The reduction (special fiber) $E_{\mathfrak p}$ of (the Neron model of) an elliptic curve $E$ modulo a prime ${\mathfrak p}$ is one of:
- good reduction = stable reduction = $E_{\mathfrak p}$ is non-singular
- multiplicative reduction = semi-stable reduction = $E_{\mathfrak p}$ is a product of the multiplicative group times a finite group
- additive reduction = unstable reduction = $E_{\mathfrak p}$ is a product of the additive group times a finite group
Of course, this trichotomy is a reflection of the fact that there are only three sorts of connected one-dimensional Lie groups, namely the additive group, the multiplicative group, and the compact case (elliptic curves).
