PlanetMath: L-series of an elliptic curve

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L-series of an elliptic curve

(Definition)

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

$\displaystyle y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$

with coefficients $a_i\in\mathbb{Z}$ . For

a prime in $\mathbb{Z}$ , define

as the number of points in the reduction of the curve modulo

, this is, the number of points in:

$\displaystyle \{O\}\cup\{(x,y)\in{\mathbb{F}_p}^2\colon y^2+a_1xy+a_3y-x^3-a_2x^2-a_4x-a_6\equiv 0\ mod\ p\}$

where

is the point at infinity. Also, let

. We define the local part at

of the L-series to be:

$\displaystyle L_p(T) = \begin{cases}1-a_pT+pT^2 \text{, if $E$ has good reducti... ...ction at $p$},\ 1 \text{, if $E$ has additive reduction at $p$}. \end{cases}$

Definition 1 The L-series of the elliptic curve is defined to be:

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

where the product is over all primes.

Note: The product converges and gives an analytic function for all . This follows from the fact that $\mid a_p \mid \leq 2\sqrt{p}$ . However, far more is true:

Theorem 1 (Taylor, Wiles) The L-series has an analytic continuation to the entire complex plane, and it satisfies the following functional equation. Define

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

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

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

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.

Bibliography

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.
4: Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.

"L-series of an elliptic curve" is owned by alozano. [ full author list (2) | owner history (1) ]

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Other names:

L-function of an elliptic curve

Also defines:

L-series of an elliptic curve, local part of the L-series, root number

Keywords:

L-function, L-series, elliptic curve

Attachments:: Birch and Swinnerton-Dyer conjecture (Conjecture) by alozano

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Cross-references: complex multiplication, Birch and Swinnerton-Dyer conjecture, gamma function, conductor, functional equation, complex plane, entire, analytic continuation, analytic function, converges, product, infinity, curve, reduction, points, number, prime, coefficients, Weierstrass equation, elliptic curve
There are 2 references to this entry.

This is version 5 of L-series of an elliptic curve, born on 2003-08-06, modified 2006-11-09.
Object id is 4560, canonical name is LSeriesOfAnEllipticCurve.
Accessed 6204 times total.

Classification:

AMS MSC:

14H52 (Algebraic geometry :: Curves :: Elliptic curves)

Pending Errata and Addenda

None.[ View all 1 ]

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