Semistable abelian variety

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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

Spec(R)

(cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism

Spec(F) → Spec(R)

gives back A. The Néron model is a smooth group scheme, so we can consider A0, 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, A0k 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 A0k is a semiabelian variety, then A has semistable reduction at the prime corresponding to k. If F is global, then A is semistable if it has good or semistable reduction at all primes.

The semistable reduction theorem of Alexander Grothendieck states that an abelian variety acquires semistable reduction over a finite extension of F.

Semistable elliptic curve[edit]

A semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type.[1] Suppose E is an elliptic curve defined over the rational number field 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 Ep 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.[2] Deciding whether this condition holds is effectively computable by Tate's algorithm.[3][4] 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.[5][4]

References[edit]

  1. ^ Husemöller (1987) pp.116-117
  2. ^ Husemoller (1987) pp.116-117
  3. ^ Husemöller (1987) pp.266-269
  4. ^ a b Tate, John (1975), "Algorithm for determining the type of a singular fiber in an elliptic pencil", in Birch, B.J.; W., Modular Functions of One Variable IV, Lecture Notes in Mathematics 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 
  5. ^ This is implicit in Husemöller (1987) pp.117-118