# bad reduction

## 1 Singular Cubic Curves

Let $E$ be a cubic curve over a field $K$ with Weierstrass equation $f(x,y)=0$, where:

 $f(x,y)=y^{2}+a_{1}xy+a_{3}y-x^{3}-a_{2}x^{2}-a_{4}x-a_{6}$
 $\partial f/\partial x(P)=\partial f/\partial y(P)=0$

and so we can write the Taylor expansion  of $f(x,y)$ at $(x_{0},y_{0})$ as follows:

 $\displaystyle f(x,y)-f(x_{0},y_{0})$ $\displaystyle=$ $\displaystyle\lambda_{1}(x-x_{0})^{2}+\lambda_{2}(x-x_{0})(y-y_{0})+\lambda_{3% }(y-y_{0})^{2}-(x-x_{0})^{3}$ $\displaystyle=$ $\displaystyle[(y-y_{0})-\alpha(x-x_{0})][(y-y_{0})-\beta(x-x_{0})]-(x-x_{0})^{3}$

for some $\lambda_{i}\in K$ and $\alpha,\beta\in\bar{K}$ (an algebraic closure  of $K$).

###### Definition 1.

The singular point $P$ is a node if $\alpha\neq\beta$. In this case there are two different tangent lines to $E$ at $P$, namely:

 $y-y_{0}=\alpha(x-x_{0}),\quad y-y_{0}=\beta(x-x_{0})$

If $\alpha=\beta$ then we say that $P$ is a cusp, and there is a unique tangent line at $P$.

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 $E/K$ be given by a Weierstrass equation, and let $\Delta$ be the discriminant    and $c_{4}$ as in the definition of $\Delta$. Then:

1. 1.

$E$ is singular if and only if $\Delta=0$,

2. 2.

$E$ has a node if and only if $\Delta=0$ and $c_{4}\neq 0$,

3. 3.

$E$ has a cusp if and only if $\Delta=0=c_{4}$.

## 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 $\widetilde{E}$ over the finite field  $\mathbb{F}_{p}$ (the field with $p$ elements).

###### Definition 2.
1. 1.

If $\widetilde{E}$ is a non-singular curve then $\widetilde{E}$ 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. 2.

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

3. 3.

If $\widetilde{E}$ 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 $\mathbb{F}_{p}$split multiplicative (and non-split otherwise).

From Proposition 1 we deduce the following:

###### Corollary 1.

Let $E/\mathbb{Q}$ be an elliptic curve with coefficients in $\mathbb{Z}$. Let $p\in\mathbb{Z}$ be a prime. If $E$ has bad reduction at $p$ then $p\mid\Delta$.

Examples:

1. 1.

$E_{1}\colon y^{2}=x^{3}+35x+5$ has good reduction at $p=7$.

2. 2.

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

3. 3.

The elliptic curve $E_{2}\colon y^{2}=x^{3}-x^{2}+35$ has bad multiplicative reduction at $5$ and $7$. The reduction at $5$ is split, while the reduction at $7$ is non-split. Indeed, modulo $5$ we could write the equation as $[(y-0)-2(x-0)][(y-0)+2(x-0)]-x^{3}$, 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, . Springer-Verlag, New York, 1986.
• 3 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, 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 2013-03-22 13:49:21 Last modified on 2013-03-22 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