Local zeta-function
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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 Fk of F. The analogy with the Riemann zeta function
- <math>\zeta(s)<math>
comes via consideration of the logarithmic derivative
- <math>\zeta'(s)/\zeta(s)<math>.
Given F, there is, up to isomorphism, just one field Fk with
- [Fk: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
- Nk
of solutions in Fk; 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 Nk 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
- Nk = qk + 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 <math>\zeta(s)<math> and <math>\zeta(s)\zeta(s-1)<math>.
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.