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Tate-Shafarevich group

For \(A\) an abelian variety or a group scheme define over a field \(k\), this is the group of 1-cocycles in \(H^1(G_K; X)\) which become boundaries at every place: \begin{align*}\sha(X_{/ {k}} ) = \cap_{v\in \mathrm{Pl}\qty{K} } \ker \qty{ H^1(G_k; X) \to H^1(G_{{ k_{\widehat{v}} }}; X)}\end{align*} Measures the extent to which the Hasse principle holds for equations with coefficients in \(k\).

\(\sha(K)\) is precisely the group of \(K\)-torsors under \(E\) that fail the local-global principle.

For \(X = A[n]\) the \(n{\hbox{-}}\)torsion on an abelian variety, related to the Selmer group:

Conjecture

• Conjecture: \begin{align*}{\sharp}\sha(X_{/ {k}} ) < \infty\end{align*}
• Known for some $E \in \mathrm{Ell} $ with CM and \(\operatorname{rank}E \leq 1\).

Consequences