## 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. 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 $\Delta=0=c_{4}$.

###### 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. 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 $\mathbb{F}_{p}$ 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. 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.