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 to $B$, 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. 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

\begin{align*} q^{s} \cdot |B(\mathbb{F}_{q}^s)| & = \sum_{x_0, \ldots,x_n \in \mathbb{F}_{q^s}^{\ast}} \psi_s(x_{0} \cdot f (x_1,\ldots,x_n)) + (q^{s} - 1)^{n} \\ & = \sum_{x_0,\ldots, x_n \in \mathbb{F}_{q^s}^{\ast}} \prod_{j=0}^{s-1} \psi(x_{0} \cdot f(x_1,\ldots,x_n))^{q^j} + (q^{s} - 1)^{n}. \end{align*}

Here, $\psi_s: \mathbb{F}_{p}^{\ast} \to \mathbb{C}_{p}$ 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 $\theta(t) \in \mathbb{C}_{p}[\![t]\!]$ 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*} \mathfrak{A}_{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 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 $\mathfrak{A}_{q} \circ \mathfrak{G}: h \mapsto \mathfrak{A}_{q}(\mathfrak{G}h)$ 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, $\mathfrak{A}_{q} \circ \mathfrak{G}$ 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)

\begin{equation*} (q^{s}-1)^{n}\mathrm{Tr}[(\mathfrak{A}_{q} \circ \mathfrak{G})^{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}. \end{equation*}

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 $\mathfrak{A}_{q}\circ \mathfrak{G}$). 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 $\mathfrak{A}_{q} \circ \mathfrak{G}$ 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 $\mathcal{E}$. Berthelot shows that $\mathcal{E}$ is indeed overconvergent, so Dwork’s method works on the nose, and yields that the zeta function associated to $\mathcal{E}$ 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 Picad-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, the infamous “Dwork’s trick” (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 associated 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