What is the spectral interpretation of the arithmetic zeta function?

Question

I recently stumbled upon the slides of a talk 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.

Answer 1

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 and Milne's Étale cohomology book, or any other book on étale cohomology that covers the Weil conjectures.