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bad reduction (Definition)

Singular Cubic Curves

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

$\displaystyle f(x,y)=y^2+a_1xy+a_3y-x^3-a_2x^2-a_4x-a_6$
which has a singular point $ P=(x_0,y_0)$. This is equivalent to:
$\displaystyle \partial f/ \partial x(P)=\partial f/ \partial y(P)=0$
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:
$\displaystyle y-y_0=\alpha(x-x_0),\quad y-y_0=\beta(x-x_0)$
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 $ \cite{silverman}$, chapter III, Proposition 1.4, page 50. $ \qedsymbol$

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:

$\displaystyle y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$
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. 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. 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$).

Bibliography

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.



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See Also: elliptic curve, j-invariant, Hasse's bound for elliptic curves over finite fields, the torsion subgroup of an elliptic curve injects in the reduction of the curve, the arithmetic of elliptic curves, singular points of plane curve

Also defines:  bad reduction, good reduction, cusp, node, multiplicative reduction, additive reduction
Keywords:  reduction, elliptic curve, split, non-split

Attachments:
criterion of Néron-Ogg-Shafarevich (Theorem) by alozano
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Cross-references: additive, multiplicative, reduction, slopes, non-singular, finite field, equation, prime, coefficients, minimal model, number field, proposition, discriminant, differentiate, singular, simple, elliptic curve, tangent lines, algebraic closure, Taylor expansion, equivalent, singular point, Weierstrass equation, field, curve
There are 54 references to this entry.

This is version 9 of bad reduction, born on 2003-08-05, modified 2006-03-16.
Object id is 4553, canonical name is BadReduction.
Accessed 14720 times total.

Classification:
AMS MSC: 14H52 (Algebraic geometry :: Curves :: Elliptic curves)
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