bad reduction

1 Singular Cubic Curves

Let $E$ be a cubic curve over a field $K$ with Weierstrassequation $f(x,y)=0$, 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 $\alpha ,\beta \in \overline{K}$ (analgebraic closure of $K$).

Definition 1.

The singular point $P$ is a node if $\alpha \mathrm{\ne }\beta$. In thiscase 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 aunique tangent line at $P$.

Note: See the entry for elliptic curve for examples of cusps andnodes.

There is a very simple criterion to know whether a cubic curve inWeierstrass form is singular and to differentiate nodes fromcusps:

Proposition 1.

Let $E\mathrm{/}K$ be given by a Weierstrass equation, and let $\mathrm{\Delta }$ bethe 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 anynumber field $K$, but we choose $\mathbb{Q}$ for simplicity in theexposition). 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 reducingeach of the coefficients $a_i$ modulo $p$ we obtain the equationof a cubic curve $\tilde{E}$ over the finite field${𝔽}_p$ (the field with $p$ elements).

Definition 2.

1. If $\tilde{E}$ is a non-singular curve then $\tilde{E}$ is an elliptic curve over ${𝔽}_p$ and we say that $E$has good reduction at $p$. Otherwise, we say that $E$ has badreduction at $p$.

2. If $\tilde{E}$ has a cusp then we say that $E$ hasadditive reduction at $p$.

3. If $\tilde{E}$ has a node then we say that $E$ hasmultiplicative reduction at $p$. If the slopes of the tangentlines ($\alpha$ and $\beta$ as above) are in ${𝔽}_p$ thenthe reduction is said to be split multiplicative (and non-splitotherwise).

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 badreduction 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 reductionis additive (since modulo $5$ we can write the equation as${[(y-0)-0(x-0)]}^2-x^3$ and the slope is $0$).

3. The elliptic curve $E_2:y^2=x^3-x^2+35$ has badmultiplicative reduction at $5$ and $7$. The reduction at $5$ is split,while the reduction at $7$ is non-split. Indeed, modulo $5$ we couldwrite the equation as $[(y-0)-2(x-0)][(y-0)+2(x-0)]-x^3$, beingthe slopes $2$ and $-2$. However, for $p=7$ the slopes are not in${𝔽}_7$ ($\sqrt{-1}$ is not in ${𝔽}_7$).