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 ζ(s) comes via consideration of the logarithmic derivative ζ'(s)/ζ(s).

Given F, there is, up to isomorphism, just one field F_k with [F_k:F] = k, for k = 1,2, ... . Given 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) = 1 + N₁.t + N₂.t² + ... .

The correct definition for Z(t) is to make -Z'/Z equal to G; this fixes G up to multiplication by a constant, and we assume Z(0) = 1.

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) = 1/(1 - t)

is the expansion of a geometric series. In this case

Z(t) = 1/(1 - t)

also, as one checks directly.

To take something more interesting, take V to 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) = 1/(1 - t) + 1/(1 - qt).

In this case we have

Z(t) = 1/((1 - t)(1 - qt)}

(currently something wrong with the formulae).

The relationship between the definitions of G and Z can be explained in a number of ways.

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.

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