Crux of Dwork's proof of rationality of the zeta function?

Question

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?

Answer 1

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).

Answer 2

See Terry Tao's blog post. A very simple proof of a slightly weaker result is given by Mike Larsen.

Answer 3

Sasha has already pointed you to the primary source I used some years ago for my "expository" undergraduate thesis on Dwork's Theorem: Koblitz's p-adic Numbers, p-adic Analysis, and Zeta-Functions, which is available in 166 searchable pages here.

I've uploaded a copy of my thesis, which should constitute a relatively easy to digest ~ 50 page write-up of the proof. There is no significant difference from Koblitz's book; I worked out some of the pieces that were left as exercises or examples (e.g., showing that the zeta function for $f = x_1 x_4 - x_2 x_3 - 1$ is rational i.e. $\frac{1-qT}{1-q^3 T}$, which I re-produced in MO 117904) and omitted some of the proofs of cited theorems (e.g., the $p$-adic Weierstrass Preparation Theorem) with pointers to Koblitz's book or Gouvêa's intro to $p$-adic numbers when necessary.

As I understood matters almost a decade ago, there were a couple of important linear maps mentioned by Koblitz (included in my section 3.2) $G$ and $T_q$, which were composed to give the trace operator, $\Psi$, already mentioned. I introduce a slight bit of terminology ("admissible" in Definition 3.14) to discuss the trace of a linear operator from a certain infinite dimensional vector space to itself, and one of the key facts drawn upon in working towards the end of the proof is the power series identity:

$$ \det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right) $$

for a linear operator $\psi$ going from a vector space to itself. I re-mention that identity here, because I asked about its history in MSE 277124 for which Markus Scheuer gave a great answer. This identity arises in proving that a sufficient-to-consider modification of the zeta function $Z'(T)$ is $p$-adic meromorphic, which is established through a bit of induction, analysis of the above identity, and the inclusion/exclusion principle (the last of which re-arises, as Sasha mentions, in proving Dwork's theorem for affine hypersurfaces can be extended to projective varieties).

I do not believe that there is anything novel in my undergrad write-up, but I think it is possible that its presentation as being intended for college juniors/seniors could make the proof, hence the underlying intuition, easier to grok.