The Complex $L$-series $L(E,s)$

In Section 1.1 we defined a function $\tilde{L}(E,s)$, which encoded information about $E(\mathbb{F}_p)$ for all but finitely many primes $p$. In this section we define the function $L(E,s)$, which includes information about all primes, and the function $\Lambda(E,s)$ that also includes information ``at infinity''.

Let $E$ be an elliptic curve over $\mathbb{Q}$ defined by a minimal Weierstrass equation

$$y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.$$ (1.3.1)

A minimal Weierstrass equation in one for which the $a_i$ are all integers and the discriminant $\Delta\in\mathbb{Z}$ is minimal amongs all discriminants of Weierstrass equations for $E$ (again, see [Sil92] for the definition of the discriminant of a Weierstrass equation, and also for an explicit description of the allowed transformations of a Weierstrass equation).

For each prime number $p\nmid \Delta$, the equation (1.3.1) reduces modulo $p$ to define an elliptic $E_{\mathbb{F}_p}$ over the finite field $\mathbb{F}_p$. Let

$$a_p = p + 1 - \char93 E(\mathbb{F}_p).$$

For each prime $p\mid \Delta$, we use the following recipe to define $a_p$. If the singular curve $E_{\mathbb{F}_p}$ has a cuspidal singularity, e.g., is $y^2 = x^3$, then let $a_p = 0$. If it has a a nodal singularity, e.g., like $y^2=x^3+x^2$, let $a_p=1$ if the slope of the tangent line at the singular point is in $\mathbb{F}_p$ and let $a_p=-1$ if the slope is not in $\mathbb{F}_p$. Summarizing:

$$a_p = \begin{cases} 0 & \text{if the reduction is cuspidal ... ...thbb{F}_p$-rational (\lq\lq non-split multiplicative'')} \end{cases}$$

Even in the cases when $p\mid \Delta$, we still have

$$a_p = p + 1 - \char93 E(\mathbb{F}_p).$$

When $E$ has additive reduction, the nonsingular points form a group isomorphic to $(\mathbb{F}_p,+)$, and there is one singular point, hence $p+1$ points, so

$$a_p = p+1 - (p+1))) = 0.$$

When $E$ has split multiplicative reduction, there is 1 singular point plus the number of elements of a group isomorphic to $(\mathbb{F}_p^*, \times)$, so $1 + (p-1) = p$ points, and

$$a_p = p+1-p = 1.$$

When $E$ has non-split multiplicative reduction, there is $1$ singular point plus the number of elements of a group isomorphic $(\mathbb{F}_{p^2}^*/\mathbb{F}_p^*, \times)$, i.e., $p+2$ points, and

$$a_p = p+1 - (p+2) = -1.$$

The definition of the full $L$-function of $E$ is then

$$L(E,s) = \prod_{p\mid \Delta} \frac{1}{1-a_p p^{-s}} \cdot ... ...-s} + p\cdot p^{-2s}}. = \sum_{n=1}^{\infty} \frac{a_n}{n^s}.$$

If in addition we add in a few more analytic factors to the $L$-function we obtain a function $\Lambda(E,s)$ that satisfies a remarkably simple functional equation. Let

$$\Gamma(z) = \int_{0}^{\infty} t^{z-1} e^{-t} dt$$

be the $\Gamma$-function (e.g., $\Gamma(n) = (n-1)!$), which defines a meromorphic function on $\mathbb{C}$, with poles at the non-positive integers.

Theorem 1.6 (Hecke, Wiles et al.)   There is a unique positive integer $N=N_E$ and sign $\varepsilon =\varepsilon _E\in\{\pm 1\}$ such that the function

$$\Lambda(E,s) = N^{s/2}\cdot (2\pi)^{-s}\cdot \Gamma(s)\cdot L(E,s)$$

extends to a complex analytic function on all $\mathbb{C}$ that satisfies the functional equation

$$\Lambda(E,2-s) = \varepsilon \cdot \Lambda(E,s),$$ (1.3.2)

for all $s\in \mathbb{C}$.

The integer $N=N_E$ is called the conductor of $E$ and $\varepsilon =\varepsilon _E$ is called the sign in the functional equation for $E$ or the root number of $E$. One can prove that the primes that divide $N$ are the same as the primes that divide $\Delta$. Moreover, for $p\geq 5$, we have that

$$\ord _p(N) = \begin{cases} 0,& \text{if $p\nmid \Delta$},\\... ... 2,&\text{if $E$ has additive reduction at $p$}. \end{cases}$$

There is a geometric algorithm called Tate's algorithm that computes $N$ in all cases and $\varepsilon$.

Example 1.7   Consider the elliptic curve $E$ defined by

$$y^2 + y = x^3 + 50x + 31.$$

The above Weierstrass equation is minimal and has discriminant

$$-1 \cdot 5^{6} \cdot 7^{2} \cdot 11.$$

sage: e = EllipticCurve('1925d'); e
Elliptic Curve defined by y^2 + y = x^3 + 50*x + 31 over Rational Field
sage: e.is_minimal()
True
sage: factor(e.discriminant())
-1 * 5^6 * 7^2 * 11

At $5$ the curve has additive reduction so $a_5 = 0$. At $7$ the curve has split multiplicative reduction so $a_7 = 1$. At $11$ the curve has nonsplit multiplicative reduction, so $a_{11} = -1$. Counting points for $p=2,3$, we find that

$$L(E,s) = \frac{1}{1^{-s}} + \frac{3}{3^{-s}} + \frac{-2}{4... ...{6}{9^{-s}} + \frac{-1}{11^{-s}} + \frac{-6}{12^{-s}} + \cdots$$

sage: [e.ap(p) for p in primes(14)]
[0, 3, 0, 1, -1, 4]

Corollary 1.8   Let $E$ be an elliptic curve over $\mathbb{Q}$, let $\varepsilon \in \{1, -1\}$ be the sign in the functional equation (1.3.2), and let $r_{E,{\rm an}} = \ord _{s=1} L(E,s)$. Then

$$\varepsilon = (-1)^{r_{E,{\rm an}}}.$$

Conjecture 1.9 (The Parity Conjecture)   Let $E$ be an elliptic curve over $\mathbb{Q}$, let $r_{E,{\rm an}}$ be the analytic rank and $r_{E,\alj }$ be the algebraic rank. Then

$$r_{E,\alj } \equiv r_{E,{\rm an}} \pmod{2}.$$

Jan Nekovar has done a huge amount of work toward Conjecture 1.9; in particular, he proves it under the (as yet unproved) hypothesis that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)$ is finite (see Section 2.2 below).