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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
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Is Ш a good parameter for the failure of Global-Local principle for abelian varieties?
(Comparing to class group cases: we have an isomorphism
$Cl(K)\rightarrow \prod \left(K^\times \backslash K_p^\times /O_p^\times \right)$ for a number field $K$.
Similarly, for an elliptic curve $E/ …