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

Semistable elliptic curve
A semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type. Suppose $E$ is an elliptic curve defined over the rational number field $$\mathbb{Q}$$. It is known that there is a finite, non-empty set S of prime numbers $p$ for which $E$ has bad reduction modulo $p$. The latter means that the curve $$E_p$$ obtained by reduction of $E$ to the prime field with $p$ elements has a singular point. Roughly speaking, the condition of multiplicative reduction amounts to saying that the singular point is a double point, rather than a cusp. Deciding whether this condition holds is effectively computable by Tate's algorithm. Therefore in a given case it is decidable whether or not the reduction is semistable, namely multiplicative reduction at worst.

The semistable reduction theorem for $E$ may also be made explicit: $E$ acquires semistable reduction over the extension of $F$ generated by the coordinates of the points of order 12.