Suppose that V is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements. In number theory, the local zeta function Z(V, s) of V (sometimes called the congruent zeta function) is defined as
where Nm is the number of points of V defined over the degree m extension Fqm of Fq. By the variable transformation , then it is defined by
as the formal power series of the variable u.
Equivalently, sometimes it is defined as follows:
In other word, the local zeta function Z(V,u) with coefficients in the finite field F is defined as a function whose logarithmic derivative generates the numbers Nm of the solutions of equation, defining V, in the m degree extension Fm.
Given F, there is, up to isomorphism, just one field Fk with
for k = 1, 2, ... . Given a set of polynomial equations — or an algebraic variety V — defined over F, we can count the number
of solutions in Fk and create the generating function
The correct definition for Z(t) is to make log Z equal to G, and so
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
equals the generating function
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
is the expansion of a logarithm (for |t| < 1). In this case we have
for |t| small enough.
In this case we have
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. 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.
For the definition and some examples, see also.
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 and .
Riemann hypothesis for curves over finite fields
For projective curves C over F that are non-singular, it can be shown that
with P(t) a polynomial, of degree 2g where g is the genus of C. Rewriting
the Riemann hypothesis for curves over finite fields states
For example, for the elliptic curve case there are two roots, and it is easy to show the absolute values of the roots are q1/2. 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.
General formulas for the zeta function
Here is a separated scheme of finite type over the finite field F with elements, and Frobq is the geometric Frobenius acting on -adic étale cohomology with compact supports of , the lift of to the algebraic closure of the field F. This shows that the zeta function is a rational function of .
An infinite product formula for is
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 , 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