Let \(B\) be a smooth affine algebraic variety over \(\mathbb{F}_{q}\). Let \(\mathcal{E}\) be an “overconvergent F-crystal” on \(B\). Then Dwork’s method shows that the zeta function associated with \(\mathcal{E}\), when viewed as a \(\mathbb{C}_{p}\)-valued function on the open unit disk, admits a meromorphic continuation on \(\mathbb{A}^{1,\text{an}}_{\mathbb{C}_{p}}\).

In Dwork’s original situation [Dwork, B. (1960). “On the rationality of the zeta
function of an algebraic variety”, Amer. J. Math., 82 631–648.], \(\mathcal{E}\)
is taken to be the “structure F-crystal” \(\mathcal{O}_{B}\). In this case, the
Weil conjecture predicts the zeta function is in fact a rational function, hence
must a fortiori be meromorphic. But Dwork *starts* by showing the meromorphy,
*then* bootstraps the meromorphy to rationality using an old result of
E. Borel. So the technical core of this paper is the proof of meromorphy
of the zeta function. Notably this is the *first* proof of Weil’s conjecture on
rationality. (Later Artin-Grothendieck also achieved this by building up the
étale cohomology, and applying the trace formula therein — fulfilling the
naive approach that one would imagine in a day-dream.)

Roughly speaking, Dwork showed that the zeta function can be written as an
alternating product of Fredholm determinants of endomorphisms on some \(p\)-adic
Banach space (the ambient Banach space, as it turns out, is the ring of
convergent analytic functions on a disk with radius larger than \(1\)).
According to Serre’s result [Serre, J. (1962). “Endomorphismes
complètement continus des espaces de Banach \(p\)-adiques”. Inst. Hautes Études
Sci. Publ. Math., (12), 69–85.] the Fredholm determinants are \(p\)-adic *entire
functions*. (Needless to say, Dwork didn’t have this abstract theorem at hand
when he wrote the paper, he built every brick by hand.) Hence the meromorphy
follows.

Let me be more precise about Dwork’s proof of meromorphy for the constant crystal. As we have said, the zeta function in this case computes the number of points of the base space \(B\):

\begin{equation*} \zeta(B, \mathcal{O}_{B}; t) = \exp \left( \sum_{s=1}^{\infty} \frac{|B(\mathbb{F}_{q^s})|}{s} \cdot t^{s}\right). \end{equation*}
By dévissage we could assume that \(B\) is a hypersurface in
\(\mathbb{G}_{\text{m}}^{n}\) defined by a *polynomial* \("f(x_1,\ldots,x_n)\) of
\(n\) variables. In this case, the number of points is given by the classical
formula

Here, \(\psi_s: \mathbb{F}_{p^s}^{\ast} \to \mathbb{C}_{p}\) is a nontrivial group
homomorphism (a nontrivial “additive character”), and \(\psi = \psi_1\).
In order to perform the calculation in a \(p\)-adic fashion Dwork ingeniously
defines a power series, now known as a *splitting function*, \(\theta(t) \in \mathbb{C}_{p}[\![t]\!]\) such that

- \(\theta\) converges on a disk of radius \(r > 1\), and
- for the Teichmüller lift \(\mathrm{Teich}(x)\) of \(x \in \mathbb{F}_{q}\), \(\theta(\mathrm{Teich}(x)) = \psi(x)\).

Thus, if \(F\) is the polynomial in \(\mathbb{C}_{p}[t_1, \ldots, t_n]\) whose coefficients are the Teichmüller lifts of those of \(f\), and if \(\mathfrak{G}(t_0,t_1,\ldots,t_n) = \theta(t_{0}F(t_1,\ldots,t_n))\), then \(\mathfrak{G}\) is convergent on some polydisk in \(\mathbb{C}_{p}^{n}\) of radius \(r > 1\) and the displayed equation on point-counting becomes

\begin{equation*} q^{s} |X(\mathbb{F}_{q^s})| = \sum_{\substack{(x_0,\ldots,x_n) \in \mathbb{C}_{p}^{\ast(n+1)} \\ x_i^{q_s} = x_i}} \prod_{j=0}^{s-1} \mathfrak{G}(x_0,\ldots,x_n)^{q^j} + (q^{s} -1)^{n}. \end{equation*}Dwork then considers the operator

\begin{equation*} \Psi_{q} : \mathbb{C}_{p}[\![t_0,\ldots, t_n]\!] \to \mathbb{C}_{p}[\![t_1, \ldots, t_n]\!] \end{equation*}such that

\begin{equation*} \sum_{\mathbf{w}\in{}\mathbb{Z}_{\geq{}0}^{n+1}}a_{\mathbf{w}}\prod_{i=0}^{n}t_i^{w_i} \mapsto \sum_{\mathbf{w}} a_{q\mathbf{w}}\prod_{i}t_{i}^{w_i}. \end{equation*}
(*Note*. This operator is some sort of the inverse of the “Frobenius”. This is
also considered in Atkin’s 1969 paper “Congruence Hecke
Operators” in “Number Theory”, Proceedings of symposia in pure mathematics,
Volume 12. Atkin’s paper is certainly later than Dwork’s, but some people
attributed this idea back to Atkin.)
It turns out that operator \(\Psi_{q}\circ \mathfrak{G}: h \mapsto \Psi_{q}(\mathfrak{G}h)\) can be defined
for those \(h\) that are convergent on the polydisk of radius \(r > 1\) and is
*completely continuous* in the sense of Serre (loc. cit.). Therefore,
\(\Psi_{q} \circ \mathfrak{G}\) is an “operator with trace” (or a
“nuclear operator” in the sense of Grothendieck’s thesis).

On the other hand, a formal computation shows that (*Dwork’s trace formula*)

Therefore one concludes that the zeta function equals an alternating product of
a rational function (the piece coming from the various \((q^s - 1)^n\) and an
*entire function* (the Fredholm determinant of the operator
\(\Psi_{q}\circ \mathfrak{G}\)). Hence the zeta function is a \(p\)-adic
meromorphic function on the space \(\mathbb{C}_{p}\). This concludes an overview
of Dwork’s argument on the meromorphy of zeta.

Dwork’s method is somehow robust, and can be generalized to the case when \(\mathcal{E}\) is any “overconvergent” crystal with a Frobenius structure. The role of being overconvergent can be visualized already in the previous argument: we need the operator \(\Psi_{q} \circ \mathfrak{G}\) to act on the space of functions on some disk that contains all the Teichmüller points, hence the radius of convergence of \(\mathfrak{G}\) must be slightly bigger than \(1\).

An important special situation is when \(f: X \to B\) is a smooth, projective family of algebraic varieties on \(B\). Then there is a natural F-crystal on \(B\) defined by the crystalline direct image, which captures the crystalline cohomology of the fibers of \(f\). Let us call it \(\mathcal{E}\). Lazda shows that \(\mathcal{E}\) is indeed overconvergent (confirming the conjecture of Berthelot), so Dwork’s method works on the nose, and yields that the zeta function associated with \(\mathcal{E}\) is a \(p\)-adic meromorphic function on the space \(\mathbb{C}_{p}\). On the one hand, the F-crystal \(\mathcal{E}\) is an algebraic incarnation of the classical Picard-Fuchs equation (or the Gauss-Manin system). When \(f\) lifts to characteristic 0, this system of differential equations is closely related to the deformation of period integrals of the fibers (in the language advocated by of P. Griffiths, a “variation of Hodge structure”). Therefore one expects to see some connections between the transcendental theory of periods and the algebraic expression of its zeta function.

The robust method however breaks down when dealing with F-crystals that are
not necessarily overconvergent. It might be strange to the reader since it seems
that one would not
be interested in non-overconvergent objects, as one knows the crystals
coming from geometry *are* overconvergent. The answer is that, while studying the
crystals coming from geometric objects, some subcrystals naturally appeared.
Given \(\mathcal{E}\) an F-crystal, a fixed point argument (cf.
[Katz, N. (1973). “Travaux de Dwork”. Séminaire Bourbaki, 24ème année
(1971/1972)], §4) extracts a natural and important subcrystal
\(\mathcal{E}^{\text{unit-root}}\) out of \(\mathcal{E}\), when we restrict
outselves to some suitable open subscheme of \(B\). It is known as the
“unit-root part” of the crystal \(\mathcal{E}\). For instance, when
\(\mathcal{E}\) is the crystal of the first cohomology of the Legendre family of
elliptic curves \(X \to B\), then there is an open subset \(B' \subset B\),
corresponding to the “nonsupersingular members” of \(X/B\), and
\(\mathcal{E}^{\text{unit-root}}\) is a rank 1 crystal on \(B'\) which, fiber-by-fiber,
captures the unique Frobenius eigenvalue of the first cohomology of the fiber
that is a \(p\)-adic unit. Already in the simplest situation we can
appreciate the arithmetic significance of \(\mathcal{E}^{\text{unit-root}}\).

In the Legendre case, Dwork shows that the theory of “excellent lifting of
Frobenius” of Serre-Tate allows one to prove that
\(\mathcal{E}^{\text{unit-root}}\) above is overconvergent (with respect to the
excellent lifting). Unfortunately, in general the unit-root crystal
\(\mathcal{E}^{\text{unit-root}}\) *may not be overconvergent* with
respect to *any* lifting of Frobenius on any lifting of \(B\)! See [Sperber, S.
(1980). “Congruence properties of the hyper-Kloosterman sum”. Compositio Math.,
40(1), 3–33.], §3 for Sperber’s proof of the nonexistence when
the F-crystal is related to a certain Kloosterman sum on \(\mathbb{G}_{\text{m}}\).

Dwork’s conjecture (now a theorem of Daqing Wan) is then, despite that \(\mathcal{E}^{\text{unit-root}}\) is possibly non-overconvergent, it’s zeta function is nevertheless \(p\)-adically meromorphic on the whole space \(\mathbb{C}_{p}\)! Dwork himself attacked this problem by means of the so-called “excellent lifting of Frobenius” method. But as we have mentioned, there exist examples where no matter how one chooses the lifting of Frobenius, the crystal is never overconvergent. Therefore Dwork’s excellent lifting method breaks down in general. Wan’s method is different from the classical ideas. We probably will talk about his proof in a future post.

*Date*. September 13, 2017