This is a followup to my question [here](https://mathoverflow.net/questions/249523/crux-of-dworks-proof-of-rationality-of-the-zeta-function).


Here is a note of Michael Larsen where he gives a very simple proof of a slightly weaker result than Dwork's rationality of the zeta function.


<http://mlarsen.math.indiana.edu/~larsen/papers/zeta.pdf>


As the question title suggests, what is needed to beef this up into a complete proof a la Dwork of rationality of the zeta function?



Weaker version of Dwork's rationality of zeta function, what is needed to beef up into a complete proof?
Not quite an answer, but a heuristics from the point of view of Weil philosophy about why the rationality mod $p$ is much easier.


Weil reduced the rationality of zeta-function to the existence of a good cohomology theory. The crucial property of a cohomology which satisifies the Weil axioms is that it has zero-characteristics coefficients. This is needed because Lefshetz formula is an identity in the coefficients ring, so to compute the number of $\mathbb{F}\_q$-points precisely, we need that coefficents ring does not have any $\mathbb{Z}$-torsion.


But if you are interested only in the number of points modulo $p$, a cohomology theory with $\mathbb{F}\_p$-coefficients which satisefies the Lefshetz trace foormula would fit(all the formulas for zeta functions should just be reduced modulo $p$). 


It is very easy to give an example of such cohomology theory. It is $X\mapsto H^i(X,\mathcal{O}\_X)$. Holomorphic Lefshetz trace formula applied to Frobenius gives $$\sum\limits\_{x\in X(\mathbb{F}\_{p^n})}\frac{1}{\det(1-Fr|\_{T\_{x,X}})}=\sum\limits\_i(-1)^i\mathrm{Tr}\, Fr|\_{H^i(X,\mathcal{O}\_X)}$$ But Frobenius is inseparable, so its differential is zero and the LHS is stil $\# X(\mathbb{F}\_{p^n})$ so we get the desired trace formula modulo $p$.


So, the rationality mod $p$ should be much easier. You can also trace the analogy between Chevalley-Warning method and holomorphic Lefshetz trace formula in the proof of equivalence of different definitions of supersingular elliptic curve(see i.e. Hartshorne).


Historically, the rationality mod $p$ was certainly known already to Serre, since he considered Witt cohomology $H^i(X,W\mathcal{O}\_X)$ as a candidate for Weil cohomology.


