semistable elliptic curve

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 $\mathrm(R)$ (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism
$\mathrm(F) \to \mathrm(R)$
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 $F$.

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 is an elliptic curve defined over the rational number field $\mathbb$. 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. Deciding whether this condition holds is effectively computable by 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 may also be made explicit: acquires semistable reduction over the extension of generated by the coordinates of the points of order 12.This is implicit in Husemöller (1987) pp.117-118

References

*
*
* {{cite book , first=Serge , last=Lang , authorlink=Serge Lang , title=Survey of Diophantine geometry , url=https://archive.org/details/surveydiophantin00lang_347 , url-access=limited , publisher=Springer-Verlag , year=1997 , isbn=3-540-61223-8 , zbl=0869.11051 , pag
70

Abelian varieties
Diophantine geometry
Elliptic curves