Local zeta function

In number theory, a local zeta-function is a generating function

Z(t)

for the number of solutions of a set of equations defined over a finite field F, in extension fields F_k of F. The analogy with the Riemann zeta-function

$\zeta (s)$

comes via consideration of the logarithmic derivative

$\zeta '(s)/\zeta (s)$.

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

[F_k:F] = k,

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

N_k

of solutions in F_k; and create the generating function

G(t) = N₁.t + N₂.t²/2 + ... .

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

Z = exp(G);

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

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

G(t) = −log(1 − t)

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

Z(t) = 1/(1 − t).

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

N_k = q^k + 1

and

G(t) = −log(1 − t) − log(1 − qt),

for |t| small enough.

In this case we have

Z(t) = 1/{(1 − t)(1 − qt)}.

The relationship between the definitions of G and Z can be explained in a number of ways. 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/p.Z as p runs over all prime numbers. In that relationship, 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). This explains too why the logarithmic derivative with respect to s is used.

With that understanding, the products of the Z in the two cases come out as $\zeta (s)$ and $\zeta (s)\zeta (s-1)$.

Riemann hypothesis for curves over finite fields

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

Z(t) = P(t)/{(1 − t)(1 − qt)},

with P(t) a polynomial, of degree 2g where g is the genus of C. The Riemann hypothesis for curves over finite fields states that the roots of P have absolute value

q^−1/2,

where q = |F|.

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

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, finally proved a generation later. See étale cohomology for the basic formulae of the general theory.