conductor of an elliptic curve

Let $E$ be an elliptic curve over $\mathbb{Q}$ . For each prime $p\in\mathbb{Z}$ define the quantity $f_{p}$ as follows:

$f_{p}=\begin{cases}0\text{, if $E$ has good reduction at $p$,}\\ 1\text{, if $E$ has multiplicative reduction at $p$,}\\ 2\text{, if $E$ has additive reduction at $p$, and $p\neq 2,3$,}\\ 2+\delta_{p}\text{, if $E$ has additive reduction at $p=2\ or\ 3$.}\end{cases}$

where $\delta_{p}$ depends on wild ramification in the action of the inertia group at $p$ of $\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the Tate module $T_{p}(E)$ .

Definition.

The conductor $N_{E/\mathbb{Q}}$ of ${E/\mathbb{Q}}$ is defined to be:

$N_{E/\mathbb{Q}}=\prod_{p}p^{f_{p}}$

where the product is over all primes and the exponent $f_{p}$ is defined as above.

Example.

Let $E/\mathbb{Q}\colon y^{2}+y=x^{3}-x^{2}+2x-2$ . The primes of bad reduction for $E$ are $p=5$ and $7$ . The reduction at $p=5$ is additive, while the reduction at $p=7$ is multiplicative. Hence $N_{E/\mathbb{Q}}=25\cdot 7=175$ .

References

1 James Milne, Elliptic Curves, http://www.jmilne.org/math/CourseNotes/math679.htmlonline 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.

Title	conductor of an elliptic curve
Canonical name	ConductorOfAnEllipticCurve
Date of creation	2013-03-22 13:49:51
Last modified on	2013-03-22 13:49:51
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	9
Author	alozano (2414)
Entry type	Definition
Classification	msc 14H52
Synonym	conductor
Related topic	EllipticCurve
Related topic	LSeriesOfAnEllipticCurve
Related topic	ArithmeticOfEllipticCurves
Defines	conductor of an elliptic curve