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Zeta functions. No.4

Yoshifumi Tsuchimoto

PROPOSITION 4.1   Let % latex2html id marker 813
$ f\in \mathbb{F}_q[X]$ be an irreducible polynomial in one variable of degree $ d$ . Let us consider $ V=\{f\}$ , an equation in one variable. Then:
  1. \begin{displaymath}
% latex2html id marker 819V(\mathbb{F}_{q^s})=
\begin{cases}
d & \text {if } d \vert s \\
0 & \text {otherwise}
\end{cases}\end{displaymath}

  2. % latex2html id marker 821
$\displaystyle Z(V/\mathbb{F}_q,T) = \frac{1}{1-T^d}
$

\fbox{projective space and projective varieties.}

DEFINITION 4.2   Let $ R$ be a ring. A polynomial $ f(X_0,X_1,\dots,X_n)\in R[X_0,X_1,\dots, X_n]$ is said to be homogenius of degree $ d$ if an equality

$\displaystyle f(\lambda X_0,\lambda X_1,\dots, \lambda X_n)
=
\lambda^d
f(X_0,X_1,\dots,X_n)
$

holds as a polynomial in $ n+2$ variables $ X_0,X_1,X_2,\dots, X_n, \lambda$ .

DEFINITION 4.3   Let $ k$ be a field.
  1. We put

    $\displaystyle \P ^n(k)=(k^{n+1}\setminus \{0\}) /k^\times
$

    and call it (the set of $ k$ -valued points of) the projective space. The class of an element $ (x_0,x_1,\dots,x_n)$ in $ \P ^n(k)$ is denoted by $ [x_0:x_1:\dots:x_n]$ .
  2. Let $ f_1,f_2,\dots, f_l \in k[X_0,\dots, X_n]$ be homogenious polynomials. Then we set

    % latex2html id marker 860
$\displaystyle V_h(f_1,\dots,f_l)=
\{
[x_0:x_1:x_2:\dots x_n] ; f_j (x_0,x_1,x_2,\dots,x_n)=0 \qquad(j=1,2,3,\dots,l)
\}.
$

    and call it (the set of $ k$ -valued point of) the projective variety defined by $ \{f_1,f_2,\dots,f_l\}$ .
(Note that the condition $ f_j(x)=0$ does not depend on the choice of the representative $ x\in k^{n+1}$ of $ [x]\in \P ^n(k)$ .)

LEMMA 4.4   We have the following picture of $ \P ^2$ .
  1. $\displaystyle \P ^2=\mathbb{A}^2\coprod \P ^1.
$

    That means, $ \P ^2$ is divided into two pieces % latex2html id marker 883
$ \{Z\neq 0\}=\complement V_h(Z)$ a nd $ V_h(Z)$ .
  2. $\displaystyle \P ^2=\mathbb{A}^2\cup \mathbb{A}^2 \cup \mathbb{A}^2.
$

    That means, $ \P ^2$ is covered by three ``open sets'' % latex2html id marker 891
$ \{Z\neq 0\}, \{Y\neq 0\}, \{X \neq 0\}$ . Each of them is isomorphic to the plane (that is, the affine space of dimension 2).

We quote the famous % latex2html id marker 739
\fbox{Weil conjecture}

CONJECTURE 4.5 (Now a theorem 1)   Let $ X$ be a projective smooth variety of dimension $ d$ . Then:
W1..
(Rationality)

$\displaystyle Z(X,T)=
\frac{P_1(X,T)P_3(X,T)\dots P_{2 d-1} (X,T)}
{P_0(X,T)P_2(X,T) \dots P_{2 d}(X,T)}
$

W2..
(Integrality) $ P_0(X,T)=1-T$ , % latex2html id marker 909
$ P_{2 d}( X,T)=1-q^d T$ , and for each $ r$ , $ P_r$ is a polynomial in $ \mathbb{Z}[T]$ which is factorized as

$\displaystyle P_r(X,T)=\prod (1-a_{r,i} T)
$

where $ a_{r,i}$ are algebraic integers.
W3..
(Functional Equation)

% latex2html id marker 921
$\displaystyle Z(X,1/q^d T)=\pm q^{d \chi/2 }T^\chi Z(t)
$

where $ \chi=(\Delta.\Delta)$ is an integer.
W4..
(Rieman Hypothesis) each $ a_{r,i}$ and its conjugates have absolute value % latex2html id marker 927
$ q^{r/2}$ .
W5..
If $ X$ is the specialization of a smooth projective variety $ Y$ over a number field, then the degeee of $ P_r(X,T)$ is equal to the $ r$ -th Betti number of the complex manifold $ Y(\mathbb{C})$ . (When this is the case, the number $ \chi$ above is equal to the ``Euler characteristic'' $ \chi=\sum_i (-1)^i b_i $ of $ Y(\mathbb{C})$ .)

It is a profound theorem, relating rational points % latex2html id marker 945
$ X(\mathbb{F}_q) $ of $ X$ over finite fields and topology of $ Y(\mathbb{C})$ .

The following proposition (which is a precursor of the above conjecture) is a special case

PROPOSITION 4.6 (Weil)   Let $ E$ be an elliptic curve over % latex2html id marker 958
$ \mathbb{F}_q$ . Then we have

% latex2html id marker 960
$\displaystyle Z(E/\mathbb{F}_q,T)
=
\frac{1-a T+ q T^2}
{(1-T)(1-q T)}.
$

where $ a$ is an integer which satisfies % latex2html id marker 964
$ \vert a\vert\leq 2 \sqrt{q}$ .

Note that for each $ E$ we have only one unknown integer $ a$ to determine the Zeta function. So it is enough to compute % latex2html id marker 970
$ \char93 E(\mathbb{F}_q)$ . to compute the Zeta function of $ E$ . (When % latex2html id marker 974
$ q=p$ then one may use the result in the preceding section.)

For a further study we recommend [1, Appendix C],[2].


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2015-05-09