| Version 4 |
Version 3 |
| Let $E$ be an elliptic curve over $\mathbb{Q}$ with Weierstrass |
Let $E$ be an elliptic curve over $\mathbb{Q}$ with Weierstrass |
| equation: |
equation: |
| $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ |
$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ |
| with coefficients $a_i\in\mathbb{Z}$. For $p$ a prime in |
with coefficients $a_i\in\mathbb{Z}$. For $p$ a prime in |
| $\mathbb{Z}$, define $N_p$ as the number of points in the |
$\mathbb{Z}$, define $N_p$ as the number of points in the |
|
reduction of the curve modulo $p$, this is, the number of points in:
|
reduction of the curve modulo $p$, this is:
|
|
$$\{O\}\cup\{(x,y)\in{\mathbb{F}_p}^2\colon y^2+a_1xy+a_3y-x^3-a_2x^2-a_4x-a_6\equiv 0\ mod\ p\}$$
|
$$N_p=\{(x,y)\in{\mathbb{F}_p}^2\colon y^2+a_1xy+a_3y-x^3-a_2x^2-a_4x-a_6\equiv 0\ mod\ p\}$$
|
|
where $O$ is the point at infinity. Also, let $a_p=p+1-N_p$. We define the \emph{local part at $p$ of
|
Also, let $a_p=p+1-N_p$. We define the \emph{local part at $p$ of |
| the L-series} to be: |
the L-series} to be: |
| $$ L_p(T) = \begin{cases} 1-a_pT+pT^2 \text{, if $E$ has good reduction at $p$}, \\ |
$$ 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 split multiplicative reduction at $p$},\\ |
| 1+T \text{, if $E$ has non-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} $$ |
1 \text{, if $E$ has additive reduction at $p$}. \end{cases} $$ |
|
|
| \begin{defn} The L-series of the elliptic curve $E$ is defined to |
\begin{defn} The L-series of the elliptic curve $E$ is defined to |
| be: |
be: |
| $$ L(E,s) = \prod_{p}\frac{1}{L_p(p^{-s})} $$ |
$$ L(E,s) = \prod_{p}\frac{1}{L_p(p^{-s})} $$ |
| where the product is over all primes in $\mathbb{Z}$. |
where the product is over all primes in $\mathbb{Z}$. |
| \end{defn} |
\end{defn} |
|
|
| Note: The product converges and gives an analytic function for all |
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 |
$Re(s)>3/2$. This follows from the fact that $\mid a_p \mid \leq |
| 2\sqrt{p}$. However, far more is true: |
2\sqrt{p}$. However, far more is true: |
|
|
| \begin{thm}[Taylor, Wiles] |
\begin{thm}[Taylor, Wiles] |
| The L-series $L(E,s)$ has an analytic continuation to the entire |
The L-series $L(E,s)$ has an analytic continuation to the entire |
| complex plane, and it satisfies the following functional equation. |
complex plane, and it satisfies the following functional equation. |
| Define |
Define |
| $$\Lambda(E,s)=({N_{E/\mathbb{Q}}})^{s/2}(2\pi)^{-s}\Gamma(s)L(E,s)$$ |
$$\Lambda(E,s)=({N_{E/\mathbb{Q}}})^{s/2}(2\pi)^{-s}\Gamma(s)L(E,s)$$ |
| where ${N_E/\mathbb{Q}}$ is the conductor of $E$ and $\Gamma$ is |
where ${N_E/\mathbb{Q}}$ is the conductor of $E$ and $\Gamma$ is |
| the Gamma function. Then: |
the Gamma function. Then: |
| $$\Lambda(E,s)=w\Lambda(E,2-s)\quad with\ w=\pm 1$$ |
$$\Lambda(E,s)=w\Lambda(E,2-s)\quad with\ w=\pm 1$$ |
| \end{thm} |
\end{thm} |
|
|
| The number $w$ above is usually called the \emph{root number} of |
The number $w$ above is usually called the \emph{root number} of |
| $E$, and it has an important conjectural meaning (see Birch and |
$E$, and it has an important conjectural meaning (see Birch and |
| Swinnerton-Dyer conjecture). |
Swinnerton-Dyer conjecture). |
|
|
| This result was known for elliptic curves having complex |
This result was known for elliptic curves having complex |
| multiplication (Deuring, Weil) until the general result was |
multiplication (Deuring, Weil) until the general result was |
| finally proven. |
finally proven. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{milne} James Milne, {\em Elliptic Curves}, \PMlinkexternal{online course |
\bibitem{milne} James Milne, {\em Elliptic Curves}, \PMlinkexternal{online course |
| notes}{http://www.jmilne.org/math/CourseNotes/math679.html}. |
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{silverman} Joseph H. Silverman, {\em The Arithmetic of Elliptic Curves}. Springer-Verlag, New York, 1986. |
| \bibitem{silverman2} Joseph H. Silverman, {\em Advanced Topics in |
\bibitem{silverman2} Joseph H. Silverman, {\em Advanced Topics in |
| the Arithmetic of Elliptic Curves}. Springer-Verlag, New York, |
the Arithmetic of Elliptic Curves}. Springer-Verlag, New York, |
| 1994. |
1994. |
| \bibitem{shimura} Goro Shimura, {\em Introduction to the |
\bibitem{shimura} Goro Shimura, {\em Introduction to the |
| Arithmetic Theory of Automorphic Functions}. Princeton University |
Arithmetic Theory of Automorphic Functions}. Princeton University |
| Press, Princeton, New Jersey, 1971. |
Press, Princeton, New Jersey, 1971. |
| \end{thebibliography} |
\end{thebibliography} |
