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

  1. ^ Grothendieck (1972) Théorème 3.6, p. 351
  2. ^ Husemöller (1987) pp.116-117
  3. ^ Husemoller (1987) pp.116-117
  4. ^ Husemöller (1987) pp.266-269
  5. ^ a b Tate, John (1975), "Algorithm for determining the type of a singular fiber in an elliptic pencil", in Birch, B.J.; Kuyk, W. (eds.), Modular Functions of One Variable IV, Lecture Notes in Mathematics, vol. 476, Berlin / Heidelberg: Springer, pp. 33–52, doi:10.1007/BFb0097582, ISBN 978-3-540-07392-5, ISSN 1617-9692, MR 0393039, Zbl 1214.14020
  6. ^ This is implicit in Husemöller (1987) pp.117-118
  • Grothendieck, Alexandre (1972). Séminaire de Géométrie Algébrique du Bois Marie - 1967-69 - Groupes de monodromie en géométrie algébrique - (SGA 7) - vol. 1. Lecture Notes in Mathematics (in French). Vol. 288. Berlin; New York: Springer-Verlag. viii+523. doi:10.1007/BFb0068688. ISBN 978-3-540-05987-5. MR 0354656.
  • Husemöller, Dale H. (1987). Elliptic curves. Graduate Texts in Mathematics. Vol. 111. With an appendix by Ruth Lawrence. Springer-Verlag. ISBN 0-387-96371-5. Zbl 0605.14032.
  • Lang, Serge (1997). Survey of Diophantine geometry. Springer-Verlag. p. 70. ISBN 3-540-61223-8. Zbl 0869.11051.