## You are here

Homebad reduction

## Primary tabs

# 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:

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

$\displaystyle=$ | $\displaystyle[(y-y_{0})-\alpha(x-x_{0})][(y-y_{0})-\beta(x-x_{0})]-(x-x_{0})^{3}$ |

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 $P$.

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

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

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

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

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

###### Definition 2.

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. If $\widetilde{E}$ has a cusp then we say that $E$ has

*additive reduction*at $p$.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 $\mathbb{F}_{p}$ then the reduction is said to be*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. $E_{1}\colon y^{2}=x^{3}+35x+5$ has good reduction at $p=7$.

2. 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, online course notes.
- 2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. 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.

## Mathematics Subject Classification

14H52*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections