The **2-torsion field** of an abelian variety is the minimal field extension over which the 2-torsion subgroup of its Mordell-Weil group is rational; it is a Galois extension of the field over which the abelian variety is defined.

The 2-torsion field of the Jacobian of a hyperelliptic curve of the form $y^2=f(x)$ is the same as the splitting field of the polynomial $f(x)$.

For abelian varieties over number fields, the 2-torsion field is specified by giving a number field of minimal degree and absolute discriminant whose Galois closure is equal to the 2-torsion field.

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- Last edited by Andrew Sutherland on 2018-06-08 05:30:07

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- 2018-06-08 05:30:07 by Andrew Sutherland (Reviewed)