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]\displaystyle{ A }[/math] defined over a field [math]\displaystyle{ F }[/math] with ring of integers [math]\displaystyle{ R }[/math], consider the Néron model of [math]\displaystyle{ A }[/math], which is a 'best possible' model of [math]\displaystyle{ A }[/math] defined over [math]\displaystyle{ R }[/math]. This model may be represented as a scheme over [math]\displaystyle{ \mathrm{Spec}(R) }[/math] (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism [math]\displaystyle{ \mathrm{Spec}(F) \to \mathrm{Spec}(R) }[/math] gives back [math]\displaystyle{ A }[/math]. The Néron model is a smooth group scheme, so we can consider [math]\displaystyle{ 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]\displaystyle{ k }[/math], [math]\displaystyle{ A^0_k }[/math] is a group variety over [math]\displaystyle{ k }[/math], hence an extension of an abelian variety by a linear group. If this linear group is an algebraic torus, so that [math]\displaystyle{ A^0_k }[/math] is a semiabelian variety, then [math]\displaystyle{ A }[/math] has semistable reduction at the prime corresponding to [math]\displaystyle{ k }[/math]. If [math]\displaystyle{ F }[/math] is a global field, then [math]\displaystyle{ 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]\displaystyle{ F }[/math].[1]
Semistable elliptic curve
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 [math]\displaystyle{ \mathbb{Q} }[/math]. 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 [math]\displaystyle{ E_p }[/math] 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.[4][5] 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.[6][5]
References
- ↑ Grothendieck (1972) Théorème 3.6, p. 351
- ↑ Husemöller (1987) pp.116-117
- ↑ Husemoller (1987) pp.116-117
- ↑ Husemöller (1987) pp.266-269
- ↑ 5.0 5.1 "Algorithm for determining the type of a singular fiber in an elliptic pencil", Modular Functions of One Variable IV, Lecture Notes in Mathematics, 476, Berlin / Heidelberg: Springer, 1975, pp. 33–52, doi:10.1007/BFb0097582, ISBN 978-3-540-07392-5, ISSN 1617-9692
- ↑ This is implicit in Husemöller (1987) pp.117-118
- Grothendieck, Alexandre (1972) (in fr). Séminaire de Géométrie Algébrique du Bois Marie - 1967-69 - Groupes de monodromie en géométrie algébrique - (SGA 7) - vol. 1. Lecture Notes in Mathematics. 288. Berlin; New York: Springer-Verlag. pp. viii+523. doi:10.1007/BFb0068688. ISBN 978-3-540-05987-5.
- Husemöller, Dale H. (1987). Elliptic curves. Graduate Texts in Mathematics. 111. With an appendix by Ruth Lawrence. Springer-Verlag. ISBN 0-387-96371-5.
- Lang, Serge (1997). Survey of Diophantine geometry. Springer-Verlag. p. 70. ISBN 3-540-61223-8. https://archive.org/details/surveydiophantin00lang_347.
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