1. Formulation

Given a finite field F, there is, up to isomorphism, just one field F_k with

[math]\displaystyle{ [ F_k : F ] = k \, }[/math],

for k = 1, 2, ... . Given a set of polynomial equations — or an algebraic variety V — defined over F, we can count the number

[math]\displaystyle{ N_k \, }[/math]

of solutions in F_k and create the generating function

[math]\displaystyle{ G(t) = N_1t +N_2t^2/2 + N_3t^3/3 +\cdots \, }[/math].

The correct definition for Z(t) is to make log Z equal to G, and so

[math]\displaystyle{ Z= \exp (G(t)) \, }[/math]

we will have Z(0) = 1 since G(0) = 0, and Z(t) is a priori a formal power series.

Note that the logarithmic derivative

[math]\displaystyle{ Z'(t)/Z(t) \, }[/math]

equals the generating function

[math]\displaystyle{ G'(t) = N_1 +N_2t^1 + N_3t^2 +\cdots \, }[/math].

2. Examples

For example, assume all the N_k are 1; this happens for example if we start with an equation like X = 0, so that geometrically we are taking V a point. Then

[math]\displaystyle{ G(t) = -\log(1 - t) }[/math]

is the expansion of a logarithm (for |t| < 1). In this case we have

[math]\displaystyle{ Z(t) = \frac{1}{(1 - t)}\ . }[/math]

To take something more interesting, let V be the projective line over F. If F has q elements, then this has q + 1 points, including as we must the one point at infinity. Therefore, we shall have

[math]\displaystyle{ N_k = q^k + 1 }[/math]

and

[math]\displaystyle{ G(t) = -\log(1 - t) -\log(1 - qt) }[/math]

for |t| small enough.

In this case we have

[math]\displaystyle{ Z(t) = \frac{1}{(1 - t)(1 - qt)}\ . }[/math]

The first study of these functions was in the 1923 dissertation of Emil Artin. He obtained results for the case of hyperelliptic curve, and conjectured the further main points of the theory as applied to curves. The theory was then developed by F. K. Schmidt and Helmut Hasse.^[1] The earliest known non-trivial cases of local zeta-functions were implicit in Carl Friedrich Gauss's Disquisitiones Arithmeticae, article 358; there certain particular examples of elliptic curves over finite fields having complex multiplication have their points counted by means of cyclotomy.^[2]

For the definition and some examples, see also.^[3]

3. Motivations

The relationship between the definitions of G and Z can be explained in a number of ways. (See for example the infinite product formula for Z below.) In practice it makes Z a rational function of t, something that is interesting even in the case of V an elliptic curve over finite field.

It is the functions Z that are designed to multiply, to get global zeta functions. Those involve different finite fields (for example the whole family of fields Z/pZ as p runs over all prime numbers). In that connection, the variable t undergoes substitution by p^−s, where s is the complex variable traditionally used in Dirichlet series. (For details see Hasse-Weil zeta-function.)

With that understanding, the products of the Z in the two cases used as examples come out as [math]\displaystyle{ \zeta(s) }[/math] and [math]\displaystyle{ \zeta(s)\zeta(s-1) }[/math].

4. Riemann Hypothesis for Curves over Finite Fields

For projective curves C over F that are non-singular, it can be shown that

[math]\displaystyle{ Z(t) = \frac{P(t)}{(1 - t)(1 - qt)}\ , }[/math]

with P(t) a polynomial, of degree 2g where g is the genus of C. Rewriting

[math]\displaystyle{ P(t)=\prod^{2g}_{i=1}(1-\omega_i t)\ , }[/math]

the Riemann hypothesis for curves over finite fields states

[math]\displaystyle{ |\omega_i|=q^{1/2}\ . }[/math]

For example, for the elliptic curve case there are two roots, and it is easy to show the absolute values of the roots are q^1/2. Hasse's theorem is that they have the same absolute value; and this has immediate consequences for the number of points.

André Weil proved this for the general case, around 1940 (Comptes Rendus note, April 1940): he spent much time in the years after that writing up the algebraic geometry involved. This led him to the general Weil conjectures, Alexander Grothendieck developed the scheme theory for the sake of resolving it and finally, Pierre Deligne had proved a generation later. See étale cohomology for the basic formulae of the general theory.

5. General Formulas for the Zeta Function

It is a consequence of the Lefschetz trace formula for the Frobenius morphism that

[math]\displaystyle{ Z(X,t)=\prod_{i=0}^{2\dim X}\det\big(1-t \mbox{Frob}_q |H^i_c(\overline{X},{\mathbb Q}_\ell)\big)^{(-1)^{i+1}}. }[/math]

Here [math]\displaystyle{ X }[/math] is a separated scheme of finite type over the finite field F with [math]\displaystyle{ q }[/math] elements, and Frob_q is the geometric Frobenius acting on [math]\displaystyle{ \ell }[/math]-adic étale cohomology with compact supports of [math]\displaystyle{ \overline{X} }[/math], the lift of [math]\displaystyle{ X }[/math] to the algebraic closure of the field F. This shows that the zeta function is a rational function of [math]\displaystyle{ t }[/math].

An infinite product formula for [math]\displaystyle{ Z(X, t) }[/math] is

[math]\displaystyle{ Z(X, t)=\prod\ (1-t^{\deg(x)})^{-1}. }[/math]

Here, the product ranges over all closed points x of X and deg(x) is the degree of x. The local zeta function Z(X, t) is viewed as a function of the complex variable s via the change of variables q^−s.

In the case where X is the variety V discussed above, the closed points are the equivalence classes x=[P] of points P on [math]\displaystyle{ \overline{V} }[/math], where two points are equivalent if they are conjugates over F. The degree of x is the degree of the field extension of F generated by the coordinates of P. The logarithmic derivative of the infinite product Z(X, t) is easily seen to be the generating function discussed above, namely

[math]\displaystyle{ N_1 +N_2t^1 + N_3t^2 +\cdots \, }[/math].