Local zeta function: Difference between revisions

Revision as of 13:54, 22 October 2018

Revision as of 22:18, 18 June 2019

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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},{\Bbbmathbb 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_q 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>.