Has square Ш (reviewed)

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.

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. For elliptic curves, only the square case can occur, but for Jacobians of genus 2 curves, both cases are known to occur.