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# Zeta functions. No.4

Yoshifumi Tsuchimoto

PROPOSITION 4.1   Let be an irreducible polynomial in one variable of degree . Let us consider , an equation in one variable. Then:

DEFINITION 4.2   Let be a ring. A polynomial is said to be homogenius of degree if an equality

holds as a polynomial in variables .

DEFINITION 4.3   Let be a field.
1. We put

and call it (the set of -valued points of) the projective space. The class of an element in is denoted by .
2. Let be homogenious polynomials. Then we set

and call it (the set of -valued point of) the projective variety defined by .
(Note that the condition does not depend on the choice of the representative of .)

LEMMA 4.4   We have the following picture of .
1. That means, is divided into two pieces a nd .
2. That means, is covered by three open sets'' . Each of them is isomorphic to the plane (that is, the affine space of dimension 2).

We quote the famous

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

W2..
(Integrality) , , and for each , is a polynomial in which is factorized as

where are algebraic integers.
W3..
(Functional Equation)

where is an integer.
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

PROPOSITION 4.6 (Weil)   Let be an elliptic curve over . Then we have

where is an integer which satisfies .

Note that for each we have only one unknown integer to determine the Zeta function. So it is enough to compute . to compute the Zeta function of . (When 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