Editing Local zeta function (section)

==General formulas for the zeta function==

It is a consequence of the [[Lefschetz trace formula]] for the [[Frobenius morphism]] that

:<math>Z(X,t)=\prod_{i=0}^{2\dim X}\det\big(1-t \mbox{Frob}_q |H^i_c(\overline{X},{\mathbb Q}_\ell)\big)^{(-1)^{i+1}}.</math>

Here <math>X</math> is a separated scheme of finite type over the finite field ''F'' with <math>q</math> elements, and Frob<sub>q</sub> is the geometric Frobenius acting on <math>\ell</math>-adic étale cohomology with compact supports of <math>\overline{X}</math>, the lift of <math>X</math> to the algebraic closure of the field ''F''.  This shows that the zeta function is a rational function of <math>t</math>.

An infinite product formula for  <math>Z(X, t)</math> is

:<math>Z(X, t)=\prod\ (1-t^{\deg(x)})^{-1}.</math>

Here, the product ranges over all closed points ''x'' of ''X'' and deg(''x'') is the degree of ''x''.
The local zeta function ''Z(X, t)'' is viewed as a function of the complex variable ''s'' via the change of 
variables ''q<sup>−s</sup>''.

In the case where ''X'' is the variety ''V'' discussed above, the closed points 
are the equivalence classes ''x=[P]'' of points ''P'' on <math>\overline{V}</math>, 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

:<math>N_1 +N_2t^1 + N_3t^2 +\cdots \,</math>.