CONGRUENT ZETA FUNCTIONS. NO.11

YOSHIFUMI TSUCHIMOTO

We quote the famous Weil conjecture

Conjecture 11.1 (Now a theorem^{1} ). Let X be a projective smooth variety of dimension d. Then:

W1. (Rationality)

Z(X, T) = \frac{P_{1}(X, T)P_{3}(X, T) … P_{2d−1}(X, T)}{P_{0}(X, T)P_{2}(X, T) … P_{2d}(X, T)}

W2. (Integrality) P_{0}(X, T) = 1 − T, P_{2d}(X, T) = 1 − q^{d}T, and for each r, P_{r} is a polynomial in ℤ[T] which is factorized as

P_{r}(X, T) = ∏(1 − a_{r,i}T)

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

Z(X, 1/q^{d}T) = ±q^{dχ/2}T^{χ}Z(t)

where χ = (Δ.Δ) is an integer. W4. (Rieman Hypothesis) each a_{r,i} and its conjugates have absolute

value 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 (ℂ). (When this is the case, the number χ above is equal to the “Euler characteristic” χ = ∑_{i}(−1)^{i}b_{i} of Y (ℂ).)

It is a profound theorem, relating rational points X(𝔽_{q}) of X over finite fields and topology of Y (ℂ).

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

Proposition 11.2 (Weil). Let E be an elliptic curve over 𝔽_{q}. Then we have

Z(E/𝔽_{q}, T) = \frac{1 − aT + qT^{2}}{(1 − T)(1 − qT)}.

where a is an integer which satisfies |a| ≤ 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 #E(𝔽_{q}). to compute the Zeta function of E. (When q = p then one may use the result in the preceding section.)

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

References

[1] R. Hartshorne, Algebraic geometry, Springer Verlag, 1977.

[2] J. S. Milne, Étale cohomology, Princeton University Press, 1980.

1There are a lot of people who contributed. See the references.