ZETA FUNCTIONS. NO.4

YOSHIFUMI TSUCHIMOTO

Proposition 4.1. Let f ∈ 𝔽_{q}[X] be an irreducible polynomial in one variable of degree d. Let us consider V = {f}, an equation in one variable. Then:

(1)

V (𝔽_{q^{s}}) = \begin{cases}d & if d|s \\ 0 & otherwise \end{cases}

(2)

Z(V/𝔽_{q}, T) = \frac{1}{1 − T^{d}}

projective space and projective varieties.

Definition 4.2. Let R be a ring. A polynomial f(X_{0}, X_{1}, …, X_{n}) ∈ R[X_{0}, X_{1}, …, X_{n}] is said to be homogenius of degree d if an equality

f(λX_{0}, λX_{1}, …, λX_{n}) = λ^{d}f(X_{0}, X_{1}, …, X_{n})

holds as a polynomial in n + 2 variables X_{0}, X_{1}, X_{2}, …, X_{n}, λ.

Definition 4.3. Let k be a field.

(1) We put

ℙ^{n}(k) = (k^{n+1} \ {0})/k^{×}

and call it (the set of k-valued points of) the projective space. The class of an element (x_{0}, x_{1}, …, x_{n}) in ℙ^{n}(k) is denoted by [x_{0} : x_{1} : ⋯: x_{n}].

(2) Let f_{1}, f_{2}, …, f_{l} ∈ k[X_{0}, …, X_{n}] be homogenious polynomials. Then we set

V_{h}(f_{1}, …, f_{l}) = {[x_{0} : x_{1} : x_{2} : … x_{n}]; f_{j}(x_{0}, x_{1}, x_{2}, …, x_{n}) = 0 (j = 1, 2, 3, …, l)}.

and call it (the set of k-valued point of) the projective variety defined by {f_{1}, f_{2}, …, f_{l}}.

(Note that the condition f_{j}(x) = 0 does not depend on the choice of the representative x ∈ k^{n+1} of [x] ∈ ℙ^{n}(k).)

Lemma 4.4. We have the following picture of ℙ^{2}.

YOSHIFUMI TSUCHIMOTO

(1)

ℙ^{2} = 𝔸^{2} ∐ ℙ^{1}.

That means, ℙ^{2} is divided into two pieces {Z ≠ 0} = ∁V_{h}(Z) a nd V_{h}(Z).

(2)

ℙ^{2} = 𝔸^{2} ∪ 𝔸^{2} ∪ 𝔸^{2}.

That means, ℙ^{2} is covered by three “open sets” {Z ≠ 0}, {Y ≠ 0}, {X ≠ 0}. Each of them is isomorphic to the plane (that is, the affine space of dimension 2).

We quote the famous Weil conjecture

Conjecture 4.5 (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 of the above theorem.

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

Proposition 4.6 (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.