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# Zeta functions. No.5

Yoshifumi Tsuchimoto

We quote the famous

CONJECTURE 5.1 (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 5.2 (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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2013-07-15