Local zeta function
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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.