Riemann hypothesis for curves over finite fields
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 extensionFqm of Fq, and V is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements. 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.
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