As the question title suggests, what is the crux of Dwork's proof of the rationality of the zeta function? What is the intuition behind the proof, what are the key steps that the proof boils down to?



Crux of Dwork's proof of rationality of the zeta function?
There is an excellent book by Neal Koblitz "p-adic numbers, p-adic analysis and zeta-functions" were the Dwork's proof is stated in a very detailed way, including all preliminaries from p-adic analysis. Let me sketch this proof in comparison with Weil's program of proving his conjecture.


First, any variety $X$ can be covered by affine charts $U\_i$ with all intersections $U\_{i\_1}\cap \dots\cap U{i\_k}$ being affine. The zeta-function of $X$ can be expressed in terms of zeta-functions of these affine varieties using inclusion-exclusion formula(I omit the base field and the formal variable to shorten notations) $$Z\_X=\prod\limits\_{i\_1<\dots<i\_k}Z\_{U\_{i\_1}\cap\dots \cap U\_{i\_K}}^{(-1)^{k+1}}$$ Thus, it is enough to prove rationality for affine varieties. Next, any affine variety is an intersection of hypersurfaces and the union of hypersurfaces is a hypersurface, so again inclusion-exclusion formula(with $\cap$ and $\cup$ swapped) reduces the problem to hypersurfaces.


Now, the idea of Dwork is to prove first that $Z\_X$ is $p$-adic meromorphic on $\mathbb{C}\_p$ for a hypersurface $X\subset \mathbb{A}^n$. He proceeds by induction on dimension. We cut a hypersurface $f(X)=0$ into lower-dimensional hypersurfaces $f(X)=0,x\_i=0$(to be precise, this variety may happen to be the whole $\mathbb{A}^{n-1}$ but this is even better since the rationality for affine space is obvious) and open subvariety $f(X),x\_i\neq 0$ for all $i$. By the same inclusion-exclusion argument and induction, it is enough to prove rationality for this open variety.


As far as I understand, the following computation is the main insight of Dwork which resembles the Weil's idea. He expresses number of points of this variety over $\mathbb{F}\_{q^k}$ in terms of trace of $\Psi^k$ where $\Psi$ is a certain linear operator. That gives an expression for zeta-function in terms of characteristic "polynomial" of $\Psi$. In contrast to Frobenius on Weil cohomology, $\Psi$ acts on an infinite-dimensional space, so its characteristic polynomial is not a polynomial, but rather a meromorphic series(one should do some work to make sense of determinant and trace of an infinite-dimensional operator -- this is perfectly carried out in Koblitz's book). This proves that $Z\_X$ is $p$-adically meromorphic.


Finally, any meromorphic series with integral coefficients and properly bounded coefficients(for $Z\_X$ the bound comes from $\# X(\mathbb{F}\_{q^k})\leq q^{nk}$) is a rational function(this follows from p-adic Weierstrass preparation theorem and characterization of rational functions series as those admitting a linear recurrence relation on coefficients). 


