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Hasse Principle
Examples
Hasse principle
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2021-03-17
What is the Hasse principle?
- rational points
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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 \(k\).
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AWS 2022: Automorphic Forms Beyond \(\operatorname{GL}_2\)
The Hasse principle