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".

Formulation

The analogy with the
Riemann zeta-function

:$zeta(s)\; ,$

comes via consideration of the logarithmic derivative

:$zeta\text{'}(s)/zeta(s)\; ,$.

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

:$G(t)\; =\; N\_1t\; +N\_2t^2/2\; +\; N\_3t^3/3\; +cdots\; ,$.

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

:$Z=\; exp\; (G(t))\; ,$

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

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

:"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")}.

Motivations

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 2"g" 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.