conductor of an elliptic curve
Let be an elliptic curve over . For each prime define the quantity as follows:
where depends on wild ramification in the action of the inertia group at of on the Tate module .
Definition.
The conductor of is defined to be:
where the product is over all primes and the exponent is defined as above.
Example.
Let . The primes of bad reduction for are and . The reduction at is additive, while the reduction at is multiplicative. Hence .
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 |