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

Otherwise $E$ has bad, i.e., if the reduction of \(E\) modulo \(\mathfrak{p}\) is singular. There 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 if the two tangents at the node are defined over $\F_q$ and non-split otherwise.

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

Knowl status:
  • Review status: reviewed
  • Last edited by John Jones on 2018-06-19 18:30:05
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