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Zeta functions. No.5

Yoshifumi Tsuchimoto

We quote the famous % latex2html id marker 576
\fbox{Weil conjecture}

CONJECTURE 5.1 (Now a theorem 1)   Let $ X$ be a projective smooth variety of dimension $ d$ . Then:

$\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)}

(Integrality) $ P_0(X,T)=1-T$ , % latex2html id marker 657
$ 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.
(Functional Equation)

% latex2html id marker 669
$\displaystyle Z(X,1/q^d T)=\pm q^{d \chi/2 }T^\chi Z(t)

where $ \chi=(\Delta.\Delta)$ is an integer.
(Rieman Hypothesis) each $ a_{r,i}$ and its conjugates have absolute value % latex2html id marker 675
$ q^{r/2}$ .
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 693
$ 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 5.2 (Weil)   Let $ E$ be an elliptic curve over % latex2html id marker 706
$ \mathbb{F}_q$ . Then we have

% latex2html id marker 708
$\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 712
$ \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 718
$ \char93 E(\mathbb{F}_q)$ . to compute the Zeta function of $ E$ . (When % latex2html id marker 722
$ 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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