Semistable abelian variety explained

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 morphismSpec (F) \to Spec (R) gives back A . The Néron model is a smooth group scheme, so we can consider 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 , A ^{0} _{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 A ^{0} _{k} is a semiabelian variety, then A has semistable reduction at the prime corresponding to k . If F is a global field, then 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 .^[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 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 for which has bad reduction modulo . The latter means that the curve E _{p} obtained by reduction of to the prime field with 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] 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 may also be made explicit: acquires semistable reduction over the extension of generated by the coordinates of the points of order 12.^[5]

References

• Book: Grothendieck , Alexandre . Alexandre Grothendieck . Séminaire de Géométrie Algébrique du Bois Marie - 1967-69 - Groupes de monodromie en géométrie algébrique - (SGA 7) - vol. 1 . Lecture Notes in Mathematics . 288 . 1972 . . Berlin; New York . fr . viii+523 . true . 10.1007/BFb0068688 . 978-3-540-05987-5 . 0354656.
• Book: Husemöller, Dale H. . Elliptic curves . . . 111 . . 1987 . 0-387-96371-5 . 0605.14032 .
• Book: Lang, Serge . Serge Lang

. Serge Lang . Survey of Diophantine geometry . limited . . 1997 . 3-540-61223-8 . 0869.11051 . 70 .

Notes and 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. This is implicit in Husemöller (1987) pp.117-118