In mathematics, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as where V is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements and Nk is the number of points of V defined over the finite field extension Fqk of Fq.
Equivalently, the local zeta function is sometimes defined as follows: In other words, the local zeta function Z(V, t) with coefficients in the finite field Fq is defined as a function whose logarithmic derivative generates the number Nk of solutions of the equation defining V in the degree k extension Fqk.
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 set log Z equal to G, so and Z(0) = 1, since G(0) = 0, and Z(t) is a priori a formal power series.
Therefore, we have and for |t| small enough, and therefore The first study of these functions was in the 1923 dissertation of Emil Artin.
[2] The earliest known nontrivial 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.
In practice it makes Z a rational function of t, something that is interesting even in the case of V an elliptic curve over a finite field.
The global products of Z in the two cases used as examples in the previous section therefore come out as
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.
Alexander Grothendieck developed scheme theory for the purpose of resolving these.
(See étale cohomology for the basic formulae of the general theory.)
It is a consequence of the Lefschetz trace formula for the Frobenius morphism that Here
, 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