L-series of an elliptic curve
Let be an elliptic curve over with Weierstrass equation:
with coefficients . For a prime in , define as the number of points in the reduction of the curve modulo , this is, the number of points in:
where is the point at infinity. Also, let . We define the local part at of the L-series to be:
Definition.
The L-series of the elliptic curve is defined to be:
where the product is over all primes.
Note: The product converges and gives an analytic function for all . This follows from the fact that . However, far more is true:
Theorem (Taylor, Wiles).
The L-series has an analytic continuation to the entire complex plane, and it satisfies the following functional equation. Define
where is the conductor of and is the Gamma function. Then:
The number above is usually called the root number of , and it has an important conjectural meaning (see Birch and Swinnerton-Dyer conjecture).
This result was known for elliptic curves having complex multiplication (Deuring, Weil) until the general result was finally proven.
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.
- 4 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.
Title | L-series of an elliptic curve |
Canonical name | LseriesOfAnEllipticCurve |
Date of creation | 2013-03-22 13:49:43 |
Last modified on | 2013-03-22 13:49:43 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 8 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 14H52 |
Synonym | L-function of an elliptic curve |
Related topic | EllipticCurve |
Related topic | DirichletLSeries |
Related topic | ConductorOfAnEllipticCurve |
Related topic | HassesBoundForEllipticCurvesOverFiniteFields |
Related topic | ArithmeticOfEllipticCurves |
Defines | L-series of an elliptic curve |
Defines | local part of the L-series |
Defines | root number |