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'L-series of an elliptic curve'
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| Title of object: |
L-series of an elliptic curve |
| Canonical Name: |
LSeriesOfAnEllipticCurve |
| Type: |
Definition |
| Created on: |
2003-08-06 10:54:04 |
| Modified on: |
2003-08-07 10:34:35 |
| Classification: |
msc:14H52 |
| Keywords: |
L-function, L-series, elliptic curve |
| Defines: |
L-series, local part of the L-series, root number |
| Synonyms: |
L-series of an elliptic curve=L-function of an elliptic curve |
Revision comment (for changes between this and next version):
| Changes for correction #2673 ('also defines "L-series"?'). |
Preamble:
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Content:
Let $E$ be an elliptic curve over $\mathbb{Q}$ with Weierstrass
equation:
$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$
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:
$$N_p=\{(x,y)\in{\mathbb{F}_p}^2\colon y^2+a_1xy+a_3y-x^3-a_2x^2-a_4x-a_6\equiv 0\ mod\ p\}$$
Also, let $a_p=p+1-N_p$. We define the \emph{local part at $p$ of
the L-series} to be:
$$ L_p(T) = \begin{cases} 1-a_pT+pT^2 \text{, if $E$ has good reduction at $p$}, \\
1-T \text{, if $E$ has split multiplicative reduction at $p$},\\
1+T \text{, if $E$ has non-split multiplicative reduction at $p$},\\
1 \text{, if $E$ has additive reduction at $p$}. \end{cases} $$
\begin{defn} The L-series of the elliptic curve $E$ is defined to
$$ L(E,s) = \prod_{p}\frac{1}{L_p(p^{-s})} $$
where the product is over all primes in $\mathbb{Z}$.
\end{defn}
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:
\begin{thm}[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:
$$\Lambda(E,s)=w\Lambda(E,2-s)\quad with\ w=\pm 1$$
\end{thm}
The number $w$ above is usually called the \emph{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.
\begin{thebibliography}{9}
\bibitem{milne} James Milne, {\em Elliptic Curves}, \PMlinkexternal{online course
notes}{http://www.jmilne.org/math/CourseNotes/math679.html}.
\bibitem{silverman} Joseph H. Silverman, {\em The Arithmetic of Elliptic Curves}. Springer-Verlag, New York, 1986.
\bibitem{silverman2} Joseph H. Silverman, {\em Advanced Topics in
the Arithmetic of Elliptic Curves}. Springer-Verlag, New York,
1994.
\bibitem{shimura} Goro Shimura, {\em Introduction to the
Arithmetic Theory of Automorphic Functions}. Princeton University
Press, Princeton, New Jersey, 1971.
\end{thebibliography} |
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