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