L-series of an elliptic curve

Let E be an elliptic curveMathworldPlanetmath over with Weierstrass equation:


with coefficients ai. For p a prime in , define Np as the number of points in the reduction of the curve modulo p, this is, the number of points in:


where O is the point at infinity. Also, let ap=p+1-Np. We define the local part at p of the L-series to be:

Lp(T)={1-apT+pT2, if E has good reduction at p,1-T, if E has split multiplicative reduction at p,1+T, if E has non-split multiplicative reduction at p,1, if E has additive reduction at p.

The L-series of the elliptic curve E is defined to be:


where the product is over all primes.

Note: The product converges and gives an analytic functionMathworldPlanetmath for all Re(s)>3/2. This follows from the fact that ap2p. However, far more is true:

Theorem (Taylor, Wiles).

The L-series L(E,s) has an analytic continuation to the entire complex planeMathworldPlanetmath, and it satisfies the following functional equation. Define


where NE/Q is the conductorPlanetmathPlanetmathPlanetmath of E and Γ is the Gamma functionDlmfDlmfMathworldPlanetmath. Then:


The number w above is usually called the root number of E, and it has an important conjectural meaning (see Birch and Swinnerton-Dyer conjecture).

This result was known for elliptic curves having complex multiplicationMathworldPlanetmath (Deuring, Weil) until the general result was finally proven.


  • 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