Reduction type (reviewed)

The reduction type of an elliptic curve $E$ defined over a number field \(K\) at a prime \(\mathfrak{p}\) of \(K\) depends on the reduction $\tilde E$ of $E$ modulo $\mathfrak{p}$. Let $\F_q$ be the ring of integers of $K$ modulo $\mathfrak{p}$, a finite field of characteristic $p$.

$E$ has good reduction at $\mathfrak p$ if $\tilde E$ is non-singular over $\F_q$. The reduction type is ordinary if $\tilde E$ is ordinary (equivalently, $\tilde E(\overline{\F_q})$ has $p$-torsion) and supersingular otherwise.

On the other hand, if the reduction of \(E\) modulo \(\mathfrak{p}\) is singular, then $E$ has bad reduction. There are two types of bad reduction are as follows.

$E$ has multiplicative reduction at $\mathfrak p$ if $\tilde E$ has a nodal singularity. It is called split multiplicative reduction if the two tangents at the node are defined over $\F_q$ and non-split multiplicative reduction otherwise.

$E$ has additive reduction at $\mathfrak p$ if $\tilde E$ has a cuspidal singularity.