I recently stumbled upon the [slides of a talk](http://kskedlaya.org/slides/nagoya2010.pdf) given by Kedlaya, in which the following appears:



> 
> For $X$ of finite type over $F\_q$, a Weil cohomology theory, mapping $X$ to
>  certain vector spaces $H^i(X)$ over a field of characteristic zero, provides a
>  spectral interpretation of $\zeta(X,s)$ via the formula:
>  $$\zeta(X,s)=\prod\_i \det(1-q^{-s}\text{Frob}\_q,H^i(X))^{(-1)^{i+1}}$$
> 
> 
> 


Where $\zeta(X,s)$ is the arithmetic zeta function of a scheme $X$ of finite type over $\mathbb{Z}$. 


This is *incredibly* intriguing to me, and seems to be intimately related to some objects I'm trying to study - however, this slide really doesn't give me enough information to pull out some *juicy math*. Where does this formula come from (it seems very close to but not equivalent to the product in the definition of an Artin L-Function), and how does it provide a spectral interpretation of the arithmetic zeta function? 


If the answer is too big to be self-contained, I'd also appreciate a reference or some buzzwords to google. 



What is the spectral interpretation of the arithmetic zeta function?
First, it's worth mentioning that the above theorem refers to the (local) Hasse-Weil zeta function of a scheme $X$ of finite type over $\mathbb{F}\_q$ rather than the arithmetic zeta function of a scheme $X$ of finite type over $\mathbb{Z}$. Certainly these are related: for $X$ is a scheme of finite type over $\mathbb{Z}$ the arithmetic zeta function $\zeta(X,s)$ is the product of the (local) Hasse-Weil zeta functions $\zeta(X\_{\mathbb{F}\_p},s)$ for primes p, along with some Archimedean factor. 


Now what Kedlaya is referring to is Grothendieck's cohomological interpretation of the (local) Hasse-Weil zeta function $\zeta(X,s)$ for $X$ a scheme of finite type over $\mathbb{F}\_q$ and the first Weil conjecture, that $\zeta(X,s)$ is a rational function given as the product of characteristic polynomials of the (geometric) Frobenius acting on the $\ell$-adic cohomology of $X$. Put another way, $\zeta(X,s)$ has a spectral interpretation in terms of Frobenius eigenvalues. The idea is that since $\zeta(X,s)$ is a generating function for the number of $\mathbb{F}\_{q^n}$-rational points of $X$, and since the number of $\mathbb{F}\_{q^n}$-rational points of $X$ is given as the alternating sum of traces of the (geometric) Frobenius acting on the $\ell$-adic cohomology of $X$ by the Grothendieck-Lefschetz trace formula, it takes only a bit of algebra to conclude the cohomological interpretation of $\zeta(X,s)$. 


A good introductory reference is [Zeta Functions in Algebraic Geometry](http://www.math.lsa.umich.edu/~mmustata/zeta_book.pdf) and Milne's [Étale cohomology book](http://www.jmilne.org/math/CourseNotes/LEC.pdf), or any other book on étale cohomology that covers the Weil conjectures. 


