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.
- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-01-07 06:14:59
- 2020-01-07 06:14:59 by Andrew Sutherland (Reviewed)
- 2020-01-06 17:11:14 by Andrew Sutherland
- 2018-05-23 16:56:15 by John Cremona (Reviewed)