Has square Ш (reviewed)

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.