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. Let A⁰ denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a residue field k, A⁰_k 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 A⁰_k 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.

[edit] Semistable elliptic curve

A semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type. 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 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. Deciding whether this condition holds is effectively computable according to Tate's algorithm. 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 by the coordinates of the points of order 2 and 3.

[edit] References

• Serge Lang (1997). Survey of Diophantine geometry. Springer-Verlag. p. 70. ISBN 3-540-61223-8.