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The Tate-Shafarevic group Ш of an abelian variety $A$ over a number field is a torsion group that classifies the principal homogenous spaces of $A$ and can be defined in terms of Galois cohomology; it is conjectured to be finite, but in most cases this is not known.

When $A$ is the Jacobian of a curve then the order of Ш is either a square, or twice a square (assuming it is finite); this is a theorem of Poonen and Stoll [arXiv:math/9911267, 10.2307/121064]. For elliptic curves, only the square case can occur, but for Jacobians of genus 2 curves, both cases are known to occur.

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• Last edited by Andrew Sutherland on 2020-01-07 06:14:59
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