conductor of an elliptic curveConductorOfAnEllipticCurve2003-08-07 10:19:422007-07-02 16:03:07Definitionconductor of an elliptic curveconductorelliptic curveL-series% this is the default PlanetMath preamble. as your knowledge
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\theoremstyle{definition}
\newtheorem*{exa}{Example}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 $T_p(E)$.
\begin{defn}
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.
\end{defn}
\begin{exa}
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
$N_{E/\mathbb{Q}}=25\cdot 7 = 175$.
\end{exa}
\begin{thebibliography}{9}
\bibitem{milne} James Milne, {\em Elliptic Curves}, \PMlinkexternal{online course
notes}{http://www.jmilne.org/math/CourseNotes/math679.html}.
\bibitem{silverman} Joseph H. Silverman, {\em The Arithmetic of Elliptic Curves}. Springer-Verlag, New York, 1986.
\bibitem{silverman2} Joseph H. Silverman, {\em Advanced Topics in
the Arithmetic of Elliptic Curves}. Springer-Verlag, New York,
1994.
\end{thebibliography}