Semistable abelian variety

In mathematics, a semistable abelian variety in Diophantine geometry is an abelian variety defined over a global or local field with reduction modulo all primes of restricted type.

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 "A"⁰ denote the open subgroup scheme of the Néron model whose fibres are the connected components. 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 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".

emistable elliptic curve

A semistable elliptic curve may 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". This can be explained by saying 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. Which it is, is something 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.

References

*