Local zeta-function

In number theory, the local zeta function Z(V, s) of V (sometimes called the congruent zeta function) is defined as

where N_m is the number of points of V defined over the degree m extension F_q^m of F_q, and V is a non-singular n-dimensional projective algebraic variety over the field F_q with q elements. By the variable transformation ${\displaystyle u=q^{-s}}$, 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 N_m of the solutions of equation, defining V, in the m degree extension F_m.

Given F, there is, up to isomorphism, just one field F_k 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 F_k 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.

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