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 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, 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.

Semistable elliptic curve

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 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.^[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

• Husemöller, Dale H. (1987). Elliptic curves. Graduate Texts in Mathematics 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.