bad reduction
1 Singular Cubic Curves
Let $E$ be a cubic curve over a field $K$ with Weierstrass equation $f(x,y)=0$, where:
$$f(x,y)={y}^{2}+{a}_{1}xy+{a}_{3}y{x}^{3}{a}_{2}{x}^{2}{a}_{4}x{a}_{6}$$ 
which has a singular point^{} $P=({x}_{0},{y}_{0})$. This is equivalent^{} to:
$$\partial f/\partial x(P)=\partial f/\partial y(P)=0$$ 
and so we can write the Taylor expansion^{} of $f(x,y)$ at $({x}_{0},{y}_{0})$ as follows:
$f(x,y)f({x}_{0},{y}_{0})$  $=$  ${\lambda}_{1}{(x{x}_{0})}^{2}+{\lambda}_{2}(x{x}_{0})(y{y}_{0})+{\lambda}_{3}{(y{y}_{0})}^{2}{(x{x}_{0})}^{3}$  
$=$  $[(y{y}_{0})\alpha (x{x}_{0})][(y{y}_{0})\beta (x{x}_{0})]{(x{x}_{0})}^{3}$ 
for some ${\lambda}_{i}\in K$ and $\alpha ,\beta \in \overline{K}$ (an
algebraic closure^{} of $K$).
Definition 1.
The singular point $P$ is a node if $\alpha \mathrm{\ne}\beta $. In this case there are two different tangent lines to $E$ at $P$, namely:
$$y{y}_{0}=\alpha (x{x}_{0}),y{y}_{0}=\beta (x{x}_{0})$$ 
If $\alpha \mathrm{=}\beta $ then we say that $P$ is a cusp, and there is a unique tangent line at $P$.
Note: See the entry for elliptic curve^{} for examples of cusps and nodes.
There is a very simple criterion to know whether a cubic curve in Weierstrass form is singular and to differentiate nodes from cusps:
Proposition 1.
Let $E\mathrm{/}K$ be given by a Weierstrass equation, and let $\mathrm{\Delta}$ be the discriminant^{} and ${c}_{\mathrm{4}}$ as in the definition of $\mathrm{\Delta}$. Then:

1.
$E$ is singular if and only if $\mathrm{\Delta}=0$,

2.
$E$ has a node if and only if $\mathrm{\Delta}=0$ and ${c}_{4}\ne 0$,

3.
$E$ has a cusp if and only if $\mathrm{\Delta}=0={c}_{4}$.
Proof.
See [2], chapter III, Proposition^{} 1.4, page 50. ∎
2 Reduction of Elliptic Curves
Let $E/\mathbb{Q}$ be an elliptic curve (we could work over any number field^{} $K$, but we choose $\mathbb{Q}$ for simplicity in the exposition). Assume that $E$ has a minimal model with Weierstrass equation:
$${y}^{2}+{a}_{1}xy+{a}_{3}y={x}^{3}+{a}_{2}{x}^{2}+{a}_{4}x+{a}_{6}$$ 
with coefficients in $\mathbb{Z}$. Let $p$ be a prime in $\mathbb{Z}$. By reducing each of the coefficients ${a}_{i}$ modulo $p$ we obtain the equation of a cubic curve $\stackrel{~}{E}$ over the finite field^{} ${\mathbb{F}}_{p}$ (the field with $p$ elements).
Definition 2.

1.
If $\stackrel{~}{E}$ is a nonsingular curve then $\stackrel{~}{E}$ is an elliptic curve over ${\mathbb{F}}_{p}$ and we say that $E$ has good reduction at $p$. Otherwise, we say that $E$ has bad reduction at $p$.

2.
If $\stackrel{~}{E}$ has a cusp then we say that $E$ has additive reduction at $p$.

3.
If $\stackrel{~}{E}$ has a node then we say that $E$ has multiplicative reduction at $p$. If the slopes of the tangent lines ($\alpha $ and $\beta $ as above) are in ${\mathbb{F}}_{p}$ then the reduction^{} is said to be split multiplicative (and nonsplit otherwise).
From Proposition 1 we deduce the following:
Corollary 1.
Let $E\mathrm{/}\mathrm{Q}$ be an elliptic curve with coefficients in $\mathrm{Z}$. Let $p\mathrm{\in}\mathrm{Z}$ be a prime. If $E$ has bad reduction at $p$ then $p\mathrm{\mid}\mathrm{\Delta}$.
Examples:

1.
${E}_{1}:{y}^{2}={x}^{3}+35x+5$ has good reduction at $p=7$.

2.
However ${E}_{1}$ has bad reduction at $p=5$, and the reduction is additive^{} (since modulo $5$ we can write the equation as ${[(y0)0(x0)]}^{2}{x}^{3}$ and the slope is $0$).

3.
The elliptic curve ${E}_{2}:{y}^{2}={x}^{3}{x}^{2}+35$ has bad multiplicative reduction at $5$ and $7$. The reduction at $5$ is split, while the reduction at $7$ is nonsplit. Indeed, modulo $5$ we could write the equation as $[(y0)2(x0)][(y0)+2(x0)]{x}^{3}$, being the slopes $2$ and $2$. However, for $p=7$ the slopes are not in ${\mathbb{F}}_{7}$ ($\sqrt{1}$ is not in ${\mathbb{F}}_{7}$).
References
 1 James Milne, Elliptic Curves, http://www.jmilne.org/math/CourseNotes/math679.htmlonline course notes.
 2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. SpringerVerlag, New York, 1986.
 3 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. SpringerVerlag, New York, 1994.
 4 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.
Title  bad reduction 
Canonical name  BadReduction 
Date of creation  20130322 13:49:21 
Last modified on  20130322 13:49:21 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  12 
Author  alozano (2414) 
Entry type  Definition 
Classification  msc 14H52 
Related topic  EllipticCurve 
Related topic  JInvariant 
Related topic  HassesBoundForEllipticCurvesOverFiniteFields 
Related topic  TorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve 
Related topic  ArithmeticOfEllipticCurves 
Related topic  SingularPointsOfPlaneCurve 
Defines  bad reduction 
Defines  good reduction 
Defines  cusp 
Defines  node 
Defines  multiplicative reduction 
Defines  additive reduction 