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The order of the Tate-Shafarevich group of the Jacobian of a curve 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 2020-10-09 18:21:28
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