Semistable abelian variety

In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field.

For an abelian variety ${\displaystyle A}$ defined over a field ${\displaystyle F}$ with ring of integers ${\displaystyle R}$, consider the Néron model of ${\displaystyle A}$, which is a 'best possible' model of ${\displaystyle A}$ defined over ${\displaystyle R}$. This model may be represented as a scheme over ${\displaystyle \mathrm {Spec} (R)}$ (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism ${\displaystyle \mathrm {Spec} (F)\to \mathrm {Spec} (R)}$ gives back ${\displaystyle A}$. The Néron model is a smooth group scheme, so we can consider ${\displaystyle A^{0}}$, the connected component of the Néron model which contains the identity for the group law. This is an open subgroup scheme of the Néron model. For a residue field ${\displaystyle k}$, ${\displaystyle A_{k}^{0}}$ is a group variety over ${\displaystyle k}$, hence an extension of an abelian variety by a linear group. If this linear group is an algebraic torus, so that ${\displaystyle A_{k}^{0}}$ is a semiabelian variety, then ${\displaystyle A}$ has semistable reduction at the prime corresponding to ${\displaystyle k}$. If ${\displaystyle F}$ is a global field, then ${\displaystyle A}$ is semistable if it has good or semistable reduction at all primes.

The fundamental semistable reduction theorem of Alexander Grothendieck states that an abelian variety acquires semistable reduction over a finite extension of ${\displaystyle F}$.