main result of this article, whose truth depends on the finiteness of Shafarevich-Tate groups involved (in general, this property is not checked yet) is a formula expressing the ratio of the product of the orders of the Shafarevich-Tate groups of \(A\) and \(A^\chi\) over \(K\), and the order of the Shafarevich-Tate group of \(A\) over \(L\). The author proves that: \[ \frac{| \text{Sh}(A/K)| | \text{Sh}(A^\chi/K)| }{| \text{Sh}(A/L)| }= \frac{| \widehat{H}^0(G,A'(L))| | H^1(G,A(L))| }{\prod_{v\in M_K}| H^1(G_{v_L},A(L_{v_L}))| }, \] where \(A'\) is the dual of \(A\) and \(v_{L}\) is a fixed place above \(v\) for each \(v\in M_K\) place of \(K\). The product is well defined because all but finitely many factors are \(\not=1\). Both sides of the equality are powers of \(2\).
scientific article; zbMATH DE number 2114649
scientific article