L-series of an elliptic curve

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

$$y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6$$

with coefficients $a_i\in \mathbb{Z}$. For $p$ a prime in$\mathbb{Z}$, define $N_p$ as the number of points in thereduction of the curve modulo $p$, this is, the number of points in:

$$\{O\}\cup \{(x,y)\in 𝔽_p{}^2:y^2+a_{1}xy+a_{3}y-x^3-a_2{x}^2-a_{4}x-a_6\equiv 0modp\}$$

where $O$ is the point at infinity. Also, let $a_p=p+1-N_p$. We define the local part at $p$ ofthe L-series to be:

$$L_p(T)=\{\begin{array}{cc}1-a_{p}T+p{T}^2\text{, if }E\text{ has good reduction at }p,\hfill & \\ 1-T\text{, if }E\text{ has split multiplicative reduction at }p,\hfill & \\ 1+T\text{, if }E\text{ has non-split multiplicative reduction at }p,\hfill & \\ 1\text{, if }E\text{ has additive reduction at }p.\hfill & \end{array}$$

Definition.

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

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

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 $\mid a_p\mid \le 2\sqrt{p}$. However, far more is true:

Theorem (Taylor, Wiles).

The L-series $L\mathit{}\mathrm{(}E\mathrm{,}s\mathrm{)}$ has an analytic continuation to the entirecomplex plane, and it satisfies the following functional equation.Define

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

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

$$\mathrm{\Lambda }(E,s)=w\mathrm{\Lambda }(E,2-s)\mathit{\hspace{1em}}withw=\pm 1$$

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

This result was known for elliptic curves having complexmultiplication (Deuring, Weil) until the general result wasfinally proven.