Local zeta function
In number theory, 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.[1]
Making the variable transformation t = q−s, gives
as the formal power series in the variable .
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.
Motivations[edit]
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.
The local Z zeta functions are multiplied to get global zeta functions,
These generally involve different finite fields (for example the whole family of fields Z/pZ as p runs over all prime numbers).
In these fields, the variable t is substituted by p−s, where s is the complex variable traditionally used in Dirichlet series. (For details see Hasse–Weil zeta function.)
The global products of Z in the two cases used as examples in the previous section therefore come out as and after letting .
