Semistable elliptic curve
In mathematics, a semistable elliptic curve in diophantine geometry is 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 of prime numbers p, for which E has bad reduction modulo p. This can be explained by saying that the curve Ep 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.
To be more accurate, one should cite the existence of the Néron model of E, which is a 'best possible' model of E defined over the integers Z. This model may be represented as a scheme over
- Spec(Z)
(cf. spectrum of a ring) for which the 'generic fibre' constructed by means of the morphism
- Spec(Q) → Spec(Z)
gives back E. The other fibres, at the points of Spec(Z) corresponding to prime numbers p, are elliptic curves unless p is in S. For those exceptional cases the fibre is still a group scheme, either the multiplicative group or additive group defined over Z/pZ. 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.
