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 <math>A</math> defined over a field <math>F</math> with ring of integers <math>R</math>, consider the Néron model of <math>A</math>, which is a 'best possible' model of <math>A</math> defined over <math>R</math>. This model may be represented as a scheme over <math>\mathrm{Spec}(R)</math> (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism <math>\mathrm{Spec}(F) \to \mathrm{Spec}(R) </math> gives back <math>A</math>. The Néron model is a smooth group scheme, so we can consider <math>A^0</math>, 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 <math>k</math>, <math>A^0_k</math> is a group variety over <math>k</math>, hence an extension of an abelian variety by a linear group. If this linear group is an algebraic torus, so that <math>A^0_k</math> is a semiabelian variety, then <math>A</math> has semistable reduction at the prime corresponding to <math>k</math>. If <math>F</math> is a global field, then <math>A</math> 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 <math>F</math>.<ref>Grothendieck (1972) Théorème 3.6, p. 351</ref>
Semistable elliptic curveEdit
A semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type.<ref>Husemöller (1987) pp.116-117</ref> Suppose Template:Math is an elliptic curve defined over the rational number field <math>\mathbb{Q}</math>. It is known that there is a finite, non-empty set S of prime numbers Template:Math for which Template:Math has bad reduction modulo Template:Math. The latter means that the curve <math>E_p</math> obtained by reduction of Template:Math to the prime field with Template:Math 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.<ref>Husemoller (1987) pp.116-117</ref> Deciding whether this condition holds is effectively computable by Tate's algorithm.<ref>Husemöller (1987) pp.266-269</ref><ref name=TL/> 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 Template:Math may also be made explicit: Template:Math acquires semistable reduction over the extension of Template:Math generated by the coordinates of the points of order 12.<ref>This is implicit in Husemöller (1987) pp.117-118</ref><ref name=TL>Template:Citation </ref>