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conductor of an elliptic curve (Definition)

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

\begin{displaymath}f_p = \begin{cases} 0 \text{, if $E$ has good reduction at $p... ...t{, if $E$ has additive reduction at $p=2\ or\ 3$.} \end{cases}\end{displaymath}
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 1   The conductor $ N_{E/\mathbb{Q}}$ of $ {E/\mathbb{Q}}$ is defined to be:
$\displaystyle N_{E/\mathbb{Q}}=\prod_p p^{f_p}$
where the product is over all primes and the exponent $ f_p$ is defined as above.
Example 1   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$.

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.



"conductor of an elliptic curve" is owned by alozano.
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See Also: elliptic curve, L-series of an elliptic curve, the arithmetic of elliptic curves

Other names:  conductor
Also defines:  conductor of an elliptic curve
Keywords:  conductor, elliptic curve, L-series
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Cross-references: multiplicative, additive, reduction, bad reduction, exponent, product, Tate module, inertia group, action, wild ramification, prime, elliptic curve
There are 2 references to this entry.

This is version 6 of conductor of an elliptic curve, born on 2003-08-07, modified 2007-07-02.
Object id is 4563, canonical name is ConductorOfAnEllipticCurve.
Accessed 4560 times total.

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