L-series of an elliptic curve

Let $E$ be an elliptic curve over $\mathbb{Q}$ with Weierstrass equation:

$y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$

with coefficients $a_{i}\in\mathbb{Z}$ . For $p$ a prime in $\mathbb{Z}$ , define $N_{p}$ as the number of points in the reduction of the curve modulo $p$ , this is, the number of points in:

$\{O\}\cup\{(x,y)\in{\mathbb{F}_{p}}^{2}\colon y^{2}+a_{1}xy+a_{3}y-x^{3}-a_{2}% x^{2}-a_{4}x-a_{6}\equiv 0\ mod\ p\}$

where $O$ is the point at infinity. Also, let $a_{p}=p+1-N_{p}$ . We define the local part at $p$ of the L-series to be:

$L_{p}(T)=\begin{cases}1-a_{p}T+pT^{2}\text{, if $E$ has good reduction at $p$}% ,\\ 1-T\text{, if $E$ has split multiplicative reduction at $p$},\\ 1+T\text{, if $E$ has non-split multiplicative reduction at $p$},\\ 1\text{, if $E$ has additive reduction at $p$}.\end{cases}$

Definition.

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

$L(E,s)=\prod_{p}\frac{1}{L_{p}(p^{-s})}$

where the product is over all primes.

Note: The product converges and gives an analytic function for all $Re(s)>3/2$ . This follows from the fact that $\mid a_{p}\mid\leq 2\sqrt{p}$ . However, far more is true:

Theorem (Taylor, Wiles).

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

$\Lambda(E,s)=({N_{E/\mathbb{Q}}})^{s/2}(2\pi)^{-s}\Gamma(s)L(E,s)$

where ${N_{E}/\mathbb{Q}}$ is the conductor of $E$ and $\Gamma$ is the Gamma function. Then:

$\Lambda(E,s)=w\Lambda(E,2-s)\quad with\ w=\pm 1$

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 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