**Yoshifumi Tsuchimoto**

holds as a polynomial in variables .

- We put
**projective space**. The class of an element in is denoted by . - Let
be homogenious polynomials. Then we set
**projective variety**defined by .

We quote the famous

- W1..
- (Rationality)
- W2..
- (Integrality)
,
, and for each
,
is
a polynomial in
which is factorized as
- W3..
- (Functional Equation)
- W4..
- (Rieman Hypothesis) each and its conjugates have absolute value .
- W5..
- If is the specialization of a smooth projective variety over a number field, then the degeee of is equal to the -th Betti number of the complex manifold . (When this is the case, the number above is equal to the ``Euler characteristic'' of .)

It is a profound theorem, relating rational points of over finite fields and topology of .

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

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