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The reduction type of an elliptic curve $E$ defined over $\mathbb Q$ at a prime $p$ depends on the reduction $\tilde E$ of $E$ modulo $p$. This reduction is constructed by taking a minimal Weierstrass equation for $E$ and reducing its coefficients modulo $p$ to obtain a curves over $\mathbb F_p$. The reduced curve is either smooth (non-singular) or has a unique singular point.

$E$ has good reduction at $p$ if $\tilde E$ is non-singular over $\mathbb F_p$. The reduction type is ordinary (ord) if $\tilde E$ is ordinary (equivalently, if $\tilde E(\overline{\F_p})$ has non-trivial $p$-torsion) and supersingular (ss) otherwise. The coefficient $a(p)$ of the L-function $L(E,s)$ is divisible by $p$ if the reduction is supersingular and not if it is ordinary.

$E$ has bad reduction at $p$ if $\tilde E$ is singular over $\mathbb F_p$. In this case the reduction type is further classified according to the nature of the singularity. In all cases the singularity is a double point.

$E$ has multiplicative reduction at $p$ if $\tilde E$ has a nodal singularity: the singular point is a node, with distinct tangents. It is called split if the two tangents are defined over $\mathbb F_p$ and non-split otherwise. The coefficient $a(p)$ of $L(E,s)$ is $1$ if the reduction is split and $-1$ if it is non-split.

$E$ has additive reduction at $p$ if $\tilde E$ has a cuspidal singularity: the singular point is a cusp, with only one tangent. In this case $a(p)=0$.

Knowl status:
  • Review status: reviewed
  • Last edited by John Cremona on 2022-02-04 09:02:19
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