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conductor of an elliptic curve
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(Definition)
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Let $E$ be an elliptic curve over $\mathbb{Q}$ . For each prime $p\in \mathbb{Z}$ define the quantity $f_p$ as follows:
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: $$N_{E/\mathbb{Q}}=\prod_p p^{f_p}$$ 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 $N_{E/\mathbb{Q}}=25\cdot 7 = 175$ .
- 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.
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"conductor of an elliptic curve" is owned by alozano.
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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 6460 times total.
Classification:
| AMS MSC: | 14H52 (Algebraic geometry :: Curves :: Elliptic curves) |
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Pending Errata and Addenda
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