Dwork’s Conjecture On Unit Root Zeta Functions

By DAQING WAN*

1. Introduction

In this article, we introduce a systematic new method to investigate the conjectural p-adic meromorphic continuation of Dwork’s unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic p.

After his pioneer p-adic investigation of Weil’s well known conjectures on the zeta function of an algebraic variety over a finite field, Dwork went on to study the p-adic analytic variation of a family of such zeta functions when the variety moves through an algebraic family. In the course of doing so, he was led to a new zeta function called the unit root zeta function, which goes beyond the reach of the existing theory. He conjectured [8] that such a unit root zeta function is p-adic meromorphic everywhere. These unit root zeta functions contain important arithmetic information about a family of algebraic varieties. They are truly p-adic in nature and are transcendental functions. They seem to be quite mysterious. In fact, no single “non-trivial” example has been proved to be true about this conjecture, other than the “trivial” overconvergent (or ∞ log-convergent) case for which his classical p-adic theory already applies, see [6-11] [26] for various attempts. In this article, we introduce a systematic new method to study such unit root zeta functions. Our method can be used to prove the conjecture in the case when the involved unit root F-crystal has rank one. In particular, this settles the first “non-trivial” case, the rank one unit root F-crystal coming from the family of higher dimensional Kloosterman sums. Our method further allows us to understand reasonably well about analytic variation of an arithmetic family of such rank one unit root zeta functions, motivated by the Gouvêa-Mazur conjecture about dimension variation of classical and p-adic modular forms. We shall introduce another systematic method in a future paper [29] which combined with the method in the present paper will be able to prove Dwork’s conjecture in higher rank case.

*This work was partially supported by NSF. The author wishes to thank P. Deligne and N.

Katz for valuable discussions at one stage of this work during 1993-94 period. The author would also like to thank S. Sperber for his careful reading of the manuscript and for his question on the independence of basis which led to considerable improvement in the infinite rank setup of section 2.

To explain our results in this introduction, we shall restrict our attention to the family of L-functions of higher dimensional Kloosterman sums parameterized by the one dimensional torus G_{m}. This is the most concrete but an essential family for Dwork’s conjecture. Let F_{q} be the finite field of q elements of characteristic p. Let Ψ be a fixed non-trivial complex valued additive character of the finite field F_{p}. Let n be a positive integer. For each non-zero element ȳ ∈ F_{q}^{∗} (the letter y is reserved to denote the Teichmüller lifting of ȳ) and each positive integer k, we define the n-dimensional Kloosterman sum over F_{q^{k}} to be

K_{k}(ȳ, n) = ∑_{x_{i}∈F_{q^{k}}^{∗}} Ψ(Tr_{F_{q^{k}}/F_{p}}(x_{1} + ⋯ + x_{n} + \frac{ȳ}{x_{1} ⋯ x_{n}})).

The L-function attached to the sequence K_{k}(ȳ, n) (k = 1, 2, ⋯) of character sums is defined to be the exponential generating function

L(ȳ, n, T) = exp(∑_{k=1}^{∞} \frac{K_{k}(ȳ, n)}{k}T^{k}).

It has an Euler product expansion which shows that L(ȳ, n, T) is a power series whose coefficients are algebraic integers in the p-th cyclotomic field. The product of all the p − 1 conjugates (when Ψ varies over the p − 1 non-trivial additive characters of F_{p}) of L(ȳ, n, T) gives the non-trivial part of the zeta function of the affine variety over F_{q} defined by

z^{p} − z = x_{1} + ⋯ + x_{n} + \frac{ȳ}{x_{1} ⋯ x_{n}} .

Thus, it suffices to understand the above L-function.

It is well known that L(ȳ, n, T)^{(−1)^{n−1}} is a polynomial of degree n + 1. Thus, there are n + 1 algebraic integers α_{0}(ȳ), ⋯, α_{n}(ȳ) such that

L(ȳ, n, T)^{(−1)^{n−1}} = (1 − α_{0}(ȳ)T) ⋯ (1 − α_{n}(ȳ)T).

Equivalently, for each integer k ≥ 1, we have the formula

(−1)^{n}K_{k}(ȳ, n) = α_{0}(ȳ)^{k} + ⋯ + α_{n}(ȳ)^{k}.

By Deligne’s theorem [5], these algebraic integers α_{i}(ȳ) have complex absolute value \sqrt{q}^{n}. For each prime ℓ ≠ p, these α_{i}(ȳ) are ℓ-adic unit. To describe their p-adic absolute values, we fix an embedding of the algebraic numbers Q̄ into Q̄_{p} and order the α_{i}(ȳ) such that

ord_{q}α_{0}(ȳ) ≤ ⋯ ≤ ord_{q}α_{n}(ȳ).

Then Sperber’s theorem [21] says that

ord_{q}α_{0}(ȳ) = 0, ord_{q}α_{1}(ȳ) = 1, ⋯, ord_{q}α_{n}(ȳ) = n.

This means that there is exactly one characteristic root of slope i for each integer 0 ≤ i ≤ n. In particular, there is exactly one p-adic unit root α_{0}(ȳ) (slope zero).

In a more conceptual language, the L-function L(ȳ, n, T)^{(−1)^{n−1}} is given by the characteristic polynomial of the Frobenius map at the closed point ȳ of a rank (n + 1) ordinary overconvergent F-crystal M_{n} over the torus G_{m}/F_{p}. This F-crystal M_{n} is not a unit root F-crystal. It is ordinary (Newton polygon coincides with Hodge polygon fibre by fibre). Its unit root part U_{n} is a sub F-crystal of rank one on the torus G_{m}/F_{p}, but no longer overconvergent in general. Denote the 1 × 1 matrix of the Frobenius map of U_{n} by α(y). This is a convergent (but not overconvergent in general) power series in y with p-adic integral coefficients. The constant term of α(y) is a p-adic unit and all other coefficients of α(y) are divisible by π, where π^{p−1} = −p. Let R be the ring of integers in Q_{p}(π). Then, we can write

α(y) = ∑_{i=0}^{∞} b_{i}y^{i}, b_{0} ∈ R^{∗}, b_{i} ∈ πR (i > 0), lim_{i→∞} b_{i} = 0.

This power series α(y) can be expressed explicitly in terms of certain uniquely determined solution of the hypergeometric differential equation attached to the family of Kloosterman sums. If a non-zero element ȳ ∈ F_{q}^{∗} with q = p^{a}, then the unit root α_{0}(ȳ) of the above L-function of Kloosterman sums over F_{q} is given by the p-adic analytic formula

α_{0}(ȳ) = α(y)α(y^{p}) ⋯ α(y^{p^{a−1}}),

where y is the Teichmüller lifting of ȳ. This formula is a consequence of the Hodge-Newton decomposition discovered by Dwork [7-8] in the early seventies.

To further understand how the unit root α_{0}(ȳ) (as a p-adic integer) varies when ȳ varies, one naturally introduces and considers the L-function L(U_{n}, T) attached to the unit root F-crystal U_{n}. More generally, for an integer k, we can consider the k-th tensor power U_{n}^{⊗k}. Its L-function over the prime field F_{p} is defined in the usual way by the infinite Euler product

L(U_{n}^{⊗k}, T) = ∏_{ȳ∈G_{m}} \frac{1}{(1 − T^{deg(ȳ)}α_{0}^{k}(ȳ))}

= ∏_{ȳ∈G_{m}} \frac{1}{(1 − T^{deg(ȳ)}α^{k}(y)α^{k}(y^{p}) ⋯ α^{k}(y^{p^{deg(ȳ)−1}}))},

where ȳ runs over the closed points of G_{m}/F_{p}. This L-function L(U_{n}^{⊗k}, T) can also be defined using the exponential generating function associated to the sequence of p-adic character sums arising from the p-adic character α(y). The unit root F-crystal U_{n} gives a one-dimensional continuous p-adic representation ρ_{n} of the arithmetic fundamental group π_{1}^{arith}(G_{m}/F_{p}):

ρ_{n} : π_{1}^{arith}(G_{m}/F_{p}) ⟶ GL_{1}(R),

where the torus G_{m}/F_{p} is the parameter space for the parameter ȳ. The Lfunction L(U_{n}^{⊗k}, T) is the same as the L-function L(ρ_{n}^{⊗k}, T) attached to the continuous p-adic representation ρ_{n}^{⊗k}. According to a plausible conjecture of Katz [14] (see [11] for a proof in the constant sheaf case), this mysterious unit root zeta function L(U_{n}, T) (k = 1) is also the L-function attached to the relative p-adic étale cohomology with compact support of the family of Kloosterman sheaves parametrized by the torus G_{m}. These are transcendental, non-rational functions. The meromorphic continuation of L(U_{n}^{k}, T) to the closed unit disk already implies the existence of a weak p-adic equi-distribution theorem for the p-adic “angle” of the zeros of the L-function L(ȳ, n, T) of Kloosterman sums [17]. Dwork’s unit root conjecture [8] is the following

Conjecture (Dwork). For every integer k, the unit root zeta function L(U_{n}^{⊗k}, T) is p-adic meromorphic.

For a so-called overconvergent F-crystal, the L-function is always meromorphic by Dwork’s trace formula. The difficulty about this conjecture is that the unit root F-crystal U_{n} (obtained by solving the fixed point of a contraction map in a p-adic Banach space) is no longer overconvergent in general. In the case that n = 1, Dwork [9] showed that there is an excellent lifting (Deligne-Tate lifting) of the Frobenius map x → x^{p} such that U_{1} is overconvergent with respect to the excellent lifting. Thus, the classical overconvergent theory applies and the zeta function L(U_{1}^{k}, T) is p-adic meromorphic in the special case n = 1.

Unfortunately, the so-called excellent lifting rarely exists. In fact, for each n > 1, Sperber [21] showed that there does not exist an excellent lifting of the Frobenius map such that U_{n} is overconvergent. Thus, the situation cannot be reduced to the “trivial” overconvergent case. Although it was conjectured more generally by Katz [14] that the L-function of any F-crystal is always padic meromorphic, but this was disproved in [26]. Thus, to handle Dwork’s conjecture for n > 1, one has to employ a new approach. The purpose of this article is to introduce a new method which can be used to handle the case when the unit root F-crystal is of rank one. See section 3 for a full description of our results. In particular, we obtain

Theorem 1.1. For every integer k, the k-th unit root zeta function L(U_{n}^{⊗k}, T) is p-adic meromorphic.

The general tool for p-adic meromorphic continuation of L-functions is to use Dwork’s trace formula. It expresses the unit root zeta function as an alternating product of the Fredholm determinants of several continuous operators in a p-adic Banach space. If these operators were completely continuous [20] (which is the case if the unit root F-crystal is overconvergent or more generally ∞ log-convergent), then the Fredholm determinants would be p-adic entire and hence the L-function would be meromorphic. Unfortunately, those operators do not seem to be completely continuous in general. All previous approaches to Dwork’s conjecture try to prove that the Fredholm determinants are entire. This is usually done by showing that the radius of convergence is infinite. But it may well be the case that those Fredholm determinants could be meromorphic but NOT entire. And so, the radius of convergence of those Fredholm determinants could indeed be finite because of possible poles. For an arbitrary unit root F-crystal, the Fredholm determinant is not even meromorphic, not mentioning entire, as the counterexample in [26] shows.

Our new approach takes a long detour and avoids proving the likely false statement that the Fredholm determinants are entire. The basic idea is to express each L(U_{n}^{⊗k}, T) as the limit of a sequence of functions which are alternating products of L-functions of various higher symmetric powers and exterior powers of the original overconvergent F-crystal M_{n} we started with. These later L-functions are meromorphic since their F-crystals are overconvergent. These L-functions, however, become out of control if we try to take the limit. In the case that the unit root F-crystal U_{n} is of rank one (which is always the case for our Kloosterman family), we are able to prove that our sequence of meromorphic L-functions are uniformly meromorphic. This uniformity and a continuity result allow us to take the limit and deduce that the limiting unit root zeta function L(U_{n}^{⊗k}, T) is still a meromorphic function. The actual proof decomposes L(U_{n}^{⊗k}, T) as a quotient of two functions, each of which is the limit of a sequence of entire functions. This decomposition makes it easier to take the limit. Note that our work does not imply that the above mentioned Fredholm determinants are entire. It only implies that they are meromorphic which is all one needs to prove Dwork’s conjecture.

Once we know that L(U_{n}^{⊗k}, T) is meromorphic for each k, we could ask how the unit root zeta function L(U_{n}^{⊗k}, T) varies with the integer parameter k. This can be viewed as an extension of Gouvêa-Mazur’s conjecture [12-13] about dimension variation of modular forms. The Gouvêa-Mazur conjecture corresponds exactly to the case of unit root zeta functions arising from the ordinary family of elliptic curves over a finite field, in which case the unit root F-crystal is of rank 1 and already overconvergent by Deligne-Tate lifting. The unit root zeta function in this elliptic case, more precisely the special value L(U_{n}^{⊗k}, 1), is also directly related to geometric Iwasawa theory and p-adic Lfunctions, see [3-4] [18]. Our results here are strong enough to obtain useful information about analytic variation of the family of unit root zeta functions L(U_{n}^{⊗k}, T) as the integer parameter k varies.

The definition of L(U_{n}^{⊗k}, T) and Fermat’s little theorem show that the L-function L(U_{n}^{⊗k}, T) is p-adically continuous in k. More precisely, if k_{1} and k_{2} are two integers such that

(1.1)  k_{1} ≡ k_{2} (mod (p − 1)p^{m}),

then

(1.2)  L(U_{n}^{⊗k_{1}}, T) ≡ L(U_{n}^{⊗k_{2}}, T) (mod πp^{m+1}).

To get deeper information about the zeroes of L(U_{n}^{⊗k}, T), we would like to decompose L(U_{n}^{⊗k}, T) in terms of its slopes.

For a given integer k, the p-adic meromorphic function L(U_{n}^{⊗k}, T) can be factored completely over Q̄_{p}:

L(U_{n}^{⊗k}, T) = ∏_{i≥0}(1 − z_{i}(k)T)^{±1},

where each z_{i}(k) is a p-adic integer in Q̄_{p}. For a rational number s ∈ Q, we define the slope s part of L(U_{n}^{⊗k}, T) to be

L_{s}(U_{n}^{⊗k}, T) = ∏_{ord_{p}z_{i}(k)=s}(1 − z_{i}(k)T)^{±1}.

This is a rational function. Let d_{s}(U_{n}, k) denote the degree of the slope s rational function L_{s}(U_{n}^{⊗k}, T), where the degree of a rational function means the degree of its numerator minus the degree of its denominator. This function d_{s}(U_{n}, k) of two variables is called the degree function of the meromorphic L-function L(U_{n}^{⊗k}, T). Note that d_{s}(U_{n}, k) can take negative values. A fundamental question is to understand the degree function d_{s}(U_{n}, k), see [28] for a discussion about possible upper bounds for d_{s}(U_{n}, k) in the elliptic case. Here, following the spirit of the Gouvêa-Mazur conjecture, we only investigate how the function d_{s}(U_{n}, k) varies with k when s is fixed.

For any rational number s, we define m(U_{n}, s) to be the smallest nonnegative integer m ∈ Z_{≥0} ∪ {+∞} such that whenever k_{1} and k_{2} are two integers satisfying (1.1), we have the equality

d_{s^{∗}}(U_{n}, k_{1}) = d_{s^{∗}}(U_{n}, k_{2})

for all 0 ≤ s^{∗} ≤ s. By our definition, the following trivial inequality

0 ≤ m(U_{n}, s) ≤ +∞

holds for each s. Furthermore, m(U_{n}, s) = 0 for s < 0. This is because d_{s}(U_{n}, k) = 0 for s < 0. The quantity m(U_{n}, s) gives information about how often the degree function d_{s}(U_{n}, k) changes as k varies p-adically, where the slope s is fixed. Our result implies

Theorem 1.2. The number m(U_{n}, s) is finite for every s and every n.

This gives us a control theorem as Coleman [3] did for the family of elliptic curves in his work on the Gouvêa-Mazur conjecture about dimension variation of modular forms. In particular, it implies that for each fixed s, d_{s}(U_{n}, k) is a padically locally constant function of k if we restrict k to a residue class modulo p − 1. Thus, the degree function d_{s}(U_{n}, k) is a bounded function of k for each fixed s. Our work makes it possible and meaningful to extend the Gouvêa-Mazur conjecture from the family of elliptic curves to a family of algebraic varieties with one unit root fibre by fibre. Since our method is effective, it can be used to give an explicit upper bound for m(U_{n}, s), generalizing the quadratic bound in [27] about Gouvêa-Mazur’s conjecture. We shall, however, not make our calculations explicit in this paper because we work under the weakest possible condition of c log-convergent setting, where the control theorem is already best possible in some sense. For those unit root F-crystals which are embedded in overconvergent F-crystals (such as Kloosterman sums), there is a somewhat simpler version of our method which would give a reasonably good and explicit estimate depending on several factors. In fact, in a future article on overconvergent setup, we will prove a general result which implies immediately that the quantity m(U_{n}, s) is bounded by a polynomial in s of degree n + 2. In the special case n = 1, this is a cubic bound which can be improved to be quadratic because of the existence of an excellent lifting.

An important further question is to ask about the possible slopes or the p-adic absolute values of the zeros and poles of the unit root zeta function L(U_{n}^{⊗k}, T), namely the p-adic Riemann hypothesis of the meromorphic function L(U_{n}^{⊗k}, T). Let S(k, n) be the set of slopes of the zeros and poles of the Lfunction L(U_{n}^{⊗k}, T), where the slope of a p-adic number α means ord_{p}α. This is in general an infinite set of rational numbers. The p-adic Riemann hypothesis in this case is the following conjecture. We stay with the simpler n = 1 case since so little is known.

Conjecture 1.3. Let n = 1. Then, the set S(k, 1) has a bounded denominator for each k.

One could strengthen the conjecture by requiring the denominator to be uniformly bounded for all k. No single case of the conjecture is known to be true other than the exceptional case k = 0 where L(U_{n}^{⊗0}, T) = L(1, T) = (1−T)/(1−pT) is rational. One may wonder why we put the restriction n = 1 in the above conjecture. The main reason is that for n = 1, we can at least prove the weak evidence that a suitable average version of Conjecture 1.3 is true. For n > 1, we are currently unable to prove any probabilistic version of Conjecture 1.3. Thus, it seems safer to stay with the simplest case n = 1 at the moment.

Our results extend to higher slope case (again assuming rank 1) as also conjectured by Dwork. The extension to higher slopes uses more tensor products. The first step is to separate the higher slope unit root zeta function from other smaller slope unit root zeta functions in a natural way. After the separation is done, it turns out that the higher slope unit root zeta function can also be expressed, in a more complicated way, as the limit of a sequence of meromorphic functions. The necessary uniformity and continuity result as the slope zero case can be established assuming the concerned unit root Fcrystal is of rank one. This condition is satisfied for the Kloosterman family by Sperber’s theorem.

If the unit root F-crystal has rank greater than one, the uniform part of our approach does not work. There are easy examples (constant unit root Fcrystals of rank greater than one, for instance) which show that the sequence of the L-functions of the higher symmetric powers are not uniformly meromorphic and thus one cannot pass the meromorphy property to its limit. Although one can use the results of this paper to prove Dwork’s conjecture in some higher rank cases, we shall not pursue it here as it does not seem to get very far. We shall introduce another method in a future article [29] which combined with the method in the present paper will be able to prove Dwork’s conjecture in full generality.

For simplicity, we shall work in the toric setting in this paper so that Dwork’s trace formula can be applied and so that we can go as far as possible without too much technical burden. For a general smooth base scheme, one needs to use Reich-Monsky’s generalization of Dwork’s trace formula. It would be more complicated and involve more nasty analytic arguments. We shall also forget the horizontal connection (differential equation) of an F-crystal and work in a larger category called σ-modules of possibly infinite rank with nuclear structure. Thus, our results actually go beyond, in several ways, the conjectural context of F-crystals of finite rank. In particular, our overconvergent nuclear σ-module (even of finite rank) is generally not overconvergent in Berthelot’s sense [1] if one wants to consider it as an F-crystal by adding back the implied horizontal connection of the Frobenius map. The difference is that we only assume that the Frobenius map is overconvergent. Berthelot’s rigid cohomology assumes that both the Frobenius map and the horizontal connection are overconvergent. For example, the rank one unit root F-crystal coming from the ordinary family of elliptic curves is overconvergent in our sense but not overconvergent in Berthelot’s sense.

As indicated in [25], there is a characteristic p version of Dwork’s conjecture which would be useful in studying analytic variation of L-functions of φ-sheaves and Drinfeld modules [22]. Our method here shows that the characteristic p Dwork conjecture holds in rank one case as well, where the base scheme is allowed to be an arbitrary separated scheme of finite type over F_{q}. In characteristic p situation, the general base scheme case can be reduced to the toric case by a simple trick because the Frobenius lifting problem disappears in characteristic p. Although the characteristic p version of Dwork’s conjecture is technically simpler, it requires the same amount of original ideas. In particular, we do not know how to use the method of the present paper to prove the characteristic p Dwork conjecture if the unit root part has rank greater than one.

From structural point of view, our result simply says that Dwork’s unit root zeta function attached to the rank one unit root part of an ordinary overconvergent F-crystal of finite rank is a finite alternating product of Lfunctions of certain overconvergent “F-crystals of infinite rank”. This gives an explicit formula for Dwork’s unit root zeta function in terms of L-functions of overconvergent “F-crystals of infinite rank”. We shall show [29] that a more complicated formula holds for a higher rank unit root zeta function, but we will need infinitely many overconvergent “F-crystals of infinite rank”. To handle these new objects (overconvergent “F-crystals of infinite rank”) called overconvergent nuclear σ-modules in this paper, one often needs to be careful about various convergent conditions. The nuclear condition (some sort of infinite Hodge structure) can be viewed as a weak finiteness condition. Our nuclear operators are not completely continuous in the usual sense (in fact our Banach space does not even have an orthonormal basis). However, they can be viewed as the dual of completely continuous operators. Working in the dual non-completely continuous but nuclear category has some significant advantages in our proofs. Our nuclear condition insures that everything works as expected. There is another reason which will be made more explicit in a future paper. Namely, even if one starts with an infinite rank σ-module for the more familiar Banach space with an orthonormal basis, in extending the more abstract Monsky trace formula to infinite rank situation, one would encounter a suitable dual which will land in the category of Banach spaces without an orthonormal basis. Thus, it makes sense to deal with them at the begining.

The setting of this paper starts immediately with infinite rank case. However, those who are interested only in Dwork’s original conjecture can assume that all our objects are in the familiar finite rank case. This is enough to prove the rank one case of Dwork’s conjecture in his original finite rank setting. To study further analytic variation of unit root zeta functions, one can still stay in the finite rank case but it is a little awkward to do it because one has to take the limit of a discontinuous family. We get around the difficulty by working with “essentially continuous families”. Thus, for the analytic variation part, we actually have two proofs. One uses the notion of “essential continuity” and stays in finite rank case. The other uses continuous families but works in infinite rank setting. The infinite rank setting also gives us the flexibility to get around the difficulty caused by the (unknown) conjectural generic finite dimensionality of the relative rigid cohomology of a family of varieties f : X → Y over a finite field. This generic finite dimensionality is needed in reducing the geometric version of Dwork’s conjecture to the more general version about L-functions of F-crystals if one insists to stay in finite rank case. The conjectural finite dimensionality has recently been proved by Berthelot [2] when the parameter space Y is a point. The generic finite dimensionality of the relative rigid cohomology in general is still open. Ultimately, we believe that one should work with the infinite rank setting because that is essentially where the unit root zeta function sits in and that is where the unit root zeta function can be best understood. For these reasons, we have chosen to work directly with infinite (countable) rank setting throughout the paper.

In addition to the example of higher dimensional Kloosterman family described in this introduction, we note that there are many other families which have exactly one p-adic unit root fibre by fibre. For instance, if f(x, y) is an arbitrary Laurent polynomial over F_{q} in two sets of variables x = (x_{1}, ⋯, x_{n}) and y = (y_{1}, ⋯, y_{m}), then we can view f(x, y) as a family of Laurent polynomials in x parametrized by the parameter y varying in the m-dimensional torus. For each closed point y in the m-torus, we can form a sequence of exponential sums of the Laurent polynomial f(x, y) on the n-torus and hence get an L-function L(f(x, y), T) for each closed point y of the m-torus. The family of L-functions L(f(x, y), T)^{(−1)^{n−1}} parametrized by y of the m-torus has exactly one p-adic unit root fibre by fibre. Our result applies to this much more general situation. In the n-dimensional Kloosterman family discussed above, the Laurent polynomial f(x, y) is simply given by x_{1} + ⋯ + x_{n} + y/x_{1} ⋯ x_{n} with m = 1.

The content is organized as follow. Section 2 introduces the basic notion of a nuclear σ-module and define its L-function. This can be viewed as an extension of F-crystals without connection from finite rank to infinite rank. Section 3 reviews the Hodge-Newton decomposition of an ordinary F-crystal in the context of nuclear σ-modules. This allows us to reformulate Dwork’s conjecture in a more general framework where adequate tools can be developed. Section 4 gives a basic decomposition formula for Dwork’s k-th power L-function of a nuclear σ-module in terms of more natural L-functions of various higher symmetric powers and exterior products. This formula is our starting point. It already gives a non-trivial result about meromorphic continuation of Dwork’s unit root zeta function. Section 5 studies continuous and uniform families of p-adic meromorphic functions. It is in this section that various basic uniform results are proved. Section 6 begins the proof of the rank one slope zero part of Dwork’s conjecture by taking limit and showing various continuity results. Section 7 extends our slope zero result to higher slope case (still rank one). Section 8 describes an alternating approach to the limiting procedure. It includes an explicit formula for Dwork’s unit root zeta function in rank one case as well as a slightly more general rank one result which is crucial in our forthcoming work on the higher rank case of Dwork’s conjecture.

2. Nuclear σ-modules and L-functions

In previous introduction section, the base field for the unit root zeta function is the prime field F_{p}. From now on, we will work with the slightly more general base field F_{q}. Let K be a fixed finite extension of the field Q_{p} of p-adic rational numbers with residue field F_{q}. Let π be a fixed uniformizer of K. Let R be the ring of integers in K. Namely, the ring R consists of those elements a ∈ K such that ord_{π}a ≥ 0. We normalize the p-adic (π-adic) absolute value on K by defining |π|_{π} = 1/p. Let n be a fixed positive integer. We shall consider various nuclear σ-modules over the n-dimensional torus G_{m}^{n}, which are certain Banach modules with a nuclear action of a semi-linear operator. Throughout the paper, our base space will be the n-dimensional torus G_{m}^{n}.

First, we define various coefficient rings. Let

A_{0} = { ∑_{u∈Z^{n}} a_{u}X^{u}|a_{u} ∈ R, lim_{|u|→∞} a_{u} = 0}

be the ring of convergent Laurent series over R, where for a lattice point u = (u_{1}, ⋯, u_{n}) ∈ Z^{n} we define X^{u} = X_{1}^{u_{1}} ⋯ X_{n}^{u_{n}} and |u| = |u_{1}| + ⋯ + |u_{n}|. Let σ be the R-linear Frobenius map on A_{0} defined by:

σ(X^{u}) = X^{qu}.

Thus, one can think of the pair (A_{0}, σ) as a natural p-adic lifting of the characteristic p coordinate ring of the n-torus (G_{m}^{n}, σ̄) over F_{q}, where σ̄ is the q-th power Frobenius map. The ring A_{0} is p-adically complete. However, it is too large for L-functions and cohomological purpose. We shall frequently need to work in various subrings of A_{0} which are not complete but weakly complete in some sense in order to get interesting results about L-functions.

The overconvergent subring A^{†} of A_{0} consists of those elements satisfying

lim_{|u|→∞} inf \frac{ord_{π}a_{u}}{|u|} > 0.

For a real number 0 ≤ c ≤ ∞, let A_{c} be the c log-convergent subring of A_{0} consisting of those elements satisfying

lim_{|u|→∞} inf \frac{ord_{π}a_{u}}{log_{q} |u|} ≥ c.

It is clear that for c_{1} > c_{2} > 0, we have the inclusion relation

R[X] ⊂ A^{†} ⊂ A_{∞} ⊂ A_{c_{1}} ⊂ A_{c_{2}} ⊂ A_{0}.

All of these rings are known to be Noetherian [24]. They are not p-adically complete except for the largest one A_{0}. The algebra A_{0} is a complete normed algebra under the Gauss norm:

∥ ∑_{u} a_{u}X^{u}∥ = max_{u} |a_{u}|_{π}.

Thus, the ring A_{0} becomes a Banach algebra over R and the monomials {X^{u}} form an orthonormal basis of A_{0} over R. A Banach module M over A_{0} is an ultra normed complete A_{0}-module, whose norm is also denoted by ∥ ∥. It satisfies the relations

∥m∥ = 0 if and only if m = 0,

∥am∥ = ∥a∥∥m∥, for a ∈ A_{0}, m ∈ M,

∥m_{1} + m_{2}∥ ≤ max(∥m_{1}∥, ∥m_{2}∥), for m_{1}, m_{2} ∈ M.

A (formal) basis of M over A_{0} is a subset {e_{i} : i ∈ I} of M for some index set I, such that every element in M can be written uniquely in the form ∑ a_{i}e_{i} with a_{i} ∈ A_{0}. Thus, a Banach module M over A_{0} with a basis is the space of sequences {a_{i}, i ∈ I} with entries in A_{0}. An orthonormal basis of M over A_{0} is a subset {e_{i} : i ∈ I} of M for some index set I, such that every element in M can be written uniquely in the form ∑ a_{i}e_{i} with a_{i} ∈ A_{0} and the additional condition lim_{i} ∥a_{i}∥ = 0. Thus, a Banach module M over A_{0} with an orthonormal basis is the space of convergent sequences with entries in A_{0}. Note that a Banach module with a basis does not necessarily have an orthonormal basis. For example, the formal power series ring R[[x]] is a Banach space with the basis {x^{u}} over R but it does not have an orthonormal basis over R.

For a finite rank free module M over A_{0}, basis and orthonormal basis are the same concept. Any free A_{0}-module M of finite rank can be viewed as a Banach module of finite rank by suitably extending the norm on A_{0} to M, depending on a chosen basis e⃗. We remind the reader that our definition of a (formal) basis in infinite rank case may be different from the traditional usage of the notion “basis”, where it is assumed that the sum ∑ a_{i}e_{i} is a finite combination. We do allow formal infinite combination. The word “basis” in this paper will always refer to formal basis. We shall be interested in the Banach module over A_{0} of the form

M = {∑_{i} a_{i}e_{i}|a_{i} ∈ A_{0}},

where e⃗ = {e_{1}, ⋯, e_{i}, ⋯} is a basis indexed by a set I of at most countable cardinality (rank). We assume that the norm on M is defined by

∥ ∑_{i} a_{i}e_{i}∥ = max_{i} ∥a_{i}∥.

If the rank is infinite, this Banach module M has a basis over A_{0} but does not have an orthonormal basis. If (b_{i,j}) is a matrix with entries b_{i,j} in A_{0} or R, we use ord_{π}(b_{i,j}) to denote min_{i,j} ord_{π}b_{i,j}. Let c(I) be the set of convergent sequences over A_{0} indexed by I. Let b(I) be the set of sequences over A_{0} indexed by I. The index set I will be identified with a subset of the positive integers.

We first study which transition matrix gives a new basis for M over A_{0}.

Lemma 2.1. Let f⃗ = e⃗U be a set of elements in M indexed by I, where U is a matrix whose rows and columns are both indexed by I. Then, f⃗ is a basis of M over A_{0} if and only if U is invertible over A_{0} and the row vectors of both U and U^{−1} are in c(I). In this case, we also have the norm relation

∥ ∑_{i} a_{i}f_{i}∥ = max_{i} ∥a_{i}∥.

Proof. Write

f_{j} = ∑_{i} u_{ij}e_{i}.

If f⃗ is a basis of M over A_{0}, then U is invertible and the infinite sum

∑_{j} f_{j} = ∑_{i}(∑_{j} u_{ij})e_{i}

is convergent. Thus, the row vectors of U are in c(I). The same statement holds for U^{−1} since e⃗ = f⃗U^{−1} is a basis.

Conversely, assume that the row vectors of U and U^{−1} are in c(I). Then, for any column vector a⃗ ∈ b(I), the substitution

e⃗ a⃗ = f⃗(U^{−1}a⃗)

is well defined. This means that each element of M can be written as a combination of the elements in f⃗. If we have two such expressions of the same element:

f⃗ a⃗ = f⃗ b⃗,

the substitution

e⃗(Ua⃗) = e⃗(Ub⃗)

is well defined. Since e⃗ is a basis, we must have

Ua⃗ = Ub⃗.

Since the row vectors of U^{−1} are in c(I), we can multiply the above equation by U^{−1} on the left side. This implies that a⃗ = b⃗. Thus, f⃗ is a basis.

To prove the norm relation, it suffices to prove for a column vector a⃗ ∈ b(I) not divisible by π, we have ∥f⃗ a⃗∥ = ∥e⃗(Ua⃗)∥ = 1. Since U is invertible and a⃗ is not divisible by π, it follows that Ua⃗ is not divisible by π. This implies that ∥e⃗(Ua⃗)∥ = 1. The proof is complete.

Definition 2.2. A nuclear σ-module over A_{0} is a pair (M, ϕ), where M is a Banach module over A_{0} with a basis e⃗ = (e_{1}, e_{2}, ⋯) indexed by I of at most countable cardinality such that ∥ ∑_{i} a_{i}e_{i}∥ = max_{i} ∥a_{i}∥, and ϕ is an infinite σ-linear map:

ϕ : M ⟶ M, ϕ(∑_{i≥1} a_{i}e_{i}) = ∑_{i≥1} σ(a_{i})ϕ(e_{i}), a_{i} ∈ A_{0}

satisfying the nuclear condition

(2.1)  lim_{i→∞} ∥ϕ(e_{i})∥ = 0.

Note that the map ϕ in Definition 2.2 is automatically continuous since ϕ is bounded. In fact, ∥ϕ(m)∥ ≤ 1 for all m ∈ M. We emphasize that our Banach module M has a basis but does not have an orthonormal basis in the infinite rank case. Thus, specifying the image of ϕ at the basis elements e_{i} is not enough to define an infinite σ-linear map. This is because the infinite sum ϕ(∑ a_{i}e_{i}) = ∑_{i} σ(a_{i})ϕ(e_{i}) may not be defined when we write out ϕ(e_{i}). Our condition that lim_{i} ∥ϕ(e_{i})∥ = 0 insures that the infinite sum ∑_{i} σ(a_{i})ϕ(e_{i}) is indeed well defined. Although our nuclear σ-module is defined in terms of a chosen basis, the resulting concept is actually independent of the chosen basis. Namely, we have

Lemma 2.3. If (M, ϕ) is nuclear with respect to one basis e⃗, then (M, ϕ) is nuclear with respect to any other basis f⃗.

Proof. We need to show that

(2.2)  lim_{i→∞} ∥ϕ(f_{i})∥ = 0.

Write

f_{j} = ∑_{i} u_{ij}e_{i}.

Since ϕ is nuclear with respect to e⃗, for any ϵ > 0, there is an integer N > 0 such that for all integers i > N, we have ∥ϕ(e_{i})∥ < ϵ. This implies that

∥ϕ(f_{j})∥ = ∥ ∑_{i} u_{ij}^{σ}ϕ(e_{i})∥
≤ max(ϵ, ∥u_{1j}^{σ}ϕ(e_{1})∥, ⋯, ∥u_{Nj}^{σ}ϕ(e_{N})∥)
≤ max(ϵ, ∥u_{1j}∥, ⋯, ∥u_{Nj}∥).

Since for each fixed i, we have by Lemma 2.1 that

lim_{j} ∥u_{ij}∥ = 0.

There is then an integer N_{1} > 0 such that for all integers 1 ≤ i ≤ N and all integers j > N_{1}, we have ∥u_{ij}∥ < ϵ. This proves that for all j > max(N_{1}, N), we have

∥ϕ(f_{j})∥ < ϵ.

The lemma is proved.

If M is of finite rank, the nuclear condition (2.1) is automatically satisfied. Thus, in finite rank case, we may drop the word “nuclear” and simply say a σ-module. Intuitively, one could think of our nuclear map ϕ as the dual of a family of completely continuous semi-linear operators (parametrized by the variable X) with an orthonormal basis. In fact, if B(X) is the matrix of ϕ under a basis, then the transpose of B(X) can be viewed as the matrix of a family of completely continuous operators of a Banach space with an orthonormal basis. Thus, the standard p-adic spectral theory applies fibre by fibre, although the fibres are generally not completely continuous in the usual sense.

A morphism between two nuclear σ-modules (M, ϕ) and (N, ψ) is an A_{0}- linear map of A_{0}-module

θ : M ⟶ N

such that θ∘ϕ = ψ∘θ. In this way, the category of nuclear σ-modules is defined. In particular, it makes sense to talk about isomorphic nuclear σ-modules. It is easy to check that the usual direct sum of two nuclear σ-modules

(M, ϕ) ⊕ (N, ψ) = (M ⊕ N, ϕ ⊕ ψ)

is again a nuclear σ-module. To extend the usual tensor operations to nuclear σ-modules, we need to introduce the concepts of formal tensor product, formal symmetric powers and formal exterior powers. We shall restrict to the definition of formal tensor product since the other concepts are completely similar.

Definition 2.4. Let (M, ϕ) be a nuclear σ-module with a basis e⃗. Let (N, ψ) be a nuclear σ-module with a basis f⃗. Let

ϕ(e_{i}) = ∑_{k} u_{ki}e_{i}, ψ(f_{j}) = ∑_{k} v_{kj}f_{j}.

Define the formal tensor product of (M, ϕ) and (N, ψ) to be the nuclear σmodule (M ⊗ N, ϕ ⊗ ψ), where M ⊗ N denotes the Banach A_{0}-module with the basis

e⃗ ⊗ f⃗ = (⋯, e_{i} ⊗ f_{j}, ⋯)

and ϕ ⊗ ψ is determined by the relation

ϕ ⊗ ψ(e_{i} ⊗ f_{j}) = ϕ(e_{i}) ⊗ ψ(f_{j})

= (∑_{k_{1}} u_{k_{1}i}e_{k_{1}}) ⊗ (∑_{k_{2}j} v_{k_{2}j}f_{k_{2}})

= ∑_{k_{1},k_{2}} u_{k_{1}i}v_{k_{2}j}(e_{k_{1}} ⊗ f_{k_{2}}).

It is easy to check that

lim_{i+j→∞} ∥ϕ ⊗ ψ(e_{i} ⊗ f_{j})∥ = 0.

Thus, the formal tensor product (M ⊗ N, ϕ ⊗ ψ) is indeed a nuclear σ-module.

Note that we are using the same usual tensor notation for the formal tensor product. In fact, we shall often call a formal tensor product just as a tensor product. This should not cause confusion since all tensor products will be formal tensor products in this paper. In the finite rank case, of course, the formal tensor product agrees with the usual tensor product. Similarly, one can define formal symmetric powers (Sym^{k}M, Sym^{k}ϕ) and formal exterior powers (∧^{k}M, ∧^{k}ϕ), denoted by the same classical notations. All symmetric powers and all exterior powers in this paper are in the formal sense. Our definition of the formal tensor product (M ⊗ N, ϕ ⊗ ψ) is based on a basis e⃗ of M and a basis f⃗ of N. The resulting concept is actually independent of the choice of the bases. Namely,

Lemma 2.5. The formal tensor product (M, ϕ) and (N, ψ) is independent of the choice of the basis e⃗ of M and the basis f⃗ of N.

Proof. Let e⃗^{∗} (resp. f⃗^{∗} ) be another basis of M (resp. N). Write

e⃗^{∗} = e⃗U, lim_{j→∞} ∥u_{ij}∥ = 0,

f⃗^{∗} = e⃗V, lim_{j→∞} ∥v_{ij}∥ = 0.

We use Lemma 2.1 to show that e⃗^{∗} ⊗ f⃗^{∗} is a basis of the formal tensor product M ⊗ N. The transition matrix U ⊗ V between e⃗ ⊗ f⃗ and e⃗^{∗} ⊗ f⃗^{∗} is clearly invertible. We need to check the convergent property of the row vectors in the transition matrix. Since

\begin{aligned}e_{i}^{∗} ⊗ f_{j}^{∗} & = (∑_{k_{1}} u_{k_{1}i}e_{k_{1}}) ⊗ (∑_{k_{2}} v_{k_{2}j}f_{j}) \\ & = ∑_{k_{1},k_{2}} u_{k_{1}i}v_{k_{2}j}(e_{k_{1}} ⊗ f_{k_{2}}), \end{aligned}

and

lim_{i+j→∞} ∥u_{k_{1}i}v_{k_{2}j}∥ = 0,

we deduce that the row vectors of the transition matrix U ⊗ V are convergent sequences. Similarly, the row vectors of the inverse transition matrix U^{−1}⊗V^{−1} are convergent sequences. This proves that e⃗^{∗} ⊗ f⃗^{∗} is indeed a basis of the formal tensor product M ⊗ N.

It remains to show that the nuclear map ϕ⊗ψ on the formal tensor product M ⊗ N is independent of the choice of the basis. Namely, we want to show that if

∑_{i,j} a_{ij}(e_{i} ⊗ f_{j}) = ∑_{i,j} a_{ij}^{∗}(e_{i}^{∗} ⊗ f_{j}^{∗}),

then we must have

∑_{i,j} σ(a_{ij})ϕ(e_{i}) ⊗ ψ(f_{j}) = ∑_{i,j} σ(a_{ij}^{∗})ϕ(e_{i}^{∗}) ⊗ ψ(f_{j}^{∗}).

Replacing e⃗^{∗} by e⃗U (resp. f⃗^{∗} by f⃗V), we need to check that if

∑_{i,j} a_{ij}(e_{i} ⊗ f_{j}) = ∑_{k_{1},k_{2}}(∑_{i,j} a_{ij}^{∗}u_{k_{1}i}v_{k_{2}j})e_{k_{1}} ⊗ f_{k_{2}},

then

∑_{i,j} σ(a_{ij})ϕ(e_{i}) ⊗ ψ(f_{j}) = ∑_{k_{1},k_{2}}(∑_{i,j} σ(a_{ij}^{∗}u_{k_{1}i}v_{k_{2}j}))ϕ(e_{k_{1}}) ⊗ ψ(f_{k_{2}}).

But this follows from the infinite σ-linearity of ϕ and ψ. The lemma is proved.

Corollary 2.6. The category of nuclear σ-modules is closed under direct sums, (formal) tensor product, symmetric powers and exterior powers.

The σ-linear map ϕ in a nuclear σ-module (M, ϕ) can also be viewed as an infinite A_{0}-linear map ϕ : M^{(σ)} → M of Banach A_{0}-modules, where M^{(σ)} is the (formal) pull back of M by σ : A_{0} → A_{0}. That is,

M^{(σ)} = {∑_{i} b_{i} ⊗ e_{i} | b_{i} ∈ A_{0}},

where for a_{i} ∈ A_{0}, we have the relation

∑ σ(a_{i}) ⊗ e_{i} = ∑_{i} a_{i}e_{i}.

The map ϕ is called the Frobenius map of the nuclear σ-module M. We shall often just write M or ϕ for the pair (M, ϕ). To be more specific about the nuclear condition of the nuclear map ϕ with respect to a basis e⃗ of M, we shall introduce several notions attached to a given basis.

Definition 2.7. Let (M, ϕ) be a nuclear σ-module with a given basis e⃗. For each integer 1 ≤ i < ∞, let d_{i} be the smallest positive integer d such that for all j > d, we have

ϕ(e_{j}) ≡ 0 (mod π^{i}).

This is a finite integer for each i since

lim_{j} ∥ϕ(e_{j})∥ = 0.

For 0 ≤ i < ∞, define

h_{i} = d_{i+1} − d_{i}, d_{0} = 0.

The sequence h = h(e⃗) = {h_{0}, h_{1}, ⋯} is called the basis sequence of ϕ with respect to the basis e⃗. If we let M_{(i)} to be the Banach A_{0}-module with the basis {e_{d_{i}+1}, e_{d_{i}+2}, ⋯}, then we have a decreasing basis filtration of Banach A_{0}-submodules:

M = M_{(0)} ⊃ M_{(1)} ⊃ ⋯ ⊃ M_{(j)} ⊃ ⋯ ⊃ 0, ∩_{j}M_{(j)} = 0,

where each M_{(i)} has a basis, each quotient M_{(i)}/M_{(i+1)} is a finite free A_{0}- module and

ϕ(M_{(i)}) ⊆ π^{i}M, rank(M_{(i)}/M_{(i+1)}) = h_{i}.

Equivalently, the matrix B of ϕ with respect to e⃗ (defined by ϕ(e⃗) = e⃗B) is of the form

B = (B_{0}, πB_{1}, π^{2}B_{2}, ⋯, π^{i}B_{i}, ⋯),

where each block B_{i} is a matrix over A_{0} with h_{i} (finitely many) columns.

We now turn to discussing the convergent condition of ϕ as a “power series” in term of X, generalizing the overconvergent and c log-convergent condition of an element in A_{0}.

Definition 2.8. Let B(X) be the matrix of a nuclear σ-module (M, ϕ) under a basis e⃗. Write

B(X) = ∑_{u∈Z^{n}} B_{u}X^{u},

where the coefficients B_{u} are (possibly infinite) matrices with entries in R. We say that (M, ϕ) is convergent if the matrix B(X) is convergent. Namely, B(X) satisfies the condition

(2.3)  lim_{|u|→∞} ord_{π}B_{u} = ∞.

Similarly, we say that (M, ϕ) is overconvergent if the matrix B(X) is overconvergent. Namely, B(X) satisfies the condition

(2.4)  lim_{|u|→∞} inf \frac{ord_{π}B_{u}}{|u|} > 0.

Likewise, we say that (M, ϕ) is c log-convergent if the matrix B(X) is c logconvergent. Namely, B(X) satisfies the condition

(2.5)  lim_{|u|→∞} inf \frac{ord_{π}B_{u}}{log_{q} |u|} ≥ c.

Note that our definition of the convergence property depends on the choice of the basis e⃗. It may happen that the matrix B(X) under one basis is overconvergent (resp. c log-convergent) but the matrix under a new basis is NOT overconvergent (resp. c log-convergent). Our definition is that as long as there is one basis for which the matrix is overconvergent (resp. c log-convergent), then the σ-module (M, ϕ) is called overconvergent (resp., c log-convergent). In the case of finite rank, an overconvergent σ-module is then a σ-module which can be defined over A^{†} and then by base extension A^{†} → A_{0}. In the case of infinite rank, an overconvergent nuclear σ-module is stronger than a nuclear σ-module which can be defined over A^{†}. It may happen that the matrix B(X) = (b_{i,j}(X)) has all of its entries b_{i,j}(X) in A^{†} without satisfying the uniformity requirement (2.4). Similar statements hold for c log-convergent nuclear σ-modules as well.

The matrix B(X) of a nuclear σ-module under a basis can be written in the form

B(X) = ∑_{u∈Z^{n}} B_{u}X^{u}, B_{u} = (b_{w_{1},w_{2}}(u)),

where w_{1} denotes the row index and w_{2} denotes the column index of the constant matrix B_{u}. Conversely, such a matrix power series B(X) is the matrix of some c log-convergent nuclear σ-module under some basis if and only if B(X) satisfies the following two conditions. First, we have the c log-convergent condition for B(X) as a power series in X:

lim_{|u|→∞} inf \frac{ord_{π}B_{u}}{log_{q} |u|} ≥ c.

Second, we have the nuclear condition for B(X) to be the matrix of ϕ under a nuclear basis. This condition can be rephrased as follow: For any positive number C > 0, there is an integer N_{C} > 0 such that for all column index w_{2} > N_{C}, we have

ord_{π}b_{w_{1},w_{2}}(u) ≥ C,

uniformly for all u and all w_{1}.

As stated in Coroolary 2.6, the category of nuclear σ-modules is closed under direct sum, (formal) tensor product, symmetric product and exterior product. The same statement holds for the category of convergent nuclear σ-modules (resp. c log-convergent nuclear σ-modules, resp. overconvergent nuclear σ-modules). The category of convergent nuclear σ-modules includes all σ-modules of finite rank. In the case that the finite rank map ϕ : M^{(σ)} → M is an isogeny, namely when ϕ : M^{(σ)} ⊗ Q → M ⊗ Q is an isomorphism of A_{0} ⊗ Qmodules, the σ-module (M, ϕ) more or less corresponds to an F-crystal on the toric space G_{m}^{n}. If, in addition, the finite rank map ϕ : M^{(σ)} → M is an isomorphism of A_{0}-modules, the σ-module (M, ϕ) corresponds exactly to a unit root F-crystal on G_{m}^{n}. The implied condition to get the horizontal connection can be easily solved. As Katz puts it, the connection comes for free. The notion of overconvergent nuclear σ-modules of finite rank arises naturally from various relative p-adic (but not p-adic étale) cohomology groups of a family of varieties over finite fields. The notion of overconvergent nuclear σ-modules of infinite rank already arises if one works on the chain level of p-adic cohomology groups of a family of varieties over a finite field.

From now on, we shall exclusively work with a nuclear σ-module (M, ϕ) and often identify it with its matrix B(X) under a basis. We can define an L-function for any nuclear σ-module. For a geometric point x̄ ∈ G_{m}^{n} of degree d over F_{q}, let x be the Teichmüller lifting of x̄. The fibre M_{x} at x is a Banach module with a countable basis over R[x], where R[x] is the ring obtained from R by adjoining the coordinates of x. Via the R-algebra homomorphism R[X] → R[x] defined by X → x, the map σ acts on R[x] by σ(x) = x^{q}. The fibre map

ϕ_{x} : M_{x} ⟶ M_{x}

is σ-linear (not linear in general). But its d-th iterate ϕ_{x}^{d} is R[x]-linear (since σ^{d}(x) = x) and can be defined over R. Furthermore, ϕ_{x}^{d} is the dual of a completely continuous operator in the sense of Serre [20]. This implies that ϕ_{x}^{d} itself is nuclear over R[x]. Thus, the characteristic series det(I − ϕ_{x}^{d}T) (or the Fredholm determinant) of ϕ_{x}^{d} acting on M_{x} (descended from R[x] to R) is well defined with coefficients in R. It is a p-adic entire function and is independent of the choice of the geometric point x̄ associated to the closed point over F_{q} containing x̄. Thus, it makes sense to say the characteristic power series of ϕ at a closed point.

Definition 2.9. The L-function of a nuclear σ-module (M, ϕ) is defined by the Euler product

L(ϕ, T) = ∏_{x̄∈G_{m}^{n}/F_{q}} \frac{1}{det(I − ϕ_{x}^{deg(x̄)}T^{deg(x̄)})},

where x̄ runs over the closed points and x is the Teichmüller lifting of x̄.

Note that each of our Euler factors is the reciprocal of an entire power series, not necessarily a polynomial. We are working in a more general situation than the existing literature. If B(X) is the matrix of ϕ under a basis, then one checks that

L(ϕ, T) = L(B(X), T)

= ∏_{x̄∈G_{m}^{n}/F_{q}} \frac{1}{det(I − B(x)B(x^{q}) ⋯ B(x^{q^{deg(x̄)−1}})T^{deg(x̄)})}.

This shows that the L-function is a power series with coefficients in R. Thus, it is trivially meromorphic in the open unit disk |T |_{π} < 1. If ϕ is convergent (which is always the case if M is of finite rank), it can be shown that the L-function is meromorphic on the closed unit disk |T |_{π} ≤ 1. Dwork’s fundamental trace formula can be extended to show that this L-function is p-adic meromorphic if ϕ is overconvergent. The optimal result is given by the following theorem, which was proved in [10] and [26] for finite rank case.

Theorem 2.10. Let (M, ϕ) be a c log convergent nuclear σ-module for some 0 < c ≤ ∞. Then the L-function L(ϕ, T) is p-adic meromorphic in the open disk |T |_{π} < p^{c}, where we define |π|_{π} = 1/p.

Actually, what we will need in this paper is a stronger family version of the above result as given later in section 5. Dwork’s conjecture says that for an invertible (M, ϕ) of finite rank embedded (see section 3) in an ordinary overconvergent σ-module of finite rank, the L-function L(ϕ, T) is meromorphic. No single non-trivial example was known previously, other than the “trivial” ∞ log convergent case already covered by the above theorem.

In order to fully describe Dwork’s conjecture, we need to define a slightly more general L-function, which could be called higher power zeta function. If ϕ is a nuclear σ-module and k is a positive integer, we define the k-th power ϕ^{k} to be ϕ iterated k-times. The pair (M, ϕ^{k}) can be viewed as a σ^{k}-module but it is not a σ-module as ϕ^{k} is σ^{k}-linear not σ-linear in general. The L-function of the k-th power ϕ^{k} is defined to be

L(ϕ^{k}, T) = ∏_{x̄∈G_{m}^{n}/F_{q}} \frac{1}{det(I − ϕ_{x}^{kdeg(x̄)}T^{deg(x̄)})}.

If B(X) is the matrix of ϕ under a basis, then one checks that

L(ϕ^{k}, T) = ∏_{x̄∈G_{m}^{n}/F_{q}} \frac{1}{det(I − B(x)B(x^{q}) ⋯ B(x^{q^{kdeg(x̄)−1}})T^{deg(x̄)})}

= ∏_{x̄∈G_{m}^{n}/F_{q}} \frac{1}{det(I − (B(x)B(x^{q}) ⋯ B(x^{q^{deg(x̄)−1}}))^{k}T^{deg(x̄)})}.

In other words, L(ϕ^{k}, T) is obtained from L(ϕ, T) by raising each reciprocal root α of the Euler factors to its k-th power α^{k}. We want to emphasize that L(ϕ^{k}, T) is different from the following function L(B^{k}, T) which is the L-function of the nuclear σ-module with matrix B^{k}:

L(B^{k}, T) = ∏_{x̄∈G_{m}^{n}/F_{q}} \frac{1}{det(I − B^{k}(x)B^{k}(x^{q}) ⋯ B^{k}(x^{q^{deg(x̄)−1}})T^{deg(x̄)})}.

They are different because matrix multiplication is in general non-commutative if B has rank greater than one.

Let (M_{1}, ϕ_{1}) and (M_{2}, ϕ_{2}) be two nuclear σ-modules. Let k_{1} and k_{2} be two positive integers. Although their powers ϕ_{1}^{k_{1}} and ϕ_{2}^{k_{2}} do not have a σ-module structure in general, we can define the L-function of the “tensor product” of these powers as follow:

L(ϕ_{1}^{k_{1}} ⊗ ϕ_{2}^{k_{2}}, T) = ∏_{x̄∈G_{m}^{n}/F_{q}} \frac{1}{det(I − ϕ_{1x}^{k_{1}deg(x̄)} ⊗ ϕ_{2x}^{k_{2}deg(x̄)}T^{deg(x̄)})}.

Namely, the Euler factor at x of L(ϕ_{1}^{k_{1}} ⊗ ϕ_{2}^{k_{2}}, T) is the “tensor product” of the corrosponding Euler factors at x of L(ϕ_{1}^{k_{1}}, T) and L(ϕ_{2}^{k_{2}}, T). The above definition works for negative integer k as well provided that ϕ is injective fibre by fibre and of finite rank. If (M, ϕ) is of rank one, then L(ϕ^{k}, T) is just the L-function L(ϕ^{⊗k}, T) of the k-th tensor (or symmetric) power of (M, ϕ). If (M, ϕ) is of rank greater than one, then the k-th power ϕ^{k} (k > 1) does not have a σ-module interpretation as matrix multiplication is non-commutative in general. Nevertheless, by the Newton-Waring formula, we see that ϕ^{k} can be viewed as a virtual σ-module, namely, an integral combination of σ-modules with both positive and negative coefficients. This implies that L(ϕ^{k}, T) can be written as an alternating product of L-functions of various σ-modules. We shall need something finer than the Newton-Warning formula in order to prove uniform results.

3. Hodge-Newton decomposition and Dwork’s conjecture

Let (M, ϕ) be a nuclear σ-module with a basis e⃗. First, we briefly recall the construction of formal exterior powers and formal symmetric powers. Let 1 ≤ i < ∞. The i-th (formal) exterior power (∧^{i}M, ∧^{i}ϕ) is the nuclear σmodule with the basis

∧^{i}e⃗ = {⋯, e_{k_{1}}∧ ⋯ ∧e_{k_{i}}, ⋯}, k_{1} < k_{2} < ⋯ < k_{i}

and with the infinite σ-linear endomorphism ∧^{i}ϕ defined by

∧^{i}(ϕ)(e_{k_{1}}∧ ⋯ ∧e_{k_{i}}) = ϕ(e_{k_{1}})∧ ⋯ ∧ϕ(e_{k_{i}}).

For i = 0, we define (∧^{0}M, ∧^{0}ϕ) to be the trivial rank one σ-module (A_{0}, σ). Similarly, the i-th (formal) symmetric power (Sym^{i}M, Sym^{i}ϕ) is the nuclear σ-module with the basis

Sym^{i}e⃗ = {⋯, e_{k_{1}} ⋯ e_{k_{i}}, ⋯}, k_{1} ≤ k_{2} ≤ ⋯ ≤ k_{i}

and with the infinite σ-linear endomorphism Sym^{i}ϕ defined by

Sym^{i}(ϕ)(e_{k_{1}} ⋯ e_{k_{i}}) = ϕ(e_{k_{1}}) ⋯ ϕ(e_{k_{i}}).

The (formal) tensor power (M^{⊗i}, ϕ^{⊗i}) is similarly constructed. We shall use the convention that Sym^{k}ϕ = 0 for all negative integers k < 0.

Next, we introduce the basis polygon and Newton polygon of a nuclear σ-module (M, ϕ) with a basis e⃗. Denote by ord(ϕ) to be the greatest integer b such that

ϕ ≡ 0 (modπ^{b}).

We say that (M, ϕ) is divisible by π if ϕ is divisible by π. If ϕ is divisible by π, then we can write ϕ = πϕ_{1}. In this case, one checks that

L(ϕ, T) = L(ϕ_{1}, πT).

Thus, for our purpose of studying L-functions, we may assume that (M, ϕ) is not divisible by π.

Definition 3.1. Let (M, ϕ) be a nuclear σ-module with a basis e⃗. Let h = h(e⃗) = (h_{0}, h_{1}, ⋯) be the basis sequence of ϕ with respect to the basis e⃗ (see Definition 2.7 ). We define the basis polygon P (e⃗) of (M, ϕ) with respect to e⃗ to be the convex closure in the plane of the following lattice points:

(0, 0), (h_{0}, 0), (h_{0} + h_{1}, h_{1}), (h_{0} + h_{1} + h_{2}, h_{1} + 2h_{2}), ⋯,
(h_{0} + ⋯ + h_{i}, h_{1} + 2h_{2} + ⋯ + ih_{i}), · · · .

Namely, the basis polygon P (e⃗) is the polygon with a side of slope i and horizontal length h_{i} for every integer 0 ≤ i < ∞.

Note that our definition of the basis polygon P (e⃗) and the basis sequence h(e⃗) depends on the given basis e⃗. The basis polygon and the basis sequence are somewhat similar to Hodge polygon and Hodge numbers. But they are different in general, even in the finite rank case. The same σ-module with different bases will give rise to different basis polygons and different basis sequences. The (generic) abstract Hodge polygon (in finite rank case) as discussed in [15] does not depend on the given filtration. One can show that the generic Hodge polygon lies on or above our basis polygon. In the ordinary case to be defined below, the basis e⃗ will be the optimal one and our basis polygon is the same as the Hodge polygon in the finite rank case. Since we are dealing with the ordinary case of Dwork’s conjecture in this paper, nothing is lost. It may be interesting to define an optimal intrinsic (necessary generic) basis polygon in general which coincides with the Hodge polygon in the finite rank case. We shall not pursue it here. Our simplified definition here is just for the purpose of quickly getting to Dwork’s conjecture without losing the key property we need to use.

Definition 3.2. For each closed point x̄ ∈ G_{m}^{n} over F_{q}, the Newton polygon of the nuclear σ-module (M, ϕ) at x̄ is the Newton polygon of the entire characteristic series det(I − ϕ_{x}^{deg(x̄)}T) defined with respect to the valuation ord_{π^{deg(x̄)}}.

A standard argument shows that the Newton polygon at each x̄ lies on or above any basis polygon P (e⃗). This is known to be true in finite rank case. Since we do not need it in this paper, we do not include the detail of the proof here.

Definition 3.3. Let e⃗ be a basis of a nuclear σ-module (M, ϕ). The basis e⃗ is called ordinary if the basis polygon P (e⃗) of (M, ϕ) coincides with the Newton polygon of each fibre (M, ϕ)_{x}, where x is the Teichmüller lifting of the closed point x̄ ∈ G_{m}^{n}/F_{q}. The basis filtration (see Definition 2.7 ) attached to e⃗ is called ordinary if e⃗ is ordinary. Similarly, the basis e⃗ is called ordinary up to slope j if the basis polygon P (e⃗) coincides with the Newton polygon of each fibre for all sides up to slope j. In particular, e⃗ is said to be ordinary at slope zero if the horizontal side of the Newton polygon at each fibre is of length h_{0}(e⃗). We say that (M, ϕ) is ordinary (resp. ordinary up to slope j, resp. ordinary at slope zero) if it has a basis which is ordinary (resp. ordinary up to slope j, resp. ordinary at slope zero).

An important property is that the category of ordinary nuclear σ-modules is closed under direct sum, tensor product, symmetric power and exterior power. This property is not hard to prove. It follows from the proof of the Hodge-Newton decomposition to be described below. Alternatively, one could check directly. For example, let (M_{1}, ϕ_{1}) be a nuclear σ-module with a basis e⃗ and with its associated basis sequence h = (h_{0}, h_{1}, ⋯). Let (M_{2}, ϕ_{2}) be a nuclear σ-module with a basis f⃗ and with its associated basis sequence g = (g_{0}, g_{1}, ⋯). Then the direct sum (M_{1} ⊕M_{2}, ϕ_{1} ⊕ϕ_{2}) is a nuclear σ-module with the basis e⃗ ⊕ f⃗. Its associated basis sequence is given by

h ⊕ g = (h_{0} + g_{0}, h_{1} + g_{1}, ⋯).

The tensor product (M_{1} ⊗ M_{2}, ϕ_{1} ⊗ ϕ_{2}) is a nuclear σ-module with basis e⃗ ⊗ f⃗. Its associated basis sequence is given by

h ⊗ g = (h_{0}g_{0}, h_{0}g_{1} + h_{1}g_{0}, h_{0}g_{2} + h_{1}g_{1} + h_{2}g_{0}, ⋯).

The symmetric square (Sym^{2}M_{1}, Sym^{2}ϕ_{1}) is a nuclear σ-module with basis Sym^{2}e⃗. Its associated basis sequence Sym^{2}h is given by

(\frac{h_{0}^{2} + h_{0}}{2} , h_{0}h_{1}, h_{0}h_{2} + \frac{h_{1}^{2} + h_{1}}{2} , h_{0}h_{3} + h_{1}h_{2}, h_{0}h_{4} + h_{1}h_{3} + \frac{h_{2}^{2} + h_{2}}{2} , ⋯).

The exterior square (∧^{2}M_{1}, ∧^{2}ϕ_{1}) is a nuclear σ-module with basis ∧^{2}e⃗. Its associated basis sequence is given by

∧^{2}h = (\frac{h_{0}^{2} − h_{0}}{2} , h_{0}h_{1}, h_{0}h_{2} + \frac{h_{1}^{2} − h_{1}}{2} , h_{0}h_{3} +h_{1}h_{2}, h_{0}h_{4} +h_{1}h_{3} + \frac{h_{2}^{2} − h_{2}}{2} , ⋯).

If e⃗ is an ordinary basis of (M_{1}, ϕ_{1}) and if f⃗ is an ordinary basis of (M_{2}, ϕ_{2}), one checks that e⃗⊕f⃗, e⃗⊗f⃗, Sym^{2}e⃗ and ∧^{2}e⃗ is, respectively, an ordinary basis of (M_{1} ⊕M_{2}, ϕ_{1} ⊕ϕ_{2}), (M_{1} ⊗M_{2}, ϕ_{1} ⊗ϕ_{2}), (Sym^{2}M_{1}, Sym^{2}ϕ_{1}) and (∧^{2}M_{1}, ∧^{2}ϕ_{1}). The associated basis sequence is respectively h ⊕ g, h ⊗ g, Sym^{2}h and ∧^{2}h. A systematic study of Newton polygons and Hodge polygons in finite rank case is given in [15].

Since A_{0} is complete, any contraction mapping from A_{0} into itself has a fixed point. This fact is all one needs to prove the following extension of Hodge-Newton decomposition, first proved by Dwork [8] in the finite rank case. The infinite rank case was already considered and used in [23] to study the Newton polygon of Fredholm determinants arising from L-functions of exponential sums.

Lemma 3.4. Let (M, ϕ) be a nuclear σ-module ordinary at slope zero. Then there is a free A_{0}-submodule M_{0} of rank h_{0} transversal to the ordinary basis filtration: M = M_{0} ⊕ M_{(1)}, such that ϕ is stable on M_{0} and (M_{0}, ϕ) is a unit root σ-module of rank h_{0}.

Proof. The proof is completely similar to Dwork’s original proof in the finite rank case. We give a brief sketch of the proof in order to have a feeling about what is going on. Under a basis e⃗ ordinary at slope zero, the matrix B of ϕ is of the form

B = ( \begin{matrix}B_{00} & πB_{01} \\ B_{10} & πB_{11} \end{matrix} ) ,

where B_{00} is a h_{0} × h_{0} matrix over A_{0}. Since the basis e⃗ of (M, ϕ) is ordinary at slope zero, the matrix B_{00} is invertible over A_{0}. Consider the new basis

(e_{1}, ⋯, e_{i}, ⋯) ( \begin{matrix}I_{00} & 0 \\ E_{10} & I_{11} \end{matrix} ) = e⃗E

of M, where I_{00} is the identity matrix of rank h_{0}, I_{11} is an identity matrix and E_{10} is some matrix to be determined. The matrix E_{10} has h_{0} columns. One calculates that the matrix of ϕ under the new basis e⃗E is given by

E^{−1}BE^{σ} = ( \begin{matrix}B_{00} + πB_{01}E_{10}^{σ} & πB_{01} \\ −E_{10}B_{00} + B_{10} + π(B_{11} − E_{10}B_{01})E_{10}^{σ} & π(B_{11} − E_{10}B_{01}) \end{matrix} ) .

To prove the theorem, we need to choose E_{10} suitably such that the above new matrix is triangular. Namely, we want

−E_{10}B_{00} + B_{10} + π(B_{11} − E_{10}B_{01})E_{10}^{σ} = 0.

This equation can be rewritten as

E_{10} = B_{10}B_{00}^{−1} + π(B_{11} − E_{10}B_{01})E_{10}^{σ}B_{00}^{−1}.

It defines a contraction map. Thus, by successive iteration or the fixed point theorem in a p-adic Banach space, we conclude that there is a unique solution matrix E_{10} with entries in the ring A_{0}. Furthermore, one checks that the map in the above change of basis satisfies the condition of lemma 2.1. Thus, e⃗E is indeed a basis with the same h_{0}. The lemma is proved. We note that the above change of basis will in general destroy any overconvergent property the original matrix B(X) might have because the solution matrix E_{10} will only be convergent even though B is overconvergent.

In the situation of Lemma 3.4, we shall denote the sub σ-module (M_{0}, ϕ) by (U_{0}, ϕ_{0}). We say that the invertible σ-module (U_{0}, ϕ_{0}) is embedded in the ambient ordinary σ-module (M, ϕ) . Alternatively, we shall say that (U_{0}, ϕ_{0}) is the unit root part (or the invertible part) of (M, ϕ). In geometric situation, the unit root part can be constructed using expansion coefficients of differential forms [16]. It can also be obtained as the relative p-adic ètale cohomology with compact support of a family of varieties defined over a finite field. Dwork’s unit root conjecture, extended to our setting, is then the following

Unit Root Conjecture. Let (M, ϕ) be an overconvergent nuclear σ-module ordinary at slope zero. Let (U_{0}, ϕ_{0}) be the unit root part of (M, ϕ). Then for every integer k, the unit root zeta function L(ϕ_{0}^{k}, T) is p-adic meromorphic.

Remark. Even though the ambient σ-module (M, ϕ) is assumed to be overconvergent, its unit root part (obtained by solving the fixed point of a contraction map) will no longer be overconvergent in general. In fact, one can only expect that the unit root part to be c log-convergent for some small real number c. It is shown in [10] that one can take c = (p − 1)/(p + 1) in finite rank case. They suggested the possibility that one might be able to take c = 1. This was later proved to be true by Sperber-Wan (unpublished). It is not very hard to construct examples to show that c = 1 is best possible.

A weaker version of our main result is the following

Unit Root Theorem. Let (M, ϕ) be a c log-convergent nuclear σmodule (M, ϕ) for some 0 < c ≤ ∞, ordinary at slope zero. Let (U_{0}, ϕ_{0}) be the unit root part of (M, ϕ). Assume that ϕ_{0} has rank one. Then for each integer k, the unit root zeta function L(ϕ_{0}^{k}, T) is p-adic meromorphic in the open disk |T |_{π} < p^{c}.

Corollary 3.5. Let (M, ϕ) be a ∞ log-convergent nuclear σ-module (M, ϕ) ordinary at slope zero. Let (U_{0}, ϕ_{0}) be the unit root part of (M, ϕ). Assume that ϕ_{0} has rank one. Then for each integer k, the unit root zeta function L(ϕ_{0}^{k}, T) is p-adic meromorphic everywhere.

This corollary shows that Dwork’s unit root conjecture is true if the unit root part has rank one. Note that our result is actually stronger, since we only assume that the ambient σ-module to be c log convergent instead of being overconvergent. The c log-convergence condition in the above theorem is best possible by the counter-example in [26]. The possibility of replacing overconvergent condition by the weaker ∞ log-convergent condition in Dwork’s conjecture (if true) was already suggested in [25]. All the above results can be extended to higher slope. The following Hodge-Newton decomposition is also due to Dwork [8] in finite rank case. It can be proved in a similar way and uses induction.

Lemma 3.6. If a nuclear σ-module (M, ϕ) is ordinary up to slope j for some integer j ≥ 0, then there is an increasing ϕ-stable filtration of free A_{0}-submodules of finite rank,

0 ⊂ M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{j} ⊂ M

which is transversal to the ordinary basis filtration: M = M_{i} ⊕ M_{(i+1)} for all 0 ≤ i ≤ j. Furthermore, for 0 ≤ i ≤ j, the quotient (M_{i}/M_{i−1}, ϕ) is a σmodule of the form (U_{i}, π^{i}ϕ_{i}), where (U_{i}, ϕ_{i}) is a unit root σ-module of rank h_{i}.

We shall call (U_{i}, ϕ_{i}) the unit root σ-module coming from the slope i part of (M, ϕ). The inductive proof shows that if (M, ϕ) is ordinary up to slope i, then we have

(3.1)  ord(∧^{h_{0}+···+h_{i}}ϕ) = h_{1} + 2h_{2} + ⋯ + ih_{i}.

Dwork [Dw3] also made the following more general conjecture.

Higher Slope Conjecture. Let (M, ϕ) be an overconvergent nuclear σ-module (M, ϕ) ordinary up to slope j for some integer j ≥ 0. Let (U_{i}, ϕ_{i}) be the unit root σ-module coming from the slope i part of (M, ϕ), where 0 ≤ i ≤ j. Then for each 0 ≤ i ≤ j and each integer k, the unit root zeta function L(ϕ_{i}^{k}, T) is p-adic meromorphic.

Similarly, we can prove the following result for higher slope zeta function.

Higher Slope Theorem. Let (M, ϕ) be a c log-convergent nuclear σ-module (M, ϕ) for some 0 < c ≤ ∞, ordinary up to slope j for some integer j ≥ 0. Let (U_{j}, ϕ_{j}) be the unit root σ-module coming from the slope j part of (M, ϕ). Assume that ϕ_{j} has rank one. Then for each integer k, the unit root zeta function L(ϕ_{j}^{k}, T) is p-adic meromorphic in the open disc |T |_{π} < p^{c}.

Corollary 3.7. Let (M, ϕ) be an ∞ log-convergent nuclear σ-module (M, ϕ) ordinary up to slope j for some integer j ≥ 0. Let (U_{j}, ϕ_{j}) be the unit root σ-module coming from the slope j part of (M, ϕ). Assume that ϕ_{j} has rank one. Then for each integer k, the unit root zeta function L(ϕ_{j}^{k}, T) is p-adic meromorphic everywhere.

4. L-functions of various tensor powers

Let E be a finite square matrix over a field of characteristic zero. Our first goal is to prove a good universal formula which expresses the characteristic polynomial det(I − T E^{k}) of the k-th power E^{k} in terms of certain alternating product of the characteristic polynomials of various tensor and exterior products of E. There are several such formulas available. The most well known one is the classical Newton-Warning formula expressing the power symmetric functions in terms of elementary symmetric functions. To get uniform results, we will need something finer than the Newton-Warning formula. Since our formulas are actually universal formulas, they hold for fields of positive characteristic as well. For simplicity of our proof, we shall assume that we are in characteristic zero. Recall that Tr(E) denotes the trace of the matrix E. We make the convention that Sym^{k}E is the zero matrix (of any size) for negative integer k < 0.

Lemma 4.1. Let E be a finite square matrix. Then, for each integer k > 0, we have the relation

(4.1)  Tr(E^{k}) = ∑_{i≥1}(−1)^{i−1}i Tr(Sym^{k−i}E) Tr(∧^{i}E).

Proof. This can be proved using a combinatorial counting argument. We shall, however, use the following more conceptual proof communicated to us by N. Katz. Taking the logarithmic derivative of the well known identity

\frac{1}{det(I − ET)} = exp(∑_{k≥1} \frac{Tr(E^{k})}{k}T^{k}),

we get

\frac{−T \frac{d}{dT} det(I − ET)}{det(I − ET)} = ∑_{k≥1} Tr(E^{k})T^{k}.

On the other hand, by looking at the eigenvalues, one easily gets

(4.2)  det(I − ET) = ∑_{i≥0}(−1)^{i} Tr(∧^{i}E)T^{i},

(4.3)  \frac{1}{det(I − ET)} = ∑_{j≥0} Tr(Sym^{j}E)T^{j}.

It follows that

(∑_{i≥0}(−1)^{i−1}i Tr(∧^{i}E)T^{i})(∑_{j≥0} Tr(Sym^{j}E)T^{j}) = ∑_{k≥1} Tr(E^{k})T^{k}.

Comparing the coefficients of T^{k} on both sides, we conclude that the lemma is true.

The next result is a further refinement in the sense that the coefficients have smaller size. This would be useful in practical calculation of the unit root zeta function.

Lemma 4.2. Let E be a finite square matrix. Then, for each integer k > 0, we have the relation

Tr(E^{k}) = Tr(Sym^{k}E) + ∑_{i≥2}(−1)^{i−1}(i − 1) Tr(Sym^{k−i}E) Tr(∧^{i}E).

Proof. Using the trivial identity

\frac{det(I − ET)}{det(I − ET)} = 1

and equations (4.2)-(4.3), we get

(∑_{i≥0}(−1)^{i} Tr(∧^{i}E)T^{i})(∑_{j≥0} Tr(Sym^{j}E)T^{j}) = 1.

Comparing the coefficients of T^{k}, we deduce that for k ≥ 1,

∑_{i≥0}(−1)^{i} Tr(Sym^{k−i}E) Tr(∧^{i}E) = 0.

Adding this equation to (4.1), we immediately get Lemma 4.2.

We now apply the above two lemmas to the tensor product E_{1}^{k_{1}} ⊗ E_{2}^{k_{2}}. Since

Tr(E_{1}^{k_{1}}) Tr(E_{2}^{k_{2}}) = Tr(E_{1}^{k_{1}} ⊗ E_{2}^{k_{2}}),

we deduce from Lemma 4.1 the following result.

Lemma 4.3. Let E_{1} and E_{2} be two finite square matrices. Then, for integers k_{1} > 0 and k_{2} > 0, we have the relation

Tr(E_{1}^{k_{1}} ⊗ E_{2}^{k_{2}}) = ∑_{i_{1},i_{2}≥1}(−1)^{i_{1}+i_{2}}i_{1}i_{2} Tr(Sym^{k_{1}−i_{1}}E_{1}) Tr(∧^{i_{1}}E_{1})×
Tr(Sym^{k_{2}−i_{2}}E_{2}) Tr(∧^{i_{2}}E_{2}).

If one uses Lemma 4.2 instead of Lemma 4.1, one gets a more efficient but less compact formula for Tr(E_{1}^{k_{1}} ⊗ E_{2}^{k_{2}}). We omit it. The above results on matrix trace can now be transformed to similar results for characteristic polynomials.

Lemma 4.4. Let E be a finite square matrix. Then, for each integer k > 0, we have the relation

det(I − T E^{k}) = ∏_{i≥1} det(I − T (Sym^{k−i}E ⊗ ∧^{i}E))^{(−1)^{i−1}i}.

Proof. For each j ≥ 1, Lemma 4.1 shows that

Tr(E^{jk}) = ∑_{i≥1}(−1)^{i−1}i Tr(Sym^{k−i}(E^{j}) ⊗ ∧^{i}(E^{j}))

= ∑_{i≥1}(−1)^{i−1}i Tr((Sym^{k−i}E ⊗ ∧^{i}E)^{j}).

These relations together with (4.3) immediately imply the lemma. The proof is complete.

A more general formula is to consider the tensor product of powers of two matrices. The following formula is a consequence of Lemma 4.3.

Lemma 4.5. Let E_{1} and E_{2} be two finite square matrices. Then, for integers k_{1} > 0 and k_{2} > 0, we have the relation

det(I − T E_{1}^{k_{1}} ⊗ E_{2}^{k_{2}}) = ∏_{i_{1},i_{2}} det(I − T E(i_{1}, i_{2}))^{(−1)^{(i_{1}+i_{2})}i_{1}i_{2}},

where

E(i_{1}, i_{2}) = Sym^{k_{1}−i_{1}}E_{1} ⊗ ∧^{i_{1}}E_{1} ⊗ Sym^{k_{2}−i_{2}}E_{2} ⊗ ∧^{i_{2}}E_{2}.

The above formulas hold for infinite matrices as well if everything is convergent. They are universal formulas, see [26]. In fact, the same proof works if we let the entries of E to be just indeterminates. Then, the above formulas hold as an identity of two formal power series in those indeterminates and T. We now apply it to certain infinite matrices over R.

Definition 4.6. An infinite matrix E over R is called nuclear if its column vector V_{j} goes to zero as the column index j goes to infinity. Namely, lim_{j} ord_{π}V_{j} = ∞.

If E is the transpose of the matrix of a completely continuous endomorphism of a Banach space over R under a countable orthonormal row basis, then E is nuclear. The converse is also true. This shows that the characteristic series det(I − T E) is well defined and entire for a nuclear matrix E. The set of nuclear matrices is clearly stable under direct sum, tensor product, exterior product and symmetric product. Thus, we have the following specialization from the above mentioned universal formula. It can also be proved directly from Lemma 4.4 by taking limit.

Lemma 4.7. Let E be a nuclear matrix over R. Then, for each integer k > 0, we have the relation

det(I − T E^{k}) = ∏_{i≥1} det(I − T (Sym^{k−i}E ⊗ ∧^{i}E))^{(−1)^{i−1}i}.

If one uses Lemma 4.2 instead, one gets the following

Lemma 4.8. Let E be a nuclear matrix over R. Then, for each integer k > 0, we have the relation

det(I −T E^{k}) = det(I −T Sym^{k}E)×∏_{i≥2} det(I −T (Sym^{k−i}E ⊗∧^{i}E))^{(−1)^{i−1}(i−1)}.

Similarly, there are two formulas for det(I − T E_{1}^{k_{1}} ⊗ E_{2}^{k_{2}}), where E_{1} and E_{2} are nuclear matrices. We omit them. We now apply Lemma 4.7 to the L-function of a nuclear σ-module to obtain some preliminary analytic information. For each integer k ≥ 1, recall that we defined

L(ϕ^{k}, T) = ∏_{x̄∈G_{m}^{n}/F_{q}} \frac{1}{det(I − ϕ_{x}^{kdeg(x̄)}T^{deg(x̄)})}.

With the above preparation, we can now prove a simple meromorphy result for the k-th power L-function L(ϕ^{k}, T), which is again best possible by the example in [26].

Lemma 4.9. Let (M, ϕ) be a c log-convergent nuclear σ-module for some 0 < c ≤ ∞. Then for each integer k > 0, the L-function L(ϕ^{k}, T) is p-adic meromorphic in the open disk |T |_{π} < p^{c}.

Proof. Applying Lemma 4.7 (which is a universal formula) and standard representation theory, we find the following basic decomposition of L-functions:

(4.4)  L(ϕ^{k}, T) = ∏_{i≥1} L(Sym^{k−i}ϕ ⊗ ∧^{i}ϕ, T)^{(−1)^{i−1}i}.

One can also derive (4.4) by applying Lemma 4.7 directly to each Euler factor in the definition of L(ϕ^{k}, T) and using the following “notational” identities:

(4.5)  (Sym^{k}ϕ_{x})^{deg(x)} = Sym^{k}(ϕ_{x}^{deg(x)}),
(4.6)  (∧^{k}ϕ_{x})^{deg(x)} = ∧^{k}(ϕ_{x}^{deg(x)}).

In fact, both sides of (4.5) are the linear map ϕ_{x}^{deg(x)} acting on the space Sym^{k}M_{x}. The left side of (4.5) means the deg(x)-th iterate of the semi-linear map ϕ_{x} acting on (Sym^{k}M)_{x}. The right side of (4.5) means the k-th symmetric power of the linear map ϕ_{x}^{deg(x)} acting on M_{x}, which is (by the definition of symmetric power) the linear map ϕ_{x}^{deg(x)} acting on the k-th symmetric power Sym^{k}M_{x}. This proves (4.5). The proof of (4.6) is completely similar.

Note that the product in (4.4) is a finite product since Sym^{k−i}ϕ = 0 for i > k. Now, for each integer i, the map Sym^{k−i}ϕ ⊗ ∧^{i}ϕ is just the Frobenius map of the nuclear σ-module Sym^{k−i}M ⊗ ∧^{i}M, which is c log-convergent. By Theorem 2.4, each of the L-factors on the right side is p-adic meromorphic in the open disk |T |_{π} < p^{c}. Thus, the L-function on the left is also meromorphic in the same disk. The lemma is proved.

This result already gives some information about Dwork’s unit root zeta function.

Theorem 4.10. Let (U_{0}, ϕ_{0}) be the unit root part of a c log-convergent nuclear σ-module (M, ϕ) ordinary at slope zero. Then, for each integer k > 0, the unit root zeta function L(ϕ_{0}^{k}, T) is meromorphic in the open disk |T |_{π} < p^{min(k,c)}.

Proof. By Hodge-Newton decomposition (Lemma 3.4), there is a short exact sequence of nuclear σ-modules:

0 ⟶ M_{0} ⟶ M ⟶ M_{1} ⟶ 0

such that (M_{1}, ϕ_{1}) is divisible by π and (M_{0}, ϕ_{0}) is a unit root σ-module of rank h_{0}. Write ϕ_{1} = πψ. Then, we have the following decomposition of L-functions:

L(ϕ^{k}, T) = L(ϕ_{0}^{k}, T)L(ψ^{k}, π^{k}T).

Because of the shifting factor π^{k}, we see that the second factor L(ψ^{k}, π^{k}T) on the right is trivially meromorphic in the open disk |T |_{π} < p^{k}. The factor L(ϕ^{k}, T) on the left is meromorphic in the open disk |T |_{π} < p^{c} by Lemma 4.9. Thus, the first factor L(ϕ_{0}^{k}, T) on the right is meromorphic in the open disk |T |_{π} < p^{min(k,c)}. The proof is complete.

If the first non-zero slope of ϕ is s_{1} instead of 1, the same proof shows that one gets the meromorphic continuation to the larger disk |T |_{π} < p^{min (s_{1}k,c)}.

Actually, it can be improved a little bit further to the disk |T |_{π} < p^{min (s_{1}(k+1),c)}. We shall, however, not pursue this direction here.

Corollary 4.11. Let (U_{0}, ϕ_{0}) be the unit root part of an ∞ logconvergent nuclear σ-module (M, ϕ) ordinary at slope zero. Then, for each integer k > 0, the unit root zeta function L(ϕ_{0}^{k}, T) is meromorphic in the open disk |T |_{π} < p^{k}.

This simple result is the starting point for our work on Dwork’s conjecture. It shows that we get better and better meromorphic continuation for L(ϕ_{0}^{k}, T) as k grows. To get meromorphic continuation of L(ϕ_{0}^{k}, T) for a fixed k, we can try to use some sort of easily derived limiting formula such as

L(ϕ_{0}^{k}, T) = lim_{m→∞} L(ϕ_{0}^{k+(q^{m!}−1)p^{m}}, T).

In order to pass the better meromorphic property to its limit, as Deligne pointed out, we have to bound uniformly the number of zeros and poles of each member of the sequence of L-functions in any fixed finite disk. Namely, we need some type of uniformity result about meromorphy of the sequence of functions L(ϕ_{0}^{k}, T) for all large integers k. The desired uniformity is not necessarily true in general. The uniformity turns out to be true if the rank of ϕ_{0} is one. In next section, we shall prove the uniformity in the normalized rank one case.

5. Continuous and uniform families

In this section, we discuss continuous family and uniform family of meromorphic functions. Let S be a subset of a complete metric space. Let S̄ denote the closure of S. Namely, S̄ is the union of S and its limit points. In our applications, the space S will be either an infinite subset of the p-adic integers Z_{p} with the induced p-adic topology, or an infinite subset of the integers Z with the sequence topology, where the sequence topology of an infinite subset S = {k_{i}} ⊂ Z indexed by positive integers i ∈ Z_{>0} means that the distance function on S is given by

d(k_{i}, k_{j}) = |\frac{1}{i} − \frac{1}{j}|.

In the latter case, ∞ is the unique limiting point not contained in S. The set S will be our parameter space for which our parameter k varies. We shall only consider a family of power series in R[[T]] of the following form:

f(k, T) = ∑_{m≥0} f_{m}(k)T^{m}, f_{0}(k) = 1, f_{m}(k) ∈ R,

where each coefficient f_{m}(k) is a function from S to R. Note that the constant term is always 1. Furthermore, the ring R is always complete and compact with respect to the p-adic topology of R. The family f(k, T) is called continuous in k if each coefficient f_{m}(k) is a continuous function from S to R. The family f(k, T) is called uniformly continuous in k if each coefficient f_{m}(k) is a uniformly continuous function from S to R. That is, for each fixed m and each ϵ > 0, there is a real number δ(ϵ) > 0 such that whenever k_{1} and k_{2} are two elements of S satisfying

d(k_{1}, k_{2}) < δ(ϵ),

we have

|f_{m}(k_{1}) − f_{m}(k_{2})| < ϵ.

For a real number c ≥ 0, the family of functions f(k, T) is called uniformly analytic in the open disk |T |_{π} < p^{c} if

(5.1)  lim_{m→∞} inf \frac{inf_{k} ord_{π}f_{m}(k)}{m} ≥ c.

The family f(k, T) is called uniformly meromorphic in the open disk |T |_{π} < p^{c} if f(k, T) can be written as a quotient f_{1}(k, T)/f_{2}(k, T) of two families, where both f_{1}(k, T) and f_{2}(k, T) are uniformly analytic in the open disk |T |_{π} < p^{c}. It is clear that the product and quotient (if the denominator is not the zero family) of two uniformly meromorphic families (parameterized by the same parameter k) are still uniformly meromorphic.

If the set S consists of a sequence of elements {k_{i}}, we shall call the family f(k_{i}, T) a sequence of functions. The following result is an immediate consequence of the above defining inequality in (5.1).

Lemma 5.1. Let f(k_{i}, T) be a continuous sequence of power series. Assume that the sequence f(k_{i}, T) is uniformly analytic in the open disk |T |_{π} < p^{c}. If k = lim_{i} k_{i} exists in S̄, then the limit function

g(k, T) = lim_{i} f(k_{i}, T)

exists and it is analytic in the open disk |T |_{π} < p^{c}. Furthermore, if k ∈ S, we have g(k, T) = f(k, T).

Corollary 5.2. If f(k, T) is a continuous family of uniformly analytic functions in |T |_{π} < p^{c} for k ∈ S, then f(k, T) extends uniquely to a family of uniformly analytic functions in |T |_{π} < p^{c} for k ∈ S̄ (the closure of S). If, in addition, the original family f(k, T) parametrized by k ∈ S is uniformly continuous, then the extended family f(k, T) parametrized by k ∈ S̄ is uniformly continuous.

Proof. The first part is clear. The second part can be done in a standard manner. We include a detailed proof here. For k ∈ S̄, write

f(k, T) = ∑_{m≥0} f_{m}(k)T^{m}.

It suffices to prove that for each fixed m, the family f_{m}(k) parametrized by k ∈ S̄ is uniformly continuous. Our uniform continuity assumption means that for any ϵ > 0, there is a real number δ(ϵ) > 0 such that for all k_{1} and k_{2} in S, we have

|f_{m}(k_{1}) − f_{m}(k_{2})| < ϵ, whenever d(k_{1}, k_{2}) < δ(ϵ).

If we let k_{2} vary, the continuity of f_{m}(k_{2}) in k_{2} shows that for all k_{1} ∈ S and k_{2} ∈ S̄, we have

|f_{m}(k_{1}) − f_{m}(k_{2})| < ϵ, whenever d(k_{1}, k_{2}) < \frac{1}{2}δ(ϵ).

If we now let k_{1} vary, the continuity of f_{m}(k_{1}) in k_{1} shows that for all k_{1} ∈ S̄ and k_{2} ∈ S̄, we have

|f_{m}(k_{1}) − f_{m}(k_{2})| < ϵ, whenever d(k_{1}, k_{2}) < \frac{1}{4}δ(ϵ).

The corollary is proved.

To obtain similar result for meromorphic families, we need to introduce another notion about families of functions.

Definition 5.3. A family f(k, T) is called a strong family in |T |_{π} < p^{c} if f(k, T) is the quotient of two uniformly continuous families of uniformly analytic functions in |T |_{π} < p^{c}.

There is probably a better choice of terminology for the notion of a strong family. We have tried to avoid the weaker notion of a uniformly continuous family of uniformly meromorphic functions, which simply means a uniformly continuous family which is also uniformly meromorphic. In such a family, it is not immediately obvious (although probably true) that the limit function is still a meromorphic function in the expected disk. In our stronger definition, this property is built in because of Corollary 5.2. Thus, we have

Corollary 5.4. If f(k, T) is a strong family in |T |_{π} < p^{c} for k ∈ S, then f(k, T) extends uniquely to a strong family in |T |_{π} < p^{c} for k ∈ S̄ (the closure of S).

Sometimes, we shall need a slightly weaker notion of continuity. This will be needed later on. We give the definition here. A family of power series f(k, T) ∈ R[[T]] is called essentially continuous in the disk |T |_{π} < p^{c} if f(k, T) can be written (in many different ways in general) as a product of two families

f(k, T) = f_{1}(k, T)f_{2}(k, T),

where f_{1}(k, T) is a continuous family in k and f_{2}(k, T) (possibly not continuous in k) is a family of functions, analytic, without reciprocal zeros and with norm 1 in the disk |T |_{π} ≤ p^{c}. Namely, f_{2}(k, T) is a 1-unit in |T |_{π} ≤ p^{c}. Two families of functions f(k, T) and g(k, T) are called equivalent in |T |_{π} < p^{c} if f(k, T) = g(k, T)h(k, T), where h(k, T) is a 1-unit in |T |_{π} ≤ p^{c} for each k ∈ S. This is an equivalence relation. If one family f(k, T) is essentially continuous and equivalent to another family g(k, T) in |T |_{π} < p^{c}, then the second family g(k, T) is also essentially continuous in |T |_{π} < p^{c}. A family f(k, T) is said to be an essentially continuous family of uniformly analytic functions in the disk |T |_{π} < p^{c} if it is both essentially continuous and uniformly analytic in the disk |T |_{π} < p^{c}. In terms of the above decomposition, this implies that the continuous part f_{1}(k, T) is uniformly analytic in the disk |T |_{π} < p^{c} because 1/f_{2}(k, T) is automatically uniformly analytic in that disk. If, in addition, the continuous part f_{1}(k, T) is uniformly continuous, then we say that f(k, T) is essentially uniformly continous. A family f(k, T) is called an essentially strong family in the disk |T |_{π} < p^{c} if it is the quotient of two essentially uniformly continuous families of uniformly analytic functions in the disk |T |_{π} < p^{c}. In other words, an essentially strong family in the disk |T |_{π} < p^{c} is a family which is equivalent to a strong family in the disk |T |_{π} < p^{c}. Thus, the notion of an essentially strong family in |T |_{π} < p^{c} depends only on the equivalent class in |T |_{π} < p^{c} of the family f(k, T).

To say that a family f(k, T) parametrized by k ∈ S extends to an essentially strong family g(k, T) parametrized by k ∈ S̄ in |T |_{π} < p^{c}, we mean that g(k, T) is an essentially strong family and its restriction to k ∈ S is equivalent to f(k, T). In particular, even for k ∈ S, g(k, T) may be different from f(k, T). But they are equivalent in |T |_{π} < p^{c}. Thus, in general, a family f(k, T) parametrized by k ∈ S may have many different ways to extend to an essentially strong family g(k, T) parametrized by k ∈ S̄ in |T |_{π} < p^{c}. By our definition, one sees that Corollary 5.4 extends to essentially continuous families.

Corollary 5.5. If f(k, T) is an essentially strong family in |T |_{π} < p^{c} for k ∈ S, then f(k, T) extends (in many ways in general) to an essentially strong family g(k, T) in |T |_{π} < p^{c} for k ∈ S̄ (the closure of S). Any two extended essentially strong families of f(k, T) are equivalent in |T |_{π} < p^{c}.

This result is true, because f(k, T) is equivalent to a strong family in |T |_{π} < p^{c} and any strong family extends uniquely. It is clear that a finite product and quotient (if denominator is not the zero family) of strong families (resp. essentially strong families) in |T |_{π} < p^{c} for k ∈ S is again a strong family (resp. an essentially strong family).

Our main concern will be the family of L-functions arising from a family of nuclear σ-modules. We want to know when such a family of L-functions are uniformly meromorphic and when it is a strong family. Before doing so, we need to define the notion of a uniform family of σ-modules. Let B(k, X) be the matrix of a family of nuclear σ-modules (M(k), ϕ(k)) parametrized by a parameter k. There may be no relations among the ϕ(k) for different k. In particular, the rank of (M(k), ϕ(k)) could be totally different (some finite and other infinite, for instance) as k varies. Write

B(k, X) = ∑_{u∈Z^{n}} B_{u}(k)X^{u},

where each coefficient matrix B_{u}(k) has entries in R:

B_{u}(k) = (b_{w_{1},w_{2}}(u, k)), b_{w_{1},w_{2}}(u, k) ∈ R,

where w_{1} is the row index of B_{u}(k) and w_{2} is the column index of B_{u}(k).

Definition 5.6. The family B(k, X) (or the family ϕ(k)) is called uniformly c log-convergent if the following two conditions hold. First, we have

lim_{|u|→∞} inf \frac{inf_{k} ord_{π}B_{u}(k)}{log_{q} |u|} ≥ c.

Second, for any positive number C > 0, there is an integer N_{C} > 0 such that for all column number w_{2} > N_{C}, we have

ord_{π}b_{w_{1},w_{2}}(u, k) ≥ C,

uniformly for all u, k, w_{1}.

Note that the second condition above is automatically satisfied if the rank of (M(k), ϕ(k)) is uniformly bounded. This is the case if each σ-module M(k) has the same finite rank for every k. The following uniform result is of basic importance to our investigation.

Theorem 5.7. Let B(k, X) be a family of matrices parameterized by k. Assume that the family B(k, X) is uniformly c log-convergent for some 0 < c < ∞. Then for every ϵ > 0, the family of L-functions L(B(k, X), T) is uniformly meromorphic in the open disk |T |_{π} < p^{c−ϵ}. If in addition, the family L(B(k, X), T) is uniformly continuous (resp. essentially uniformly continuous) in the disk |T |_{π} < p^{c−ϵ}, then L(B(k, X), T) is a strong family (resp. an essentially strong family) in the disk |T |_{π} < p^{c−ϵ}.

Slightly weaker version of this uniform result is already stated in Theorem 5.2 of [26] in the case that the rank of the family B(k, X) is a finite constant. We shall need the above more general result which allows the rank of B(k, X) to be unbounded. We include a proof here, closely following the arguments in For each parameter k, let F (k) be the following infinite matrix

F (k) = (B_{qu−v}(k))_{v,u∈Z^{n}}

with block entries B_{qu−v}(k), where v denotes the row block index and u denotes the column block index. Note that each block entry B_{qu−v}(k) is a nuclear matrix. The Fredholm determinant det(I − T F (k)) is defined and is p-adic meromorphic in |T |_{π} < p^{c} since B(k, X) is c log-convergent. This is proved in [26] if B(k, X) has finite rank. The infinite rank case is obtained in a similar way or simply by taking the limit. It also follows from the following stronger proof in Lemma 5.8. The Dwork trace formula for the n-torus is given by

L(B(k, X), T)^{(−1)^{n−1}} = ∏_{j=0}^{n} det(I − q^{j}T F (k))^{(−1)^{j} \binom{n}{j}}.

This is known to be true if B(k, X) has finite rank. The infinite rank case is similar. It can also be obtained from the finite rank case by taking limit. The trace formula is a universal formula in some sense. Our two conditions on B(k, X) guarantee that everything is convergent as shown below. Because of the shifting factor q^{j}, for the first part of the theorem, it suffices to prove the following lemma.

Lemma 5.8. The Fredholm determinant det(I − T F (k)) is a family of uniformly analytic functions in the finite open disk |T |_{π} < p^{c−ϵ}.

Proof. For any 0 < ϵ < 2c, we define a non-negative weight function w(u) on Z^{n} by

(5.2)  w(u) = \{ \begin{matrix}(c − \frac{ϵ}{2}) log_{q} |u|, & if |u| > 0, \\ 0, & if u = 0. \end{matrix}

This function is increasing in |u|. Let G_{v,u}(k) be the twisted matrix

G_{v,u}(k) = π^{w(v)−w(u)}B_{qu−v}(k).

This is also a nuclear matrix. The entries of this matrix will be in the quotient field of R, but they are bounded. Let G(k) be the twisted infinite Frobenius matrix (G_{v,u}(k))_{v,u∈Z^{n}}. Then,

det(I − T F (k)) = det(I − T G(k)).

Lemma 5.8 follows easily from the following lemma and a standard argument on determinant expansion, see the proof in [26].

Lemma 5.9. For any ϵ > 0, all but finitely many column vectors V⃗ of G(k) satisfy the inequality

(5.3)  ord_{π}(V⃗) ≥ c − ϵ.

All (including those exceptional finitely many column vectors from (5.3 )) of V⃗ are bounded by

(5.4)  ord_{π}(V⃗) ≥ −N(ϵ),

where N(ϵ) is a finite positive constant independent of k and V⃗.

Proof. For a given ϵ > 0 with ϵ < 2c, by our uniform c log-convergent assumption, there is an integer N_{ϵ} > 0 such that for all |u| > N_{ϵ},

(5.5)  ord_{π}B_{u}(k) ≥ w(u) = (c − \frac{ϵ}{2}) log_{q} |u|.

Take a positive integer N_{ϵ}^{∗} to be so large that N_{ϵ}^{∗} > N_{ϵ} and

(5.6)  (c − \frac{ϵ}{2}) log_{q}(q − \frac{N_{ϵ}}{N_{ϵ}^{∗}}) ≥ c − ϵ.

If |u| ≤ N_{ϵ}^{∗}, we have the trivial inequality ord_{π}G_{v,u}(k) ≥ −w(u) ≥ −w(N_{ϵ}^{∗}). To prove the lemma, we first prove the claim that

ord_{π}G_{v,u}(k) ≥ c − ϵ

uniformly for all |u| > N_{ϵ}^{∗}, all v and all k. Assume |u| > N_{ϵ}^{∗}. If v = 0 or qu, one checks that ord_{π}G_{v,u}(k) ≥ w(qu) − w(u) = (c − ϵ/2) > c − ϵ and the claim is true. We now assume that |u| > N_{ϵ}^{∗} and v is different from 0 and qu. There are two cases.

If |qu − v| ≤ N_{ϵ}, then by (5.2) and (5.6), we deduce that

ord_{π}G_{v,u}(k) ≥ (c − \frac{ϵ}{2}) log_{q} \frac{|v|}{|u|}

= (c − \frac{ϵ}{2}) log_{q} \frac{q|u| + (|v| − q|u|)}{|u|}

≥ (c − \frac{ϵ}{2}) log_{q}(q − \frac{N_{ϵ}}{N_{ϵ}^{∗}})

≥ c − ϵ.

If |qu−v| > N_{ϵ}, we use the inequality ab ≥ a+b−1 (i.e., (a−1)(b−1) ≥ 0 for a, b ≥ 1) and deduce that

ord_{π}G_{v,u}(k) ≥ (c − \frac{ϵ}{2}) log_{q} \frac{|v||qu − v|}{|u|}

≥ (c − \frac{ϵ}{2}) log_{q} \frac{|v| + |qu − v| − 1}{|u|}

≥ (c − \frac{ϵ}{2}) log_{q}(\frac{q|u| − 1}{|u|})

≥ (c − \frac{ϵ}{2}) log_{q}(q − \frac{1}{N_{ϵ}^{∗}})

≥ c − ϵ.

The claim is proved. It shows that all column vectors V⃗ contained in the block column of G(k) indexed by u with |u| > N_{ϵ}^{∗} satisfies the inequality in (5.3). To finish the proof of the inequality in (5.3), we can thus restrict our attention to the column vectors of G(k) contained in the finitely many block columns of G(k) indexed by |u| ≤ N_{ϵ}^{∗}. For each such u, our second condition in Definition 5.6 shows that only a uniformly bounded number of column vectors in G(k) may not satisfy the inequality in (5.3). Since there are only finitely many such u, inequality (5.3) is proved. To get (5.4), we can assume that |u| < N_{ϵ}^{∗} by (5.3). For these small u, inequality (5.4) holds with N(ϵ) = w(N_{ϵ}^{∗}). The lemma is proved.

It remains to prove the second part of the theorem. By the inverted version of the Dwork trace formula, we have

det(I − T F (k)) = ∏_{j=0}^{∞} L(B(k, X), q^{j}T)^{(−1)^{n−1}\binom{n+j−1}{j}}.

Since the family L(B(k, X), T) is uniformly continuous (resp. essentially uniformly continuous), the above shifting factor q^{j} shows that the family det(I − T F (k)) is also uniformly continuous (resp., essentially uniformly continuous). By Lemma 5.8, the family det(I − T F (k)) is a uniformly continuous (resp. essentially uniformly continuous) family of uniformly analytic functions in |T |_{π} < p^{c−ϵ}. By Dwork’s trace formula again, the family L(B(k, X), T) is a strong (resp. essentially strong) family in |T |_{π} < p^{c−ϵ}. The theorem is proved.

In order to apply the above uniform result to the study of Dwork’s unit root zeta functions, we need to understand the uniform convergent properties for the family Sym^{k}ϕ of σ-modules arising from various symmetric power of a given nuclear σ-module (M, ϕ) as k varies. Such a family is apparently not uniformly c log-convergent in general, even if the initial one ϕ is c logconvergent. However, we have the following result, which is at the heart of our uniform proof later on.

Lemma 5.10. Let (M, ϕ) be a c log-convergent nuclear σ-module for some 0 < c < ∞, ordinary at slope zero. Let B be the matrix of ϕ under a basis which is ordinary at slope zero. Assume that h_{0} = 1 and assume that the reduction of B modulo π is a constant matrix (namely, each non-constant term of B involving the variable X is divisible by π). Then, the family Sym^{k}B parametrized by positive integer k is uniformly (c − ϵ) log-convergent for every ϵ > 0.

There are two conditions in our definition of uniform c log-convergence. The proof of the above lemma naturally splits into two parts as well, corresponding to checking each one of the two conditions. We first prove that the second condition of uniformity is satisfied. Write

Sym^{k}B = ∑_{u∈Z^{n}} B_{u}(k)X^{u},

where the coefficients B_{u}(k) are matrices with entries in R. The rank of B_{u}(k) is equal to the rank of Sym^{k}ϕ, which is unbounded in general as k varies even if the rank of B is assumed to be finite. Write

B_{u}(k) = (b_{w_{1},w_{2}}(u, k)),

where w_{1} denotes the row index and w_{2} denotes the column index of the matrix B_{u}(k).

Lemma 5.11. For any finite constant C, there is an integer C_{1} > 0 such that for all w_{2} > C_{1}, we have the inequality

ord_{π}b_{w_{1},w_{2}}(u, k) ≥ C

uniformly for all u, k, w_{1}.

Proof. Let e⃗ = {e_{1}, e_{2}, ⋯} be a basis of M ordinary at slope zero. We order the basis

{e_{i_{1}}e_{i_{2}} ⋯ e_{i_{k}} | 1 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{k}}

of the k-th symmetric product Sym^{k}M in some order compatible with the increasing size of i_{1} + ⋯ + i_{k}. For instance,

{e_{1}^{k}, e_{1}^{k−1}e_{2}, e_{1}^{k−2}e_{2}^{2}, e_{1}^{k−1}e_{3}, e_{1}^{k−2}e_{2}e_{3}, · · · .}

By definition of the map Sym^{k}ϕ, we have

(5.7)  Sym^{k}ϕ(e_{i_{1}} ⋯ e_{i_{k}}) = ϕ(e_{i_{1}})ϕ(e_{i_{2}}) ⋯ ϕ(e_{i_{k}}).

Because ϕ is nuclear, the inequality

ord_{π}ϕ(e_{i}) ≥ C

holds for all i > r, where r is a finite number depending on the given integer C. Thus, in equation (5.7), we may restrict our attention to the indices in the range

1 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{k} ≤ r.

Since h_{0} = 1 and our basis is ordinary at slope zero, ϕ(e_{i}) is divisible by π for every i ≥ 2. This shows that the product ϕ(e_{i_{1}})ϕ(e_{i_{2}}) ⋯ ϕ(e_{i_{k}}) is divisible by π^{C} if at least C of the indices in {i_{1}, ⋯, i_{k}} are greater than one. If {i_{1}, ⋯, i_{k}} is a sequence with 1 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{k} ≤ r such that at most C of the indices in it are greater than one, then we must have

1 = i_{1} = i_{2} = ⋯ = i_{k−C} ≤ i_{k−C+1} ≤ ⋯ ≤ i_{k} ≤ r.

The number of such sets {i_{1}, ⋯, i_{k}} is at most r^{C}. Thus, for w_{2} ≥ r^{C} + 1, we have the inequality

ord_{π}b_{w_{1},w_{2}}(u, k) ≥ C

uniformly for all u, k, w_{1}. The lemma is proved.

Next, we prove that the first condition of uniformity is satisfied.

Lemma 5.12. For any ϵ > 0, there is a constant N_{ϵ} > 0 such that the inequality

ord_{π}B_{u}(k) ≥ (c − ϵ) log_{q} |u|

holds uniformly for all |u| > N_{ϵ} and all k.

Proof. Let (b_{ij}(X)) be the matrix of ϕ under a basis {e_{1}, e_{2}, ⋯} which is ordinary at slope zero. Then, we can write

ϕ(e_{j}) = ∑_{i} b_{ij}(X)e_{i} = ∑_{i}(∑_{u} b_{ij}^{(u)}X^{u})e_{i}, b_{ij}(X) ∈ A_{c}.

Expand the product on the right side of (5.7), we get

ϕ(e_{j_{1}}) ⋯ ϕ(e_{j_{k}}) = ∑_{i_{1},···,i_{k}} ∑_{u^{(1)},···,u^{(k)}∈Z^{n}} b_{i_{1}j_{1}}^{(u^{(1)})}X^{u^{(1)}} ⋯ b_{i_{k}j_{k}}^{(u^{(k)})}X^{u^{(k)}}e_{i_{1}} ⋯ e_{i_{k}}.

We need to show that if |u^{(1)}| + ⋯ + |u^{(k)}| > N_{ϵ}, then

ord_{π}(b_{i_{1}j_{1}}^{(u^{(1)})} ⋯ b_{i_{k}j_{k}}^{(u^{(k)})}) ≥ (c − ϵ) log_{q}(|u^{(1)}| + ⋯ + |u^{(k)}|).

By using a smaller positive integer k if necessary, we may assume that all the exponents u^{(1)}, ⋯, u^{(k)} in a typical term of the above expansion are non-zero. In this case, the coefficients b_{i_{ℓ}j_{ℓ}}^{(u^{(ℓ)})} are divisible by π since B is a constant matrix modulo π. Thus, we can write

b_{ij}^{(u)} = πa_{ij}^{(u)}

in R for all u ≠ 0, i and j. The ring A_{c} is π-saturated, i.e., if f = πf_{1} for some f_{1} ∈ A_{0} and f ∈ A_{c}, then f_{1} is also in A_{c}. We choose N_{ϵ} sufficiently large such that the inequality

ord_{π}a_{ij}^{(u)} ≥ (c − ϵ) log_{q} \frac{|u|}{N_{ϵ}}

holds for all u ≠ 0, i and j. This choice is possible for b_{ij}^{(u)} and thus possible for a_{ij}^{(u)} as well since

ord_{π}a_{ij}^{(u)} = ord_{π}b_{ij}^{(u)} − 1.

From these, we deduce that for all u^{(ℓ)} ≠ 0, we have

ord_{π}(b_{i_{1}j_{1}}^{(u^{(1)})} ⋯ b_{i_{k}j_{k}}^{(u^{(k)})}) = k + ord_{π}(a_{i_{1}j_{1}}^{(u^{(1)})} ⋯ a_{i_{k}j_{k}}^{(u^{(k)})})

≥ k + max_{1≤ℓ≤k} ord_{π}a_{i_{ℓ}j_{ℓ}}^{(u^{(ℓ)})}

≥ k + (c − ϵ) log_{q} \frac{|u^{(1)}| + ⋯ + |u^{(k)}|}{kN_{ϵ}}

≥ k + (c − ϵ) log_{q}(|u^{(1)}| + ⋯ + |u^{(k)}|) − (c − ϵ) log_{q} kN_{ϵ}

≥ (c − 2ϵ) log_{q}(|u^{(1)}| + ⋯ + |u^{(k)}|)

uniformly for all sufficiently large |u^{(1)}| + ⋯ + |u^{(k)}| and all k. The lemma is proved.

To obtain more uniformly c log-convergent families from known ones, one can use operations such as direct sum and tensor product. Namely, the set of uniformly c log-convergent families is stable under direct sum and tensor product. This is obvious for direct sum. We shall prove the stability for the tensor operation.

Lemma 5.13. Let B(k, X) and G(k, X) be two families of uniformly c log-convergent matrices. Then their tensor product B(k, X)⊗G(k, X) is also a family of uniformly c log-convergent matrices.

Proof. Write

B(k, X) = ∑_{u∈Z^{n}} B_{u}(k)X^{u}, G(k, X) = ∑_{u∈Z^{n}} G_{u}(k)X^{u}.

Then

B(k, X) ⊗ G(k, X) = ∑_{u,v} B_{u}(k) ⊗ G_{v}(k)X^{u+v}.

For any ϵ > 0, there is an integer N_{ϵ} > 0 such that for all |u| > N_{ϵ} and all k, we have

ord_{π}B_{u}(k) ≥ (c − ϵ) log_{q} |u|, ord_{π}G_{u}(k) ≥ (c − ϵ) log_{q} |u|

and

(c − ϵ) log_{q} \frac{|u|}{2} ≥ (c − 2ϵ) log_{q} |u|.

Thus, for all |u + v| > 2N_{ϵ} and for all k, we have

ord_{π}(B_{u}(k) ⊗ G_{v}(k)) ≥ max(ord_{π}B_{u}(k), ord_{π}G_{v}(k))

≥ (c − ϵ) log_{q} \frac{|u| + |v|}{2}

≥ (c − 2ϵ) log_{q} |u + v|.

This verifies the first condition. It remains to check the second condition. Write

B_{u}(k) = (b_{w_{1},w_{2}}(u, k)), G_{u}(k) = (g_{w_{1},w_{2}}(u, k))

where w_{1} is the row index and w_{2} is the column index. For any positive number C > 0, there is an integer N_{C} > 0 such that for all column index w_{2} > N_{C}, we have

ord_{π}b_{w_{1},w_{2}}(u, k) ≥ C, ord_{π}g_{w_{1},w_{2}}(u, k) ≥ C

uniformly for all u, k, w_{1}. This shows that except for the first N_{C}^{2} columns, all column vectors V⃗ of B_{u}(k) ⊗ G_{v}(k) satisfy

ord_{π}(V⃗) ≥ C

uniformly for all u, v and k. The lemma is proved.

Corollary 5.14. Let ϕ and ψ be two c log-convergent nuclear σmodules, both ordinary at slope zero. Let B (resp. G) be the matrix of ϕ (resp. ψ) under a basis ordinary at slope zero. Assume that h_{0}(ϕ) = h_{0}(ψ) = 1. Assume further that the reductions of B and G modulo π are constant matrices. Then, the family Sym^{k}B ⊗ Sym^{k}G parametrized by k is uniformly (c − ϵ) logconvergent for every ϵ > 0.

Corollary 5.15. Let ϕ(k) be a family of uniformly c log-convergent nuclear σ-modules. Let ψ be a fixed c log-convergent σ-module. Then the twisted family ϕ(k) ⊗ ψ is also uniformly c log-convergent.

These two corollaries can be combined to give many uniformly c logconvergent families of nuclear σ-modules.

6. Unit root theorem

We are now ready to start the proof of the unit root theorem. In fact, we will prove something stronger than the unit root theorem. Namely, the unit root functions L(ϕ_{0}^{k}, T) are not just meromorphic, but in fact they form a strong family in the expected disk in p-adic topology of k.

Theorem 6.1. Let (M, ϕ) be a c log-convergent nuclear σ-module (M, ϕ) for some 0 < c < ∞, ordinary at slope zero. Let (U_{0}, ϕ_{0}) be the unit root part of (M, ϕ). Assume that the rank h_{0} of ϕ_{0} is one. Then, the family L(ϕ_{0}^{k}, T) of unit root zeta functions parametrized by k is a strong family in the open disc |T |_{π} < p^{c−ϵ} for any ϵ > 0, where k varies in any given residue class modulo q − 1 with induced p-adic topology.

Proof. Under our assumption, the matrix of ϕ under an ordinary basis has the following form

(6.1)  B = ( \begin{matrix}B_{00} & πB_{01} \\ B_{10} & πB_{11} \end{matrix} ) ,

where B_{00} is an invertible element of A_{c}, and the other B_{ij} are matrices over A_{c}. We first assume that the reduction of B modulo π is a constant matrix all of whose entries are zero except for the top left corner entry which is 1. Namely, we assume that

(6.2)  B_{00} ≡ 1 (modπ), ord_{π}B_{10} ≥ 1.

Under this simplifying condition, we show that the family L(ϕ_{0}^{k}, T) is a strong family, where k varies in the ring of integers with induced p-adic topology.

Let k be an integer and let

k_{m} = k + p^{m}.

Then as p-adic numbers, we have lim_{m→∞} k_{m} = k. Furthermore, k_{m} is positive for all large m. For any p-adic 1-unit α ∈ R, one sees easily that

lim_{m→∞} α^{k_{m}} = α^{k}.

Since ϕ_{0} is of rank 1 and B_{00} is a 1-unit fibre by fibre, the characteristic root of each Euler factor in L(ϕ_{0}, T) is a p-adic 1-unit in R. Thus, we have the following limiting formula

L(ϕ_{0}^{k}, T) = lim_{m→∞} L(ϕ_{0}^{k_{m}}, T)

= lim_{m→∞} L(ϕ^{k_{m}}, T). (6.3)

The second equality holds because lim_{m} π^{k_{m}} = 0. Let S be the sequence {k_{m}}. There are two obvious topologies on S. One is the p-adic topology which has k as the unique limiting point not in S. The other is the sequence topology which has ∞ as the unique limiting point not in S. It is easy to check that these two topologies agree on the set S, although the distance function may be a little different. Thus, one could use either topology in (6.3). We use the sequence topology to stress that m will go to infinity. To show that the limit function L(ϕ_{0}^{k}, T) is meromorphic in the expected disk, we need to show that the family L(ϕ^{k_{m}}, T) is a strong family with respect to the sequence topology.

Since Sym^{k−i}ϕ = 0 for k < i, the following fundamental decomposition formula

L(ϕ^{k}, T) = ∏_{i≥1} L(Sym^{k−i}ϕ ⊗ ∧^{i}ϕ, T)^{(−1)^{i−1}i}

is a finite product. Thus, we need to show that for each fixed i, the family L(Sym^{k_{m}−i}ϕ⊗∧^{i}ϕ, T) is a strong family, where k_{m} varies over the set of positive integers in S with the sequence topology. We first prove the uniform continuity property of this family.

Lemma 6.2. The family of L-functions L(Sym^{k_{m}−i}ϕ ⊗ ∧^{i}ϕ, T) is uniformly continuous in k_{m} with respect to the sequence topology.

Proof. With respect to the sequence topology, uniformly continuous notion is the same as the continuous notion. Thus, it suffices to prove that L(Sym^{k_{m}−i}ϕ ⊗ ∧^{i}ϕ, T) is continuous in k_{m} with respect to the sequence topology. Since there are only a finite number of closed points with a bounded degree, it suffices to check that each Euler factor is continuous in k. At a closed point x̄ ∈ G_{m}/F_{q}, let α_{0}, α_{1}, ⋯ denote the characteristic roots (counting multiplicities) of ϕ at the fibre x̄ such that α_{0} is the p-adic 1-unit and all α_{j} are divisible by π for j ≥ 1. One checks that each α_{j} for j ≥ 1 is in fact divisible by the higher power π^{deg(x)}. The local Euler factor at x of L(Sym^{k}ϕ ⊗ ∧^{i}ϕ, T) is given by the reciprocal of the product

(6.4)  E_{k}(x, T) = ∏(1 − α_{j_{1}} ⋯ α_{j_{k}}α_{ℓ_{1}} ⋯ α_{ℓ_{i}}T^{deg(x)}),

where the product runs over all

0 ≤ j_{1} ≤ ⋯ ≤ j_{k}, 0 ≤ ℓ_{1} < ⋯ < ℓ_{i}.

The above product is convergent as α_{j} is more and more divisible by π as j increases. Let B_{x} denote the diagonal matrix diag(α_{0}, α_{1}, ⋯) and let B_{x,1} denote the diagonal sub-matrix diag(α_{1}, α_{2}, ⋯). With k replaced by k_{m} − i in (6.4), one checks that

E_{k_{m}−i}(x, T) = ∏_{j=0}^{k_{m}−i} det(I − T^{deg(x)}α_{0}^{k_{m}−i−j}Sym^{j}B_{x,1} ⊗ ∧^{i}B_{x}).

Now, since α_{0} is a 1-unit, for fixed i and j, the sequence α_{0}^{k_{m}−i−j} converges to α_{0}^{k−i−j} as m goes to infinity. Since Sym^{j}B_{x,1} is divisible by π^{jdeg(x)}, we deduce that the sequence L(Sym^{k_{m}−i}ϕ ⊗ ∧^{i}ϕ, T) is continuous in k_{m} with respect to the sequence topology. The lemma is proved.

We now return to the proof of Theorem 6.1. Because of the special form of our matrix B(X), Lemma 5.10 and Corollary 5.15 show that the family Sym^{k_{m}−i}ϕ ⊗ ∧^{i}ϕ is uniformly (c − ϵ) log-convergent for each fixed i. We just proved that the L-function of this sequence is uniformly continuous. Thus, by Theorem 5.7, the family of L-functions L(Sym^{k_{m}−i}ϕ⊗∧^{i}ϕ, T) is a strong family in |T |_{π} < p^{c−ϵ} for each fixed i. By Corollary 5.4 and the limiting formula

(6.5)  L(ϕ_{0}^{k}, T) = lim_{m→∞} ∏_{i≥1} L(Sym^{k_{m}−i}ϕ ⊗ ∧^{i}ϕ, T)^{(−1)^{i−1}i},

we conclude that the limiting unit root zeta function L(ϕ_{0}^{k}, T) is meromorphic in the expected disk for each fixed integer k. Here, we are using the fact that ∧^{i}ϕ is more and more divisible by π as i grows. Also, we are using the sequence topology in the limiting formula. This finishes the proof of the meromorphy part (for each k) of Theorem 6.1.

Once we know that each L(ϕ_{0}^{k}, T) is meromorphic, we want to consider it as a family of functions parametrized by integer k ∈ Z with induced p-adic topology. We want to show that this family is a strong family. There are at least two approaches to make it work. Here we describe one of them. The other approach is given in section 8.

The point is that the family L(Sym^{k}ϕ ⊗ ∧^{i}ϕ, T) (and also the family L(ϕ^{k}, T)) is not continuous in k with respect to the p-adic topology. However, it is essentially continuous if we restrict to a fixed finite disk and remove a few small values of k. The above proof then goes through if we use essentially continuous family. As the final family L(ϕ_{0}^{k}, T) is indeed continuous with respect to the p-adic topology, we will be able to drop the word “essentially ” in the final conclusion.

To carry out this idea, we need to modify Lemma 6.2 as follows.

Lemma 6.3. Let S_{c} be the set of all positive integers k ≥ c with induced p-adic topology. Then the family of L-functions L(Sym^{k}ϕ ⊗ ∧^{i}ϕ, T) is essentially uniformly continuous in k in the disk |T |_{π} < p^{c}.

Proof. The proof is similar to the proof of Lemma 6.2. In the product decomposition

E_{k}(x, T) = ∏_{j=0}^{k} det(I − T^{deg(x)}α_{0}^{k−j}Sym^{j}B_{x,1} ⊗ ∧^{i}B_{x}),

we just take j runs up to c + 1 instead of k. Namely, we let

E_{k,c}(x, T) = ∏_{0≤j<c+1} det(I − T^{deg(x)}α_{0}^{k−j}Sym^{j}B_{x,1} ⊗ ∧^{i}B_{x}),

The function E_{k,c}(x, T) is clearly uniformly continuous in k ∈ S_{c} with respect to the p-adic topology since each of its factors is uniformly continuous in k and we have a bounded number of factors. The quotient E_{k}(x, T)/E_{k,c}(x, T) has no contribution of zeros and poles in the disk |T |_{π} ≤ p^{c}. This is because for j ≥ c + 1, Sym^{j}B_{x,1} is divisible by π^{jdeg(x)} and hence divisible by π^{(c+1)deg(x)}. This shows that E_{k}(x, T) and thus L(Sym^{k}ϕ ⊗ ∧^{i}ϕ, T) is essentially uniformly continuous in the disk |T |_{π} < p^{c} with respect to the p-adic topology of k ∈ S_{c}. The lemma is proved.

We now return to the proof of the strong family part of Theorem 6.1. Lemma 6.3 and Theorem 5.7 show that for each fixed i, the family

f(k, i, T) = L(Sym^{k}ϕ ⊗ ∧^{i}ϕ, T)

parametrized by k ∈ S_{c} is an essentially strong family in |T |_{π} < p^{c−ϵ}. For a positive integer k ∈ S_{c}, let

g(k, i, T) = lim_{m→∞} L(Sym^{k+p^{m}}ϕ ⊗ ∧^{i}ϕ, T).

Since

lim_{m→∞} k + p^{m} = k,

the Euler factor proof in Lemma 6.3 shows that the quotient f(k, i, T)/g(k, i, T) is a 1-unit in |T |_{π} ≤ p^{c}. Note that f(k, i, T) is not equal to g(k, i, T) since the family f(k, i, T) is not continuous but only essentially continuous in |T |_{π} < p^{c}. Thus, the family g(k, i, T) parametrized by k ∈ S_{c} is equivalent to the family f(k, i, T) in |T |_{π} < p^{c}. This implies that g(k, i, T) parametrized by k ∈ S_{c} is also an essentially strong family in |T |_{π} < p^{c−ϵ}. By (6.5), we have the formula for k ∈ S_{2c}:

L(ϕ_{0}^{k}, T) = h(k, T) ∏_{1≤i≤c} g(k − i, i, T),

where h(k, T) is the product of those factors with i > c and h(k, T) is a 1-unit in |T |_{π} ≤ p^{c}. Thus, the unit root family L(ϕ_{0}^{k}, T) parametrized by k ∈ S_{2c} is an essentially strong family in |T |_{π} < p^{c−ϵ}. In this way, we can write

L(ϕ_{0}^{k}, T) = f_{1}(k, T)f_{2}(k, T),

where f_{1}(k, T) is a strong family in |T |_{π} < p^{c−ϵ} and f_{2}(k, T) is a family of analytic functions but without reciprocal zeros and bounded by 1 on |T |_{π} ≤ p^{c−ϵ}. Since both L(ϕ_{0}^{k}, T) and f_{1}(k, T) are uniformly continuous in k, we deduce that the family f_{2}(k, T) is a uniformly continuous family of uniformly analytic functions (without reciprocal zeros) in |T |_{π} ≤ p^{c−ϵ}. It follows that L(ϕ_{0}^{k}, T) is a strong family parametrized by k ∈ S_{2c}. This family extends uniquely to a strong family parametrized by k varying in the topological closure of S_{2c} which is the whole space Z_{p} of p-adic integers. By uniqueness and the continuity of the family L(ϕ_{0}^{k}, T), this extended family agrees with L(ϕ_{0}^{k}, T) for all k ∈ Z_{p}. This concludes our proof under the above simpler condition (6.2) on B.

We now explain how to reduce the general case to the above simpler situation. Consider the new basis (still ordinary at slope zero)

(e_{1}, e_{2}, ⋯) ( \begin{matrix}I_{00} & 0 \\ B_{10}B_{00}^{−1} & I_{11} \end{matrix} ) = e⃗E

of M, where I_{00} is the rank 1 identity matrix and I_{11} is an identity matrix. The transition matrix E is c log-convergent since B_{10}B_{00}^{−1} is c log-convergent. One calculates that the matrix of ϕ under e⃗E is given by

E^{−1}BE^{σ} = ( \begin{matrix}B_{00} + πB_{01}(B_{10}B_{00}^{−1})^{σ} & πB_{01} \\ π(B_{11} − B_{10}B_{00}^{−1}B_{01})(B_{10}B_{00}^{−1})^{σ} & π(B_{11} − B_{10}B_{00}^{−1}B_{01}) \end{matrix} ) .

This matrix is still c log-convergent since both E and E^{σ} are c log-convergent. Let f be the polynomial which is obtained from the top left entry of the above matrix by dropping all terms divisible by π. Since ϕ is ordinary at slope zero, the polynomial f is a unit in the coordinate ring of the n-torus over F_{q}. This means that f must be a monomial whose coefficient is a p-adic unit. It is of course an invertible element in A_{c}. Pulling out the rank one factor f, we see that in c log-convergent category, the σ-module (M, ϕ) is the tensor product of a rank one unit root σ-module with matrix f and an ordinary nuclear σmodule with a matrix satisfying the above simpler condition (6.2). Namely, we have

ϕ = f ⊗ ψ,

where f is of rank 1 and ψ has a matrix satisfying (6.2). At each fibre x̄, we have

lim_{m→∞}(f(x)f(x^{q}) ⋯ f(x^{q^{deg(x̄)−1}}))^{(q−1)p^{m}} = 1.

Using the fact that f is of rank 1, by looking at the Euler factors, one easily checks that

L(ϕ_{0}^{k}, T) = lim_{m→∞} L(f^{k} ⊗ ψ^{k+(q−1)p^{m}}, T)

= lim_{m→∞} ∏_{i≥1} L(f^{k} ⊗ Sym^{k+(q−1)p^{m}−i}ψ ⊗ ∧^{i}ψ, T)^{(−1)^{i−1}i}.

We are now in a similar situation as before, except that we now have an extra rank 1 twisting factor f^{k}. For a fixed k (and fixed i, of course), the family f^{k} ⊗Sym^{k+(q−1)p^{m}−i}ψ⊗∧^{i}ψ parametrized by p^{m} is uniformly (c−ϵ) logconvergent. For a fixed k, L(f^{k} ⊗Sym^{k+(q−1)p^{m}−i}ψ⊗∧^{i}ψ, T) is also continuous with respect to the sequence topology of the sequence p^{m}. In the same way as before, we deduce that the limiting function L(ϕ_{0}^{k}, T) is meromorphic in the open disk |T |_{π} < p^{c−ϵ} for each fixed k.

To finish the proof, we have to show that L(ϕ_{0}^{k}, T) (k ∈ S) is a strong family in k with respect to the p-adic topology, where S is the integers in a fixed residue class modulo (q − 1). This is done by the following trick of change of basis. Let r denote the smallest non-negative residue of S modulo q − 1. We observe that if we change our basis e⃗ to ge⃗, where g is an invertible element in A_{c}, then the matrix of ϕ under ge⃗ will be g^{−1}Bg^{σ} = g^{σ}g^{−1}B, which is still ordinary at slope zero and c log-convergent. In the case that g is a monomial with coefficient 1, the new basis is then g^{q−1}B and thus we can remove the factor g^{q−1} without changing the L-function. Because our f is indeed a monomial, we can thus write f = ag, where a is a p-adic unit in R and g is a monomial with coefficient 1. In this way, we can replace f^{k} by a^{k}g^{r} in the above limiting formula since we can drop any power of g^{q−1}. Thus, we have shown, with k_{m} = k + (q − 1)p^{m}, that

L(ϕ_{0}^{k}, T) = ∏_{i≥1} lim_{m→∞} L(a^{k_{m}}g^{r} ⊗ Sym^{k_{m}−i}ψ ⊗ ∧^{i}ψ, T)^{(−1)^{i−1}i},

where we have replaced a^{k} by a^{k_{m}} from the previous formula. Now r is fixed and a is a constant. Corollary 5.15 shows that the family a^{k}g^{r} ⊗Sym^{k−i}ψ⊗∧^{i}ψ parametrized by k ∈ S_{c} is uniformly (c − ϵ) log-convergent, where S_{c} consists of the elements of S which are greater than c. It is clear that a^{k} is uniformly continuous in k ∈ S. One checks as before that the family of L-functions L(a^{k}g^{r} ⊗ Sym^{k−i}ψ ⊗ ∧^{i}ψ, T) is essentially uniformly continuous with respect to the p-adic topology of S_{c} in the disk |T |_{π} < p^{c−ϵ} and thus it forms an essentially strong family in the same disk. This implies that L(ϕ_{0}^{k}, T) is an essentially strong family on S_{c}. Since L(ϕ_{0}^{k}, T) is also a uniformly continuous family, it must be a strong family on S_{c}. The topological closure of S_{c} includes all S. We conclude that L(ϕ_{0}^{k}, T) is a strong family on all of S. The theorem is proved.

7. Higher slope theorem

We are now ready to prove the higher slope theorem. Again, we will prove a stronger family version. The idea here will be refined and extended in [29] to treat the higher rank case of Dwork’s conjecture.

Theorem 7.1. Let (M, ϕ) be a c log-convergent nuclear σ-module (M, ϕ) for some 0 < c < ∞, ordinary up to slope j for some integer j ≥ 0. Let (U_{i}, ϕ_{i}) be the unit root σ-module coming from the slope i part of (M, ϕ) for 0 ≤ i ≤ j. Assume that h_{j} = 1. Then the family of unit root zeta functions L(ϕ_{j}^{k}, T) parametrized by k is a strong family in the open disc |T |_{π} < p^{c−ϵ} for every ϵ > 0, where k varies in any given residue class modulo q − 1 with induced p-adic topology.

Proof. The case for j = 0 has already been proved by the previous unit root theorem. We may thus assume that j ≥ 1. It is clear that

∧^{h_{0}+···+h_{i}}ϕ ≡ 0 (mod π^{h_{1}+···+ih_{i}}).

Let ψ_{i} be the twist

(7.1)  ψ_{i} = π^{−(h_{1}+···+ih_{i})}∧^{h_{0}+···+h_{i}}ϕ.

By (3.1), this is a c log-convergent σ-module with h_{0}(ψ_{i}) = 1 and ordinary at slope zero for all 0 ≤ i ≤ j.

Lemma 7.2. For integer k > 0, we have the limiting formula

L(ϕ_{j}^{k}, T) = lim_{m→∞} L(ψ_{j−1}^{−k+(q−1)p^{m+k}} ⊗ ψ_{j}^{k+(q−1)p^{m}}, T),

where ψ_{j}^{k} is the k-th power (not tensor power) defined at the end of section 2.

Proof. Since there are only a finite number of closed points with a bounded degree, it suffices to prove the corresponding limiting formula for each Euler factor. At a closed point x̄ ∈ G_{m}/F_{q}, let π^{ih_{i}deg(x)}α_{i} be the product of the h_{i} characteristic roots of ϕ at the fibre x̄ with slope i, where 0 ≤ i ≤ j. Similarly, let γ_{i} be the unique characteristic root of ψ_{i} at the fibre x̄ which is a p-adic unit, where 0 ≤ i ≤ j. Then for each i with 0 ≤ i ≤ j, we have the relation

α_{0}α_{1} ⋯ α_{i} = γ_{i}.

In particular,

γ_{j−1}α_{j} = γ_{j}.

One checks that the elements α_{i} and γ_{i} for 0 ≤ i ≤ j are actually units in R. For each i, let β_{i} run over the non-unit characteristic roots of ψ_{i}. One computes that the local Euler factor at x of L(ψ_{j−1}^{−k+(q−1)p^{m+k}} ⊗ ψ_{j}^{k+(q−1)p^{m}}, T) is given by the reciprocal of the product

E_{m,k}(x, T) =(1 − (γ_{j−1})^{(q−1)(p^{m+k}+p^{m})}α_{j}^{k+(q−1)p^{m}}T^{deg(x)})

× ∏_{β_{j}}(1 − (γ_{j−1})^{−k+(q−1)p^{m+k}}β_{j}^{k+(q−1)p^{m}}T^{deg(x)})

× ∏_{β_{j−1}}(1 − β_{j−1}^{−k+(q−1)p^{m+k}}γ_{j}^{k+(q−1)p^{m}}T^{deg(x)})

× ∏_{β_{j−1},β_{j}}(1 − β_{j−1}^{−k+(q−1)p^{m+k}}β_{j}^{k+(q−1)p^{m}}T^{deg(x)}).

Now both β_{j−1} and β_{j} are divisible by π. Also both γ_{j−1}^{q−1} and α_{j}^{q−1} are 1-unit in R. We deduce that

lim_{m→∞} E_{m,k}(x, T) = (1 − α_{j}^{k}T^{deg(x)}).

This is the same as the reciprocal of the Euler factor of L(ϕ_{j}^{k}, T) at x. The desired limiting formula of the lemma follows.

As in the proof of the unit root theorem, we can write

(7.2)  ψ_{i} = f_{i} ⊗ φ_{i},

where f_{i} is an invertible monomial in A_{c}, φ_{i} is a c log-convergent σ-module with h_{0}(φ_{i}) = 1 and ordinary at slope zero. Furthermore, the matrix of φ_{i} under an ordinary basis satisfies the condition in (6.2). Since the f_{i} have rank one, we can pull them out of the above limiting formula and get

L(ϕ_{j}^{k}, T) = lim_{m→∞} L(Φ_{m}(k), T),

where

Φ_{m}(k) = f_{j−1}^{−k+(q−1)p^{m+k}} ⊗ f_{j}^{k+(q−1)p^{m}} ⊗ φ_{j−1}^{−k+(q−1)p^{m+k}} ⊗ φ_{j}^{k+(q−1)p^{m}}.

At each fibre x̄, we have

lim_{m→∞}(f(x)f(x^{q}) ⋯ f(x^{q^{deg(x̄)−1}}))^{(q−1)p^{m}} = 1.

Using this fact and the decomposition formula derived from Lemma 4.5, we deduce that

\begin{aligned}L(ϕ_{j}^{k}, T) & = lim_{m→∞} L((\frac{f_{j}}{f_{j−1}} )^{k} ⊗ φ_{j−1}^{−k+(q−1)p^{m+k}} ⊗ φ_{j}^{k+(q−1)p^{m}}, T) \\ & = lim_{m→∞} ∏_{ℓ_{1},ℓ_{2}≥1} L((\frac{f_{j}}{f_{j−1}} )^{k} ⊗ Ψ_{m}(k, ℓ_{1}, ℓ_{2}), T)^{(−1)^{(ℓ_{1}+ℓ_{2})}ℓ_{1}ℓ_{2}}, \end{aligned}

where Ψ_{m}(k, ℓ_{1}, ℓ_{2}) is given by

Sym^{−k+(q−1)p^{m+k}−ℓ_{1}}φ_{j−1} ⊗ ∧^{ℓ_{1}}φ_{j−1} ⊗ Sym^{k+(q−1)p^{m}−ℓ_{2}}φ_{j} ⊗ ∧^{ℓ_{2}}φ_{j}.

For fixed k, ℓ_{1} and ℓ_{2}, Corollaries 5.14-5.15 show that the family Ψ_{m}(k, ℓ_{1}, ℓ_{2}) parametrized by the sequence {m} is uniformly c log-convergent. As in Lemma 6.2, one checks that the family of L-functions L((\frac{f_{j}}{f_{j−1}} )^{k} ⊗ Ψ_{m}(k, ℓ_{1}, ℓ_{2}), T) is (uniformly) continuous with respect to the sequence topology of {m}. It is therefore a strong family by Theorem 5.7. This shows that the limiting function L(ϕ_{j}^{k}, T) is meromorphic in the expected disk for each fixed integer k > 0.

Let S be the set of integers in a fixed residue class modulo q − 1. To show that L(ϕ_{j}^{k}, T) is a strong family parametrized by S with respect to the p-adic topology, one can use the notion of an essentially strong family as in the case i = 0. Let S_{c} be the set consisting of the integers in S which are greater than c. By the proof of Lemma 7.2, we also have the slightly different limiting formula:

L(ϕ_{j}^{k}, T) = lim_{a→∞} L(ψ_{j−1}^{−k_{a}+(q−1)p^{ka}} ⊗ ψ_{j}^{k_{a}}, T),

where k_{a} is any sequence of positive integers in S_{c} such that lim_{a} k_{a} = ∞ as integers and lim_{a} k_{a} = k as p-adic integers. For instance, we can take k_{a} = k + (q − 1)p^{a}. Using the decomposition formula from Lemma 4.5 and (7.2), we get

\begin{aligned}L(ϕ_{j}^{k}, T) & = lim_{a} L((\frac{f_{j}}{f_{j−1}} )^{k_{a}} ⊗ φ_{j−1}^{−k_{a}+(q−1)p^{ka}} ⊗ φ_{j}^{k_{a}}, T) \\ & = lim_{a} ∏_{ℓ_{1},ℓ_{2}≥1} L((\frac{f_{j}}{f_{j−1}} )^{k_{a}} ⊗ Ψ(k_{a}, ℓ_{1}, ℓ_{2}), T)^{(−1)^{(ℓ_{1}+ℓ_{2})}ℓ_{1}ℓ_{2}}, \end{aligned}

where

Ψ(k_{a}, ℓ_{1}, ℓ_{2}) = Sym^{−k_{a}+(q−1)p^{ka}−ℓ_{1}}φ_{j−1} ⊗ ∧^{ℓ_{1}}φ_{j−1} ⊗ Sym^{k_{a}−ℓ_{2}}φ_{j} ⊗ ∧^{ℓ_{2}}φ_{j}.

As in Lemma 6.3, one finds that the family L((\frac{f_{j}}{f_{j−1}} )^{k_{a}} ⊗ Ψ(k_{a}, ℓ_{1}, ℓ_{2}), T) of functions parametrized by k_{a} ∈ S_{c} is essentially uniformly continuous in the expected disk. The family Ψ(k_{a}, ℓ_{1}, ℓ_{2}) parametrized by k_{a} is uniformly c logconvergent by Corollaries 5.14-5.15. Using the trick of the change of basis, we can reduce the power of (f_{j}/f_{j−1})^{k_{a}} modulo q − 1 up to a twisting constant in R. This implies that the family L((\frac{f_{j}}{f_{j−1}} )^{k_{a}} ⊗ Ψ(k_{a}, ℓ_{1}, ℓ_{2}), T) parametrized by k_{a} ∈ S_{c} is an essentially strong family. It extends to an essentially strong family on the topological closure of S_{c}. We deduce that the limiting family L(ϕ_{j}^{k}, T) is an essentially strong family on S_{c} and hence on all of S. Because L(ϕ_{j}^{k}, T) is uniformly continuous in k ∈ S, we conclude that L(ϕ_{i}^{k}, T) is a strong family for k ∈ S in the expected disk. The proof is complete.

8. Limiting σ-module and explicit formula

In the proof of the unit root theorem and the higher slope theorem, we used the notion of essential continuity to understand the variation of the unit root zeta function as k varies with respect to the p-adic topology. There is another approach which avoids the essential continuity notion and works directly with continuous families. This will be useful for further investigations as well as some other applications. The purpose of this section is to describe this second approach in the optimal c log-convergent setup. A full treatment of this approach in the simpler overconvergent setting will be given in a future paper when we need to get explicit information about zeros and poles of unit root zeta functions.

Assuming that ϕ satisfies the simpler condition in (6.2) and assume that k_{m} = k + p^{m}. As we have seen before, the key is to understand the limit function of the sequence of L-functions L(Sym^{k_{m}}ϕ, T) as k_{m} varies. In the new approach presented in this section, we show that there is a c log-convergent nuclear σ-module (M_{∞,k}, ϕ_{∞,k}) called the limiting σ-module of the sequence (Sym^{k_{m}}M, Sym^{k_{m}}ϕ) such that

(8.1)  lim_{m→∞} L(Sym^{k_{m}}ϕ, T) = L(ϕ_{∞,k}, T).

Furthermore, the family (M_{∞,k}, ϕ_{∞,k}) parametrized by k is uniformly c logconvergent. As the positive integer k varies p-adically, the family of functions L(ϕ_{∞,k}, T) parametrized by k turns out to be uniformly continuous in k. This provides another proof of the unit root theorem and the higher slope theorem. It shows that Dwork’s unit root zeta function in the ordinary rank one case is an infinite alternating product of L-functions of nuclear σ-modules with the same convergent condition as the initial ϕ. This infinite product can be taken to be a finite product if one restricts to a finite disk. In the case that the initial nuclear σ-module (M, ϕ) is of finite rank, then the infinite product is actually a finite product. This can be viewed as a structure theorem. It implies that if the initial (M, ϕ) is c log-convergent (resp. overconvergent), then Dwork’s unit root zeta function in the ordinary rank one case can be expressed in terms of L-functions of c log-convergent (resp. overconvergent) nuclear σ-modules of infinite rank.

Let (M, ϕ) be a nuclear c log-convergent σ-module satisfying the simple condition in (6.2). Let k be a positive integer. We want to identify the k-th symmetric power (Sym^{k}M, Sym^{k}ϕ) with another σ-module (M_{k}, ϕ_{k}) which is easier to take the limit as k varies p-adically. Let e⃗ = {e_{1}, e_{2}, ⋯} be a basis of (M, ϕ) ordinary at slope zero such that its matrix satisfies the condition in (6.2). Define M_{k} to be the Banach module over A_{0} with the basis f⃗(k):

{f_{i_{1}}f_{i_{2}} ⋯ f_{i_{r}}|2 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{r}, 0 ≤ r ≤ k},

where we think of 1 (corresponding to the case r = 0) as the first basis element of f⃗(k). There is an isomorphism of Banach A_{0}-modules between Sym^{k}M and M_{k}. This isomorphism is given by the map:

Υ : e_{1}^{k−r}e_{i_{1}} ⋯ e_{i_{r}} ⟶ f_{i_{1}} ⋯ f_{i_{r}}.

Thus,

Υ(e_{1}) = 1, Υ(e_{i}) = f_{i} (i > 1).

We can give a nuclear σ-module structure on M_{k}. The semi-linear map ϕ_{k} acting on M_{k} is given by the pull back Υ ∘ Sym^{k}ϕ ∘ Υ^{−1} of Sym^{k}ϕ acting on Sym^{k}M. Namely,

(8.2)  ϕ_{k}(f_{i_{1}} ⋯ f_{i_{r}}) = Υ(ϕ(e_{1}^{k−r})ϕ(e_{i_{1}}) ⋯ ϕ(e_{i_{r}})).

Under the identification of Υ, the map Sym^{k}ϕ becomes ϕ_{k}. Thus, the k-th symmetric power (Sym^{k}M, Sym^{k}ϕ) is identified with (M_{k}, ϕ_{k}). It follows that

L(Sym^{k}ϕ, T) = L(ϕ_{k}, T).

In order to take the limit of the sequence of modules M_{k}, we define M_{∞} to be the Banach module over A_{0} with the countable basis f⃗:

(8.3)  {f_{i_{1}}f_{i_{2}} ⋯ f_{i_{r}}|2 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{r}, 0 ≤ r},

where, again, we think of 1 (corresponding to the case r = 0) as the first basis element of f⃗. Note that in (8.3), there is no longer restriction on the size of r.

The module M_{∞} can be identified with the graded commutative algebra over A_{0} generated by the free elements {f_{2}, f_{3}, ⋯} which are of degree 1. Namely, as graded commutative A_{0}-algebra, we have

M_{∞} = A_{0}[[f_{2}, f_{3}, ⋯]],

where A_{0}[[f]] denotes the ring of power series over A_{0} in the infinite set of variables {f_{2}, f_{3}, ⋯}. To be more precise, we define the weight w(f_{i}) = i. Denote by I the countable set of integer vectors u = (u_{2}, u_{3}, ⋯) satisfying u_{i} ≥ 0 and almost all of the u_{i} (except for finitely many of them) are zero. For a vector u ∈ I, define

f^{u} = f_{2}^{u_{2}}f_{3}^{u_{3}} ⋯ f_{m}^{u_{m}} ⋯, w(u) = 2u_{2} + 3u_{3} + ⋯ + mu_{m} + · · · .

Then the infinite dimensional A_{0}-algebra M_{∞} can be described precisely as follows:

M_{∞} = {∑_{u∈I} a_{u}f^{u}|a_{u} ∈ A_{0}}.

The algebra M_{∞} becomes a Banach algebra over A_{0} with a basis given by (8.3) or equivalently by {f^{u}|u ∈ I}. We order the elements of I in any way which is compatible with the increasing size of w(u). If (M, ϕ) is of finite rank, say of rank r, then one checks that the algebra M_{∞} is just the formal power series ring A_{0}[[f_{2}, ⋯, f_{r}]] over A_{0}. Thus, the algebra M_{∞} is Noetherian if and only if the rank r of ϕ is finite. Note that the Banach A_{0}-module M_{∞} has a countable basis but it does not have an orthonormal basis.

For u ∈ I, the degree of f^{u} is defined to be ∥u∥ = u_{2} + u_{3} + · · ·. The degree of an element ∑ a_{u}f^{u} in M_{∞} is defined to be the largest degree of its non-zero terms. Namely, the degree is given by

sup{∥u∥ | a_{u} ≠ 0}.

The degree of an element ∑ a_{u}f^{u} could be infinite in general because there are infinitly many terms in ∑ a_{u}f^{u}. For each positive integer k, the Banach module M_{k} is the submodule of M_{∞} consisting of those elements of degree at most k. That is,

M_{k} = {f ∈ M_{∞}| deg(f) ≤ k}.

For each integer k, define

M_{∞,k} = M_{∞}.

This module is thus independent of k. To define the nuclear semi-linear map ϕ_{∞,k} on M_{∞,k} for each integer k, we take a sequence of positive integers k_{m} (for instance k_{m} = k + p^{m}) such that lim k_{m} = ∞ as integers and lim_{m} k_{m} = k as p-adic integers. Define ϕ_{∞,k} by the following limiting formula:

ϕ_{∞,k}(f_{i_{1}} ⋯ f_{i_{r}}) = lim_{m→∞} Υ(ϕ(e_{1}^{k_{m}−r})ϕ(e_{i_{1}}) ⋯ ϕ(e_{i_{r}}))

= ( lim_{m→∞} Υ(ϕ(e_{1}^{k_{m}−r})))Υ(ϕ(e_{i_{1}}) ⋯ ϕ(e_{i_{r}})). (8.4)

To show that this is well defined, we need to show that the above first limiting factor exists. Write

(8.5)  ϕ(e_{1}) = u(e_{1} + πe),

where u is a 1-unit in R and e is an element of M which is an infinite linear combination of the elements in {e_{2}, e_{3}, ⋯}. By the binomial theorem, we have

\begin{aligned}ϕ(e_{1}^{k}) & = u^{k}(e_{1} + πe)^{k} \\ & = u^{k}(e_{1}^{k} + \binom{k}{1}πe_{1}^{k−1}e + \binom{k}{2}π^{2}e_{1}^{k−2}e^{2} + ⋯). \end{aligned}

This identity implies that

lim_{m} Υ(ϕ(e_{1}^{k_{m}−r})) = u^{k−r}(1 + \binom{k − r}{1}πΥ(e) + \binom{k − r}{2}π^{2}Υ(e)^{2} + ⋯)

= u^{k−r}(1 + πΥ(e))^{k−r}. (8.6)

Thus, the map ϕ_{∞,k} is well defined for every integer k and it is independent of the choice of our chosen sequence k_{m}. Furthermore, using (8.4) and (8.6), we see that ϕ_{∞,k} makes sense for any p-adic integer k, not necessarily usual integers. With this definition, we have the following result.

Theorem 8.1. Let (M, ϕ) be a nuclear c log-convergent σ-module which is ordinary at slope zero and h_{0}(ϕ) = 1. Assume that ϕ satisfies the simpler condition in (6.2 ). Then, the family (M_{∞}, ϕ_{∞,k}) parametrized by p-adic integer k is a family of uniformly c log-convergent nuclear σ-modules. Furthermore, the family of L-functions L(ϕ_{∞,k}, T) parametrized by p-adic integer k ∈ Z_{p} is a strong family in the open disk |T |_{π} < p^{c−ϵ} for any ϵ > 0.

Proof. The proof of the uniform part is essentially the same as the proof of Lemma 5.10. The main point is that all bounds in that proof are uniformly independent of k. We omit the details. For the strong family part, by Theorem 5.7, we only need to check that the family of L-functions L(ϕ_{∞,k}, T) is uniformly continuous in k with respect to the p-adic topology. By our definition of ϕ_{∞,k} in (8.4) and (8.6), it suffices to show that

(8.7)  ∥u^{k_{1}−r}(1 + πΥ(e))^{k_{1}−r} − u^{k_{2}−r}(1 + πΥ(e))^{k_{2}−r}∥ ≤ ∥k_{1} − k_{2}∥^{c}

uniformly for all integers r, where c is some positive constant. But this follows from the binomial theorem since u is a 1-unit in R. Note that the constant c may be smaller than 1 depending on the ramification degree of R. The proof is complete.

In order to take the limit of ϕ_{k} acting on the submodule M_{k} of M_{∞} as k varies in a sequence of positive integers, we semi-linearly extend ϕ_{k} to act on M_{∞} by requiring

ϕ(f_{i_{1}} ⋯ f_{i_{r}}) = 0

for all indices 2 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{r} with i_{r} > k. In this way, ϕ_{k} induces a semi-linear endomorphism of the Banach algebra M_{∞} over A_{0}. The pair (M_{∞}, ϕ_{k}) then becomes a nuclear σ-module over A_{0}. The L-function L(ϕ_{k}, T) is the same, whether we view ϕ_{k} as acting on M_{∞} or on the submodule M_{k}. Let μ(k) be the greatest integer such that for all integers j > k, we have

ϕ(e_{j}) ≡ 0 (mod π^{μ(k)}).

The integers μ(k) are related to but different from the integers d_{k} defined in Definition 2.7. Since ϕ is nuclear, it follows that

lim_{k→∞} μ(k) = ∞.

From our definitions of ϕ_{k} and ϕ_{∞,k}, one deduces that the following congruence formula holds.

Lemma 8.2. Let k be an integer and let k_{m} = k + p^{m} > 0. Then as endomorphisms on M_{∞}, we have the congruence

ϕ_{k_{m}} ≡ ϕ_{∞,k} (mod π^{min(μ(k_{m}),m+1)}).
Proof. We need to show that for all indices 2 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{r},
ϕ_{k_{m}}(f_{i_{1}} ⋯ f_{i_{r}}) ≡ ϕ_{∞,k}(f_{i_{1}} ⋯ f_{i_{r}}) (mod π^{min(μ(k_{m}),m+1)}).

If i_{r} > k_{m}, the left side is zero and the right side is divisible by π^{μ(k_{m})}. The congruence is indeed true. Assume now that i_{r} ≤ k_{m}. By (8.2) and (8.4), it suffices to check that

(8, 8)  Υ(ϕ(e_{1}^{k_{m}−r})) ≡ lim_{a} Υ(ϕ(e_{1}^{k_{a}−r})) (mod π^{m+1}),

where k_{a} is a sequence of integers with k as its p-adic limit and with ∞ as its complex limit. But (8.8) is a consequence of (8.5) and the binomial theorem. In fact, the congruence in (8.8) can be improved to

(8, 9)  Υ(ϕ(e_{1}^{k_{m}−r})) ≡ lim_{a} Υ(ϕ(e_{1}^{k_{a}−r})) (mod πp^{m})

if the ramification index ord_{π}(p) is smaller than p. The Lemma is proved.

Corollary 8.3. Let k be an integer and let k_{m} = k + p^{m}. Then, we have the limiting formula

lim_{m→∞} L(Sym^{k_{m}}ϕ, T) = lim_{m→∞} L(ϕ_{k_{m}}, T)
= L(ϕ_{∞,k}, T).

Using these results, we obtain an explicit formula for Dwork’s unit root zeta function.

Theorem 8.4. Let (M, ϕ) be a c log-convergent nuclear σ-module for some 0 < c < ∞, ordinary at slope zero with h_{0} = 1. Let (U_{0}, ϕ_{0}) be the unit root part of (M, ϕ). Write ϕ = ag ⊗ ψ, where a is a p-adic unit in R, g is a monomial (hence in A_{c}) with coefficient 1 and ψ satisfies the simpler condition in (6.2 ). Denote by ψ_{∞,k} the limiting nuclear σ-module of the sequence Sym^{k_{m}}ψ. Then for all integers k in the residue class of r modulo (q − 1), we have the following explicit formula for Dwork’s unit root zeta function:

L(ϕ_{0}^{k}, T) = ∏_{i≥1} L(a^{k}g^{r} ⊗ ψ_{∞,k−i} ⊗ ∧^{i}ψ, T)^{(−1)^{i−1}i}.

In particular, the family L(ϕ_{0}^{k}, T) of L-functions parametrized by k in any residue class modulo (q − 1) is a strong family in the open disk |T |_{π} < p^{c−ϵ} with respect to the p-adic topology of k.

It should be noted that this product is a finite product if ϕ is of finite rank, since we have ∧^{i}ψ = 0 for i greater than the rank of ϕ. Similar formula holds for higher slope unit root zeta functions. We now make this precise. Its proof follows from the proof of Theorem 7.1.

Theorem 8.5. Let (M, ϕ) be a c log-convergent nuclear σ-module for some 0 < c < ∞, ordinary up to slope j for some integer j ≥ 0. Let (U_{i}, ϕ_{i}) be the unit root σ-module coming from the slope i part of (M, ϕ) for 0 ≤ i ≤ j. Assume h_{j} = 1. Let ψ_{i} be defined as in equation (7.1 ) for 0 ≤ i ≤ j. Write ψ_{i} = a_{i}g_{i} ⊗ φ(i), where a_{i} is a p-adic unit in R, g_{i} is a monomial in A_{c} with coefficient 1 and φ(i) satisfies the simpler condition in (6.2 ). Denote by φ_{∞,k}(i) the limiting nuclear σ-module of the sequence Sym^{k_{m}}φ(i) as m goes to infinity. Then for all integers k in the residue class of r modulo (q − 1), we have the following explicit formula for Dwork’s j-th unit root zeta function:

L(ϕ_{j}^{k}, T) = ∏_{ℓ_{1},ℓ_{2}≥1} L((\frac{a_{j}}{a_{j−1}} )^{k}(\frac{g_{j}}{g_{j−1}} )^{r} ⊗ Ψ(k, ℓ_{1}, ℓ_{2}), T)^{(−1)^{(ℓ_{1}+ℓ_{2})}ℓ_{1}ℓ_{2}},

where

Ψ(k, ℓ_{1}, ℓ_{2}) = φ_{∞,−k−ℓ_{1}}(j − 1) ⊗ ∧^{ℓ_{1}}φ_{j−1} ⊗ φ_{∞,k−ℓ_{2}}(j) ⊗ ∧^{ℓ_{2}}φ_{j}.

In particular, the family L(ϕ_{j}^{k}, T) of L-functions parametrized by k in any residue class modulo (q − 1) is a strong family in the open disk |T |_{π} < p^{c−ϵ} with respect to the p-adic topology of k.

In both Theorem 8.4 and Theorem 8.5, we could have used the decomposition formula derived from Lemma 4.8. If we do so, we would get explicit formulas with fewer cancellations but they would look more complicated (less compact). Thus, we shall omit them and use the above simpler looking formulas. For future applications to higher rank case of Dwork’s conjecture, we need a slightly more general result than Theorem 8.4. Namely, we need to twist the unit root family ϕ_{0}^{k} by a fixed nuclear σ-module φ. We state this generalization here. The proof is the same as the proof of Theorem 8.4. One simply twists the whole proof by φ. The twisted basic decomposition formula is

L(ϕ^{k} ⊗ φ, T) = ∏_{i=1}^{∞} L(Sym^{k−i}ϕ ⊗ ∧^{i}ϕ ⊗ φ, T)^{(−1)^{i−1}i}.

The required uniform result is stated in Corollary 5.15.

Theorem 8.6. Let (M, ϕ) and (N, φ) be two c log-convergent nuclear σ-modules for some 0 < c < ∞. Assume that the first one ϕ is ordinary at slope zero with h_{0} = 1. Let (U_{0}, ϕ_{0}) be the unit root part of (M, ϕ). Write ϕ = ag ⊗ ψ, where a is a p-adic unit in R, g is a monomial in A_{c} with coefficient 1 and ψ satisfies the simpler condition in (6.2 ). Denote by ψ_{∞,k} the limiting nuclear σ-module of the sequence Sym^{k_{m}}ψ. Then for all integers k in the residue class of r modulo (q − 1), we have the following explicit formula for the twisted Dwork’s unit root zeta function:

L(ϕ_{0}^{k} ⊗ φ, T) = ∏_{i≥1} L(a^{k}g^{r} ⊗ ψ_{∞,k−i} ⊗ ∧^{i}ψ ⊗ φ, T)^{(−1)^{i−1}i}.

In particular, the twisted family L(ϕ_{0}^{k} ⊗ φ, T) of L-functions parametrized by k in the residue class of r modulo (q − 1) is a strong family of meromorphic functions in the disk |T |_{π} < p^{c} with respect to the p-adic topology of k.

University of California, Irvine, CA 92697

Email: dwan@math.uci.edu

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