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.
spectrum of a ring) for which the generic fibre constructed by means of the morphism
The Néron model is a smooth group scheme, so we can consider
, 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.
, hence an extension of an abelian variety by a linear group.
If this linear group is an algebraic torus, so that
The fundamental semistable reduction theorem of Alexander Grothendieck states that an abelian variety acquires semistable reduction over a finite extension of
[1] A semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type.
[2] Suppose E is an elliptic curve defined over the rational number field
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
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.
[3] Deciding whether this condition holds is effectively computable by Tate's algorithm.
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.