In this article by Katz and Tate here, there's a nice account of Dwork's argument for showing the rationality of the zeta function part of the Weil conjectures. Here is an excerpt.
To recapitulate, we now know that the zeta function as power series has integer coefficients and that it is the ratio of two $p$-adically entire functions. We also know the zeta function has a nonzero radius of archimedean convergence (since we have the trivial archimedean bound $N_d \le (q^d - 1)^n$). Bernie's third new idea is to generalize a classical but largely forgotten result of E. Borel to show that any power series with these three properties is a rational function. Thus he proves the rationality of the zeta function.
Can anybody give a sketch of the argument for this "third new idea"? What is the crux of the proof of Dwork's generalization of Borel's result that any power series with the aforementioned three properties is a rational function? What is the intuition behind the proof, what are the key steps that the proof boils down to?
