Meromorphic continuation of L-functions of p-adic representations

By DAQING WAN*

1. Introduction Let F_{q} be the finite field of q elements of characteristic p. Let Ω_{p} be the

completion of an algebraic closure of the p-adic rational numbers Q_{p}. Let R_{p} be the ring of integers in a finite extension of Q_{p} contained in Ω_{p}. Suppose that B(X) is an r × r matrix (a_{ij}(X)) whose entries a_{ij}(X) are elements of the convergent power series ring over R_{p}:

R_{p}{{X}} = R_{p}{{X_{1}, ⋯, X_{n}}} = { ∑_{u∈N^{n}} a_{u}X^{u}|a_{u} ∈ R_{p}, lim_{|u|→∞} a_{u} = 0},

where N is the set of non-negative integers, X^{u} = X_{1}^{u_{1}} ⋯ X_{n}^{u_{n}} and |u| = u_{1} + ⋯ + u_{n}. Alternatively, the matrix B(X) can be written as a power series ∑ b_{u}X^{u} with coefficients in the (non-commutative) ring of r×r matrices M_{r×r}(R_{p}). We can then define the L-function of B(X) on the affine n-space A^{n}/F_{q} as follow: (1.1)

L(B/A^{n}, T) = ∏_{closed points x̄∈A^{n}/F_{q}} \frac{1}{det(I − T^{d(x)}B(x^{q^{d(x)−1}}) ⋯ B(x^{q})B(x))},

where x is the Teichmüller lifting of x̄ and d(x) denotes the degree of x̄ over F_{q}. This infinite product is a well defined power series with coefficients in R_{p} because there are only finitely many closed points of a given degree. If V/F_{q} is an affine variety contained in A^{n}, then one defines L(B/V, T) to be the product in (1.1) except that we now restrict x̄ to run only over the closed points of V/F_{q}. Using a standard reduction argument and Dwork’s analytic construction of additive characters, one sees that the more general L-function L(B/V, T) can be written as an L-function on the affine space A^{n} as defined in (1.1) for a larger n and a different matrix B(X). Thus we will be interested only in L(B/A^{n}, T) in this paper, except for a brief discussion of the more general case at the end of section 2.

*This work was partially supported by NSF and the UNLV Faculty Development Leave. The author wishes to thank P. Deligne for emphasizing the importance of uniform variation of L-functions, N. Katz for explaining the concrete meaning of F-crystals, and the Institute For Advanced Study for its hospitality. The author would also like to thank B. Dwork for helpful comments on an earlier version of the manuscript and for explaining to me two new shorter proofs of Theorem 1.2.

The general L-function in (1.1) was first investigated by Dwork [9-10]. If B(X) is invertible in M_{r×r}(R_{p}{{X}}⊗Q_{p}), the analytic matrix B defines (the Frobenius matrix of) an F-crystal on the affine space A^{n}/F_{q} (though not all F-crystals arise this way), see Katz [18]. The L-function in (1.1) is then the Katz L-function of the corresponding F-crystal on A^{n}. In the case that B is invertible in M_{r×r}(R_{p}{{X}}) (the so-called unit root F-crystal), the functorial equivalence between unit root F-crystals and p-adic representations of π_{1} shows that the analytic matrix B(X) defines a continuous p-adic representation of the arithmetic fundamental group:

ρ : π_{1}(A^{n}/F_{q}) ⟶ GL_{r}(R_{p}).

Equivalently, the representation ρ defines a p-adic étale sheaf on A^{n}/F_{q}. In algebraic term, the fundamental group π_{1}(A^{n}/F_{q}) is the Galois group of a separable closure of the function field F_{q}(t_{1}, ⋯, t_{n}) of A^{n}/F_{q} modulo the inertia groups at the closed points of A^{n}/F_{q}. In the case that B is invertible in M_{r×r}(R_{p}{{X}}), the L-function in (1.1) is the Artin L-function L(ρ, T) of the corresponding p-adic representation or the Grothendieck L-function of the corresponding p-adic étale sheaf. These L-functions are extremely general and include all classical zeta functions and complex L-functions of algebraic varieties over finite fields as a very special case [8][17]. They naturally arise from Dwork’s study [9-10] of p-adic analytic variation of a family of zeta functions over finite fields and Katz’s [18] more general work on F-crystals. Based on the work of Dwork, Reich [23], Monsky-Washnitzer [20-21] on such L-functions, Katz made the following conjecture (Conjecture 6.1.1 in Katz [18], see also Crew [6] and Dwork [10, p53]):

Katz Conjecture. The L-function L(B/A^{n}, T) of an F-crystal is a p-adic meromorphic function.

Essentially, the conjecture was only known to be true in the so-called overconvergent case or more general for those B(X) which decays faster than the logarithm function. Dwork’s proof in several non-trivial cases [9-11] reduces again to the overconvergent case by using a different lifting of the Frobenius map (the so-called excellent lifting or canonical lifting). Unfortunately, excellent liftings rarely exist for general F-crystals. According to Dwork-Ogus [12, p113], the first such example seems to be given by Mumford in a letter to Dwork (1972). Later, Sperber [27] showed that in the case of higher dimensional Kloosterman sums there does not exist any excellent lifting for the unit root F-crystal. Dwork-Ogus [12] and Oort-Sekiguchi [22] independently proved that under a mild assumption, a generic ordinary curve has no excellent lifting. As Dwork pointed out, excellent liftings may exist if one introduces more parameters. In fact, Dwork [10, p56] conjectured the existence of excellent liftings in this broader sense for a unit root F-crystal B(X) which arises as the unit root part of certain overconvergent F-crystals. However, very little is known about Dwork’s conjecture. Even if one assumes Dwork’s conjecture, one gets the meromorphic continuation of the L-function only for those unit root F-crystals which come from certain overconvergent F-crystals. Most unit root F-crystals do not come from any overconvergent F-crystals. According to Dwork, Adolphson (around 1975, unpublished) found an example with the property that the trace of the related Dwork operator is not well defined (more precisely, it is defined but depends on the basis chosen) and thus the related Dwork operator is not compact. This suggests some evidence that Katz’s conjecture might be false in general. However, even if an operator is not compact, it is conceivable that the Fredholm determinant (defined with respect to a fixed basis) is still meromorphic (not necessarily entire). Thus, as far as we can see, Adolphson’s work does not disprove Katz’s conjecture. Recently, Dwork-Sperber [13] proved the meromorphic continuation in a non-trivial disk for those B(X) with logarithm decay.

Our purpose of this paper is to use the Dwork trace formula to study the p-adic meromorphic continuation of the function L(B/A^{n}, T). We show that a weaker form of the Katz conjecture is true but the full form is false. To describe the results, let us write

(1.2)  B(X) = ∑_{u∈N^{n}} b_{u}X^{u}, b_{u} ∈ M_{r×r}(R_{p}).

Define

(1.3)  h(B) = lim_{|u|→∞} inf \frac{ord_{q}(b_{u})}{log_{q} |u|},

where ord_{q}(b_{u}) is the least rational number s such that q^{s} divides all entries of b_{u}. Since ord_{q}(b_{u}) is non-negative, it is clear that 0 ≤ h(B) ≤ ∞. We have the following results.

THEOREM 1.1. The L-function L(B/A^{n}, T) is a p-adic meromorphic function in the open disk ord_{q}(T) > −h(B). In particular, if h(B) = ∞, then the L-function L(B/A^{n}, T) is a p-adic meromorphic function in the entire p-adic plane.

This result is not new. It is in fact a consequence of a more general theorem of Dwork-Sperber [13] on the compliment of a hypersurface (with a mild assumption on the hypersurface), except that in [13] a step function is used instead of the much simpler logarithm function. We shall also prove a strengthened form of Theorem 1.1 for a family of L-functions, see Definition 3.7 and Theorem 5.2 for the precise formulation. Roughly speaking, if we have a family of power series B(X, y) = ∑ b_{u}(y)X^{u} parameterized by y varying in some metric space Y and if each b_{u}(y) is continuous in y such that

(1.4)  lim_{|u|→∞} inf \frac{inf_{y} ord_{q}(b_{u}(y))}{log_{q} |u|} ≥ α,

for some real number α, then the family of L-functions L(B(X, y)/A^{n}, T) is uniformly meromorphic in y on the closed disk ord_{q}(T) > −(α − ϵ) for any ϵ > 0. Furthermore, the zeroes and poles of the family L(B(X, y)/A^{n}, T) flow continuously in y on the closed disk ord_{q}(T) > −(α − ϵ) for any ϵ > 0. This type of uniform results is the basis in [28] to prove the full strength of some conjectures of Goss [15] on global L-series of Drinfeld modules. It is expected that uniform results would also play an important role in the author’s future investigation on Dwork’s unit root zeta function [10]. The importance of uniform variation was first pointed out to me by P. Deligne.

The main new result is the following counter-example which shows that Theorem 1.1 is best possible in some sense.

THEOREM 1.2. Let r = n = 1. For a positive integer h, let

B(X) = 1 + ∑_{u>0} q^{hu+1}c_{u}X^{(q^{u}−1)}, c_{u} ∈ R_{p}.

Assume that the power series ∑_{u>0} c_{u}T^{u} modulo the maximal ideal of R_{p} is not a rational function over its residue field (i.e., the coefficients c_{u} do not satisfy any linear recurrence relation in the residue field), then the L-function L(B/A, T) is meromorphic in the open disk ord_{q}(T) > −h but not meromorphic on the closed

disk ord_{q}(T) ≥ −h .

To describe the final result, recall that the degree of a rational function is the degree of its numerator minus the degree of its denominator while the total degree of a rational function is the degree of its numerator plus the degree of its denominator.

THEOREM 1.3. If B(X) = ∑_{u} b_{u}X^{u} is a polynomial in X with coefficients in M_{r×r}(R_{p}) of degree d, then the L-function L(B/A^{n}, T) is a rational function in T whose degree is the negative of the degree of det(I − T b_{0}) and whose total degree is at most

r ∑_{i=0}^{n} \binom{n}{i}\binom{n − i + [d/(q − 1)]}{n}.

Remark. In his p-adic theory, Dwork originally treated the overconvergent case, namely, those matrix B(X) such that lim_{|u|→∞} inf \frac{ord_{q}(b_{u})}{|u|} > 0. Dwork’s overconvergent result on the affine space A^{n} was generalized to more general varieties (still overconvergent matrix) by Reich [23], Monsky [21] and Boyarsky [5]. Note that the hypothesis h(B) = ∞ in Theorem 1.1 is much weaker than the overconvergent hypothesis.

If B(X) with h(B) = ∞ is not a polynomial, then L(B/A^{n}, T) is meromorphic but not rational in general. If, in addition, the Euler factors at all closed points x̄ ∈ A^{n}/F_{q} have coefficients which can be embedded in the ring of integers of a fixed number field, then the L-function L(B/A^{n}, T) can be viewed as a complex function. If this complex function has non-trivial radius of convergence, then the Borel lemma as used in Dwork’s [7] rationality proof shows that in fact L(B/A^{n}, T) is a rational function over the complex numbers. In this case, the degree and total degree of the rational function L(B/A^{n}, T) can be estimated by Bombieri’s methods [3-4]. The bound for the total degree in Theorem 1.3 is best possible in general. However, if one takes into account which terms actually occur in the polynomial expansion of B(X), then Theorem 1.3 can be improved in some cases using Adolphson-Sperber’s [1] improvement of Bombieri’s methods. The rationality result in Theorem 1.3 is natural and very useful in function field case [28]. However, in our present p-adic case, we note that the rationality result in Theorem 1.3 does not include the rationality of the classical L-function of a finite character of π_{1}(A^{n}/Fq). It would be interesting to have a more general rationality theorem which includes both the finite character case and the polynomial case treated in Theorem 1.3.

Even though Theorem 1.1 can be derived from a more general theorem of Dwork-Sperber [13], we shall include a self-contained treatment here for several reasons. First, the special affine space case treated here can be proved in a significantly simpler way. Second, this special affine space case is sufficient for applications to function fields even in the most general case [28]. Third, since we also need a stronger family version of Theorem 1.1, it is by going through the known explicit proof of Theorem 1.1 that one sees the uniform variation of L-functions. The proof of Theorem 1.2 is a little more subtle. It requires a delicate precise estimate for the coefficients of the L-function. Two shorter variations of the proof of Theorem 1.2 were kindly communicated to me by Dwork. However, we decide to use our original detailed proof.

For simplicity, as in [29] we shall directly work with (infinite) matrices instead of operators on p-adic Banach spaces. This is sufficient for meromorphic continuation, which is our main purpose here. However, to study deeper properties such as cohomological formula for L-functions (see Conjecture 6.1.2 in Katz [18]), one needs to develop a p-adic spectral theory as in Serre [26]. It is well known that Serre’s theory applies in the case h(B) = ∞. Thus it is conceivable that the Dwork-Monsky-Washnitzer overconvergent theory extends to the much broader case h(B) = ∞. For most general B, our counterexample shows that the best one can hope for is a cohomological formula which accounts for the zeros and poles of the L-function in the open disk ord_{q}(T) > −h(B).

We feel that such an optimal theory to be plausible in some sense and it would apparently have useful applications in Katz’s [19] theory of overconvergent padic modular forms, see also Gouvea [16]. A preliminary evidence is obtained in [30], where it is shown that the related subring of the power series ring is Noetherian, generalizing the result of Fulton [14].

Most results in this paper remain valid if one replaces the p-adic ring R_{p} by any complete discrete valuation ring of characteristic zero or characteristic p (the same p). This generalization is described in the last section. The characteristic p version of our results will be used in a future paper [28] to prove some conjectures of Goss on L-functions of Drinfeld modules and t-motives.

There is a striking analogue between Dwork’s p-adic theory for L-functions of F-crystals over a non-archimedean field and Ruelle’s [25] theory for dynamic zeta functions over an archimedean field. In fact, both theories essentially employ the same type of techniques, such as transfer operators, spectral theory and De Rham type cohomology. One can sometimes transport both results and proofs to each other. In view of the connection of the Ruelle zeta function with dynamic system and statistical mechanics [24], it might be natural and interesting to reformulate Dwork’s theory as a theory of p-adic dynamic system. Such reformulation of Dwork’s powerful results may have fruitful applications in statistical mechanics.

2. Reduction to torus G_{m}^{n}

For a subset S ⊆ {1, 2, ⋯, n} (possibly empty), let B_{S}(X) be the analytic matrix obtained from B(X) by setting x_{i} = 0 for all i ∉ S. The matrix B(X) can also be viewed as an analytic matrix on the torus G_{m}^{n} and we may follow (1.1) to define the L-function (2.1)

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

Define L(B_{S}/G_{m}^{|S|}, T) in a similar way for each S. One checks easily that

(2.2)  L(B/A^{n}, T) = ∏_{S⊆{1,2,···,n}} L(B_{S}/G_{m}^{|S|}, T).

Then Theorems 1.1-1.2 are reduced to the corresponding Theorems 2.1-2.2 for torus G_{m}^{n} described below. Let

(2.3)  B(X) = ∑_{u∈Z^{n}} b_{u}X^{u}, b_{u} ∈ M_{r×r}(R_{p}),

where u runs over all lattice points in Z^{n} and |u| = |u_{1}| + ⋯ + |u_{n}|. Thus, the matrix B(X) is now a Laurent series with coefficients b_{u} in M_{r×r}(R_{p}) and we are in a slightly more general situation. Define

(2.4)  h(B) = lim_{|u|→∞} inf \frac{ord_{q}(b_{u})}{log_{q} |u|}.

Since ord_{q}(b_{u}) is non-negative, it is clear that 0 ≤ h(B) ≤ ∞ and h(B_{S}) ≥ h(B) for all subsets S of {1, 2, ⋯, n}.

THEOREM 2.1. The L-function L(B/G_{m}^{n}, T) is a p-adic meromorphic function in the open disk ord_{q}(T) > −h(B).

THEOREM 2.2. Let r = n = 1. For a positive integer h, let

B(X) = 1 + ∑_{u>0} q^{hu+1}c_{u}X^{(q^{u}−1)}, c_{u} ∈ R_{p}.

Assume that the power series ∑_{u>0} c_{u}T^{u} modulo the maximal ideal of R_{p} is not a rational function over its residue field, then the L-function L(B/G_{m}, T) can not be extended to a p-adic meromorphic function on the closed disk ord_{q}(T) ≥ −h.

THEOREM 2.3. If B(X) = ∑ b_{u}X^{u} is a polynomial in X with coefficients in M_{r×r}(R_{p}) of degree d, then the L-function L(B/G_{m}^{n}, T) is a rational function in T whose degree is zero and whose total degree is at most

r2^{n}\binom{n + [d/(q − 1)]}{n}.

In the more general case case that B(X) is a Laurent polynomial in X, the L-function L(B/G_{m}^{n}, T) is still a rational function whose degree is zero and whose total degree can be estimated using the method of Adolphson-Sperber [1]. By (2.2), it is clear that Theorem 2.1 implies Theorem 1.1. For Theorem 1.2, one checks that

L(B/A, T) = det(I − T B(0))^{(−1)}L(B/G_{m}, T)

and thus L(B/G_{m}, T) is meromorphic in a given disk if and only if L(B/A, T) is meromorphic in the same disk. In the case of Theorem 1.3, for a nonempty subset S ∈ {1, 2, ⋯, n} the L-function L(B_{S}/G_{m}^{|S|}, T) has degree zero by Theorem 2.3; while for the empty set S = ϕ, the L-function L(B_{S}/G_{m}^{|S|}, T) is just the Euler factor det(I − T B(0))^{(−1)} = det(I − T b_{0})^{(−1)}. The bound for the total degree in Theorem 1.3 can not be reduced to the (larger) bound for the total degree in Theorem 2.3. It follows, instead, from a slightly more subtle trace formula (see the end of section 5 for the proof).

Remark. If B(X) is the rank one matrix corresponding to the overconvergent power series for the exponential sums associated to a polynomial of one variable, then the L-function is a rational function whose degree is not zero in general. Comparing with Theorem 2.3, one deduces the observation that Dwork’s analytic formula for additive characters can not be replaced by a polynomial.

Now, we briefly discuss the more general L-function L(B/V, T), where B(X) is the matrix given in (1.2) and V is an affine (or quasi-affine) subvariety in A^{n}/F_{q}. We emphasis that our L(B/V, T) is not the most general form. We assumed that B(X) is a global convergent power series as in (1.2). One can define L(B/V, T) for a more general Krasner analytic matrix B(X) on V, which in general can not be written as a global convergent power series in (1.2). The study of such more general L-functions needs the more general Reich-Monsky trace formula. We have essentially restricted ourself to the simplest torus and affine space cases in this paper.

THEOREM 2.4. Assume that B(X) is a convergent power series as in (1.2). Then, the L-function L(B/V, T) is a p-adic meromorphic function in the open disk

ord_{q}(T) > −h(B).

Clearly, we may assume that V is affine. Let ψ be a non-trivial additive character of F_{q}. Let V be defined by the equations g_{1}(X) = ⋯ = g_{k}(X) = 0. Dwork’s analytic formula for an additive character shows that there is an overconvergent power series ϕ(X, Z_{1}, ⋯, Z_{k}) such that

L(B/V, T) = L(B(X)ϕ(X, Z_{1}, ⋯, Z_{k})/A^{n+k}, T/q^{k}).

Applying Theorem 1.1, one gets the meromorphic continuation of L(B/V, T) only in the smaller disk ord_{q}(T) > −h(B) + k because of the shifting factor 1/q^{k}. This is weaker than the expected bound. If one uses the Dwork trace formula for the affine space A^{n} (see Lemma 4.3) which is a little bit more subtle than the simplest torus case, then it can be proved with a little more work that L(B/V, T) is actually meromorphic in the expected disk ord_{q}(T) > −h(B).

3. Trace and Fredholm Determinant

The trace and determinant of an infinite matrix can be defined in usual way as long as everything is convergent. To be self-contained and rigorous, we recall precisely what we mean by trace and Fredholm determinant.

Definition 3.1. Let M = (a_{u,v}) be an infinite matrix with entries in the quotient field of R_{p}, where the row number u and the column number v run over an ordered countable set which we identify as the set of positive integers.

(i). If lim_{u→∞} a_{u,u} = 0, we define the trace of M to be the sum

(3.1)  tr(M) = ∑_{u=1}^{∞} a_{u,u}.

(ii). If for each integer k ≥ 1, the following sum is convergent,

(3.2)  α_{k} = (−1)^{k} ∑_{u,σ} sign(σ)a_{u_{1},σ(u_{1})} ⋯ a_{u_{k},σ(u_{k})},

where the summation is over all u_{i}’s with 1 ≤ u_{1} < ⋯ < u_{k} and all permutations σ of the u_{i}’s, then we define the Fredholm determinant of M to be the power series

(3.3)  det(I − T M) = 1 + ∑_{k=1}^{∞} α_{k}T^{k}.

Note that for a fixed k, the sum α_{k} is convergent if and only if

(3.4)  lim_{u_{k}→∞} a_{u_{1},σ(u_{1})} ⋯ a_{u_{k},σ(u_{k})} = 0

uniformly for all 1 ≤ u_{1} < u_{2} < ⋯ < u_{k} and all permutations σ of {u_{1}, …, u_{k}}. Since we are in non-archimedean situation, convergence of an infinite series is independent of the order of summation.

Let Y = (Y_{u,v}) (1 ≤ u, v < +∞) be an infinity matrix all of whose coefficients are indeterminates. For each positive integer k, one can define the power Y^{k} of Y with the usual multiplication rule. The coefficients of Y^{k} are formal power series in the variables Y_{u,v}. Define tr(Y) and α_{k}(Y) as in Definition 3.1 with a_{u,v} replaced by Y_{u,v}. Then tr(Y) and α_{k}(Y) are also formal power series in the variables Y_{u,v}. It is clear that tr(M) (resp. α_{k}(M)) is defined if and only if tr(Y) (resp. α_{k}(Y) ) is convergent at the specialization Y_{u,v} = a_{u,v}.

We first prove a well known simple lemma, which is essentially a universal version of the classical Newton formula.

LEMMA 3.2. Let Y = (Y_{u,v}). Then we have the following identity of formal power series

(3.5)  det(I − T Y) = exp(− ∑_{k=1}^{∞} \frac{tr(Y^{k})}{k}T^{k}).

Proof. It is well-known that (3.5) is true for all finite matrices. Let Y (m) be the specialization of Y such that Y_{u,v} = 0 for all u and v satisfying max(u, v) > m. Then (3.5) holds for all Y (m). Let β_{k}(Y) be the coefficient of T^{k} in the expansion of the right side of (3.5) as a formal power series. We want to show that

(3.6)  β_{k}(Y) = α_{k}(Y) (k = 1, 2, …).

The terms of α_{k}(Y) and β_{k}(Y) are monomials of degree k in the Y_{u,v}’s. Since (3.6) holds for all specializations Y = Y (m), it follows that (3.6) holds in general.

COROLLARY 3.3. Let S_{k} be tr(Y^{k}) and α_{k}(Y) be defined as above. Then for all positive integers m, the expression α_{m}(Y) is a polynomial of the S_{i}’s (1 ≤ i ≤ m) with rational coefficients. Similarly, for all positive integers m, the expression S_{m} is a polynomial of the α_{i}(Y) (1 ≤ i ≤ m) with rational coefficients. More precisely, we have

(3.7) α_{m} = ∑_{i_{1}+2i_{2}+···+mi_{m}=m}(−1)^{i_{1}+···+i_{m}} \frac{S_{1}^{i_{1}} ⋯ S_{m}^{i_{m}}}{i_{1}! ⋯ i_{m}!1^{i_{1}}2^{i_{2}} ⋯ m^{i_{m}}} ,

(3.8) \frac{S_{m}}{m} = ∑_{i_{1}+2i_{2}+···+mi_{m}=m}(−1)^{i_{1}+···+i_{m}} \binom{i_{1} + ⋯ + i_{m}}{i_{1}, …, i_{m}} \frac{α_{1}^{i_{1}} ⋯ α_{m}^{i_{m}}}{i_{1} + ⋯ + i_{m}} .

Proof. By (3.5), we have

(3.9)  1 + ∑_{k=1}^{∞} α_{k}T^{k} = ∑_{m=0}^{∞} \frac{(−1)^{m}}{m!} (∑_{k=1}^{∞} \frac{S_{k}}{k} T^{k})^{m}.

The first formula follows from the power series expansion of (3.9). To prove (3.8), we take the logarithm on both sides of (3.5) and deduce that

∑_{k=1}^{∞} \frac{S_{k}}{k} T^{k} = (−1) log(1 + ∑_{m=1}^{∞} α_{m}T^{m})

= ∑_{d=1}^{∞} \frac{(−1)^{d}}{d} (∑_{m=1}^{∞} α_{m}t^{m})^{d}. (3.10)

The second formula follows from the power series expansion of (3.10).

COROLLARY 3.4. Let M = (a_{u,v}) be an infinite matrix with entries in the quotient field of R_{p}. Then det(I − T M) is defined if and only if tr(M^{k}) is defined for all positive integers k. In which case, we have

(3.11)  det(I − T M) = exp(− ∑_{k=1}^{∞} \frac{tr(M^{k})}{k}T^{k}).

Proof. det(I − T M) is defined if and only if α_{k}(M) is defined for all k ≥ 0. By Corollary 3.3, this is true if and only if all S_{k}(M) = tr(M^{k}) are defined. Equation (3.11) is then a consequence of (3.5). The corollary is proved.

We now study the meromorphic continuation of the Fredholm determinant.

Definition 3.5. Let h be a fixed real number. We define M_{h} to be the set of infinite matrices M = (a_{u,v}) such that the entries of M are bounded in the quotient field of R_{p} and almost all rows of M are divisible by q^{h−ϵ} for every ϵ > 0. More precisely, the matrix M is in M_{h} if ord_{q}(M) = inf_{u,v} ord_{q}(a_{u,v}) is bounded from below and for any given ϵ > 0, there is an integer N_{ϵ} > 0 such that for all u > N_{ϵ} and all v, the following inequality holds:

(3.12)  ord_{q}(a_{u,v}) ≥ h − ϵ.

The next result gives an estimate for the radius of convergence of the Fredholm determinant det(I − T M).

PROPOSITION 3.6. Let M = (a_{u,v}) ∈ M_{h}. Assume that the Fredholm determinant det(I − T M) is well defined. Then det(I − T M) is p-adic analytic in the open disk ord_{q}(T) > −h.

Proof. For any given ϵ > 0, there is an integer N_{ϵ} > 0 such that (3.12) holds. By definition,

det(I − T M) = 1 + ∑_{m=1}^{∞} α_{m}T^{m},

where

α_{m} = (−1)^{m} ∑_{u,σ} sign(σ)a_{u_{1},σ(u_{1})} ⋯ a_{u_{m},σ(u_{m})}.

Since the u_{i} are distinct, it follows that for m > N_{ϵ},

ord_{q}(α_{m}) ≥ inf_{u,σ}{ord_{q}(a_{u_{1},σ(u_{1})}) + ⋯ + ord_{q}(a_{u_{m},σ(u_{m})})}
≥ N_{ϵ}ord_{q}(M) + (m − N_{ϵ})(h − ϵ).

This inequality implies that

lim_{m→∞} inf \frac{ord_{q}(α_{m})}{m} ≥ h − ϵ.

We have proved that det(I −T M) is p-adic analytic in the open disk ord_{q}(T) > −h + ϵ. Since ϵ can be taken to be arbitrarily small, the proposition is proved.

Next, we consider the variation of a family of Fredholm determinants parameterized by a parameter y in some metric space Y, not necessarily compact. In all applications we have in mind, the space Y will be a subset of the ring of p-adic integers Z_{p} with its p-adic topology. Thus, we shall assume that Y is a subset of Z_{p} with its induced p-adic topology.

Definition 3.7. (i). Let h be a fixed real number. A family of infinite matrices M(y) = (a_{u,v}(y)) is called uniformly in M_{h} if the entries of M(y) are uniformly bounded in the quotient field of R_{p} and almost all rows of M(y) are uniformly divisible by q^{h−ϵ} for every ϵ > 0. More precisely, the family M(y) is uniformly in M_{h} if ord_{q}(M(y)) = inf_{u,v,y} ord_{q}(a_{u,v}(y)) is bounded from below and for any given ϵ > 0, there is an integer N_{ϵ} > 0 such that for all u > N_{ϵ}, all v and all y, the following inequality holds:

(3.13)  ord_{q}(a_{u,v}(y)) ≥ h − ϵ.

(ii). Let f(y, T) = ∑_{m≥0} f_{m}(y)T^{m} be a family of power series, where each f_{m}(y) is a function from Y to R_{p}. The family f(y, T) is called continuous if each coefficient f_{m}(y) is a continuous function. For a real number α, the family of functions f(y, T) is called uniformly analytic (or entire) on the closed disk ord_{q}(T) ≥ α if

(3.14)  lim_{m→∞} inf(inf_{y} ord_{q}(f_{m}(y)) + αm) = ∞.

In terms of the Gauss norm ||f(y, T)|| = sup_{m} |f_{m}(y)|_{q}, this means that the maximum value ||f(y, q^{α}T)|| of f(y, T) on the closed disk ord_{q}(T) ≥ α is uniformly bounded in y, where |?|_{q} is the absolute value on R_{p} normalized such that |1/q|_{q} = q. The family f(y, T) is called uniformly meromorphic on the closed disk ord_{q}(T) ≥ α if f(y, T) can be written as a quotient f_{1}(y, T)/f_{2}(y, T), where both f_{1}(y, T) and f_{2}(y, T) are uniformly analytic on the closed disk ord_{q}(T) ≥ α. It is clear that the product and quotient of two uniformly meromorphic families (parameterized by the same parameter y) are still uniformly meromorphic.

(iii). Let f(y, T) be parameterized by some metric space Y and let α be a real number. Following a terminology used by Goss, we say that the zeroes of the family f(y, T) flow continuously in y on the closed disk ord_{q}(T) ≥ α if the following two conditions hold. (1). For each y, the function f(y, T) is analytic on the closed disk ord_{q}(T) ≥ α. (2). For any ϵ > 0, there is a real number δ > 0 such that whenever |y_{1} − y_{2}|_{q} < δ, we have the inequality

||f(y_{1}, q^{α}T) − f(y_{2}, q^{α}T)|| < ϵ.

In terms of the Gauss norm, this means that ||f(y, q^{α}T)|| is uniformly continuous in y. These two conditions implies that for any given ϵ > 0, there is a real number δ > 0, such that whenever |y_{1} − y_{2}|_{q} < δ, the two functions f(y_{1}, T) and f(y_{2}, T) have the same number s of zeroes on the closed disk ord_{q}(T) ≥ α, which can be arranged as {ρ_{1}(i), ⋯, ρ_{s}(i)} for 1 ≤ i ≤ 2 such that |ρ_{j}(1) − ρ_{j}(2)|_{q} < ϵ for all 1 ≤ j ≤ s.

We say that the zeroes and poles of the family f(y, T) flow continuously in y on the closed disk ord_{q}(T) ≥ α if f(y, T) can be written as a quotient f_{1}(y, T)/f_{2}(y, T), where the zeroes of f_{1}(y, T) and the zeroes of f_{2}(y, T) flow continuously in y on the closed disk ord_{q}(T) ≥ α.

The following result is immediate.

LEMMA 3.8. Let f(y, T) be a continuous family of power series. If the family f(y, T) is uniformly analytic on the closed disk ord_{q}(T) ≥ α, then the zeroes of f(y, T) flow continuously in y on the closed disk ord_{q}(T) ≥ α. The converse is

also true if Y is compact.

The proof of Proposition 3.6 yields the following version of Proposition 3.6 for a family.

PROPOSITION 3.9. Let M(y) = (a_{u,v}(y)) be a family of infinite matrices which are uniformly in M_{h}. Assume that the Fredholm determinant det(I −T M(y)) is well defined for each y. Then for any ϵ > 0, the family det(I −T M(y)) is uniformly analytic on the closed disk ord_{q}(T) > −(h − ϵ).

4. Dwork trace formula

The L-function L(B, T) can be defined in two ways. One is via the Euler product expansion as in (2.1). The other is to use character sums via the exponential function. The second way can be obtained from (2.1) by taking the logarithmic derivative (or just using (3.5)):

L(B/G_{m}^{n}, T) = exp(∑_{k=1}^{∞} \frac{T^{k}}{k} S_{k}(B)),

where S_{k}(B) is the character sum

(4.1)  S_{k}(B) = ∑_{x^{(q^{k}−1)}=1, x∈Ω_{p}^{n}} tr(B(x^{q^{k−1}}) ⋯ B(x^{q})B(x)),

the notation tr denotes the trace of a matrix and x^{q} = (x_{1}^{q}, ⋯, x_{n}^{q}).

Similarly, the Dwork trace formula has two forms. One is via the trace of an infinite matrix. The other is via the Fredholm determinant of the same infinite matrix. This infinite matrix is simply F_{B} = (b_{qu−v})_{u,v∈Z^{n}}, called the infinite Frobenius matrix associated with B(X). Thus, the (u, v) block entry of F_{B} is the r × r matrix b_{qu−v} ∈ M_{r×r}(R_{p}) from (2.3). Since lim_{|u|→∞} b_{u} = 0, one checks immediately that the power F_{B}^{k} is well defined for each positive integer k. The Dwork trace formula says that the trace of F_{B}^{k} is also defined and related to the character sum S_{k}(B).

LEMMA 4.1 (additive form). For each integer k ≥ 1, we have

(4.2)  S_{k}(B) = (q^{k} − 1)^{n}tr(F_{B}^{k}).

Proof. For k = 1, by definition,

S_{1}(B) = ∑_{x^{(q−1)}=1} tr( ∑_{u∈Z^{n}} b_{u}x^{u})

= ∑_{u∈Z^{n}} tr(b_{u}) ∑_{x^{(q−1)}=1} x^{u}

= (q − 1)^{n} ∑_{u∈Z^{n}} tr(b_{(q−1)u}) (4.3)

= (q − 1)^{n}tr(F_{B}).

In deriving (4.3), we used the relation

(4.4)  ∑_{x_{i}^{(q−1)}=1} x_{i}^{u_{i}} = \{ \begin{matrix}q − 1, & if (q − 1) divides u_{i} \\ 0, & otherwise \end{matrix} .

For k > 1, the proof is similar. Again, by definition,

S_{k}(B) = ∑_{x^{(q^{k}−1)}=1} tr(B(x^{q^{k−1}}) ⋯ B(x^{q})B(x))

= ∑_{u^{(k−1)},···,u^{(0)}∈Z^{n}} tr(b_{u^{(k−1)}} ⋯ b_{u^{(0)}}) ∑_{x^{(q^{k}−1)}=1} x^{q^{k−1}u^{(k−1)}+···+qu^{(1)}+u^{(0)}}

= (q^{k} − 1)^{n} ∑_{q^{k−1}u^{(k−1)}+···+qu^{(1)}+u^{(0)}≡0 (mod (q^{k}−1))} tr(b_{u^{(k−1)}} ⋯ b_{u^{(0)}}), (4.5)

where the i in u^{(i)} is a superscript. Note that all sums above are convergent. To show that (4.5) is actually equal to the number (q^{k} − 1)^{n}tr(F_{B}^{k}), one simply writes all lattice point solutions of the equation

(4.6)  q^{k−1}u^{(k−1)} + ⋯ + qu^{(1)} + u^{(0)} = (q^{k} − 1)v^{(k−1)}

in the form

(4.7)  u^{(k−1)} = qv^{(k−1)} − v^{(k−2)}, ; u^{(k−2)} = qv^{(k−2)} − v^{(k−3)}, ; ⋮ ; u^{(0)} = qv^{(0)} − v^{(k−1)}.

Then

(4.8) \begin{aligned}S_{k}(B) & = (q^{k} − 1)^{n} ∑_{v^{(k−1)},···,v^{(0)}∈Z^{n}} tr(b_{qv^{(k−1)}−v^{(k−2)}} ⋯ b_{qv^{(0)}−v^{(k−1)}}) \\ & = (q^{k} − 1)^{n}tr(F_{B}^{k}) \end{aligned}

is well defined and the lemma is proved.

Remark. If B(X) = ∑_{u∈N^{n}} b_{u}X^{u}, where u runs only over the set of nonnegative integers, then we can simply take the infinite Frobenius matrix F_{B} to be (b_{qu−v})_{u,v∈N^{n}}. The point is that in this case, for u^{(k−1)}, ⋯, u^{(0)}, v^{(k−1)} ∈ N^{n}, equations (4.6) and (4.7) guarantee that all v^{(i)} ∈ N^{n}. Alternatively, if one of the lattice points v^{(i)} is not in N^{n}, then the corresponding term in (4.8) is easily seen to be zero (simply looking at the most “negative” point v^{(i)}).

The above lemma and Corollary 3.4 show that the Fredholm determinant of F_{B} is well defined. Expanding (q^{k} −1)^{n} in (4.2) and using the identity (3.5), one gets the following multiplicative form of the Dwork trace formula.

LEMMA 4.2 (multiplicative form). We have

(4.9)  L(B/G_{m}^{n}, T)^{(−1)^{n−1}} = ∏_{i=0}^{n} det(I − q^{i}T F_{B})^{(−1)^{i}\binom{n}{i}}.

Equivalently, we have

(4.10)  det(I − T F_{B}) = ∏_{i=0}^{∞}(L(B/G_{m}^{n}, q^{i}T)^{(−1)^{n−1}})^{\binom{n+i−1}{i}}.

Note that (4.10) follows from (4.2) by expanding (q^{k} − 1)^{−n}. These formulas show that the L-function is completely determined by the Fredholm determinant det(I − T F_{B}) of the infinite Frobenius matrix F_{B}. If the matrix F_{B} defines in some sense a compact operator (completely continuous in the sense of Serre [26]) of a certain p-adic Banach space, then the Fredholm determinant is p-adic entire and Lemma 4.2 gives the meromorphic continuation of the L-function to the entire p-adic plane. However, the matrix F_{B} does not in general define a compact operator. In the case when h(B) = ∞, it is well known that F_{B} does define a compact operator.

The Dwork trace formula for the affine space A^{n} can be derived from its torus version in (4.9) via a combinatorial argument using the torus decomposition (2.2). Since the affine space version will be used in section 7, we now carry out the deduction.

Let B(X) = ∑_{u∈N^{n}} b_{u}X^{u} be a power series with coefficients in M_{r×r}(R_{p}). The infinite Frobenius matrix is then given by (b_{qu−v})_{u,v∈N^{n}}. For a subset U of {1, ⋯, n}, define F_{B}^{U} to be the submatrix (b_{qu−v}) with u, v ∈ N^{n} such that u_{i} ≥ 1 and v_{i} ≥ 1 for all i ∈ U. Define F_{B}(U) to be the submatrix (b_{qu−v}) of F_{B}^{U} satisfying the further condition that u_{i} = v_{i} = 0 for all i ∉ U. If we order the m = 2^{n} subsets of {1, ⋯, n} as U_{1}, ⋯, U_{m} in such a way that |U_{j}| ≤ |U_{j+1}|, then it is easy to check that the infinite matrix F_{B} has the following triangular form

F_{B} = ( \begin{matrix}F_{B}(U_{1}) & ∗ & ⋯ & ∗ \\ 0 & F_{B}(U_{2}) & ⋯ & ∗ \\ ⋮ & ⋯ & ⋮ & ⋮ \\ 0 & 0 & ⋯ & F_{B}(U_{m}) \end{matrix} ) .

Thus,

(4.11)  det(I − T F_{B}) = ∏_{U∈{1,···,n}} det(I − T F_{B}(U)).

This decomposition is a special case of the trivial boundary decomposition from Theorem 5.1 in [29]. More general, for a subset V ∈ {1, ⋯, n}, we have

(4.12)  det(I − T F_{B}^{V}) = ∏_{V ⊆U} det(I − T F_{B}(U)).

If V is the empty set, then (4.12) reduces to (4.11).

LEMMA 4.3 (trace formula for the affine space, see (8.4) in [29]). We have

(4.13)  L(B/A^{n}, T)^{(−1)^{n−1}} = ∏_{i=0}^{n} ∏_{|U|=i} det(I − q^{n−i}T F_{B}^{U})^{(−1)^{n−i}},

where U denotes a subset of {1, ⋯, n}.

Proof. By the torus decomposition (2.2) and Lemma 4.2, one derives that

\begin{aligned}L(B/A^{n}, T)^{(−1)^{n−1}} & = ∏_{S⊆\{1,2,···,n\}} L(B_{S}/G_{m}^{|S|}, T)^{(−1)^{n−1}} \\ & = ∏_{S} ∏_{V ⊆S} det(I − q^{|V |}T F_{B_{S}})^{(−1)^{n+|S|+|V |}}. \end{aligned}

Applying (4.11) with F_{B} replaced by F_{B_{S}}, we compute that

\begin{aligned}L(B/A^{n}, T)^{(−1)^{n−1}} & = ∏_{S} ∏_{V ⊆S} ∏_{U⊆S} det(I − q^{|V |}T F_{B}(U))^{(−1)^{n+|S|+|V |}} \\ & = ∏_{V} ∏_{U} det(I − q^{|V |}T F_{B}(U))^{(−1)^{n+|V |} ∑_{U,V ⊆S}(−1)^{|S|}}. \end{aligned}

The above sum of signs over S is (−1)^{n} or zero depending on whether U contains the compliment V̄ of V or not. Applying (4.12), we then conclude that

L(B/A^{n}, T)^{(−1)^{n−1}} = ∏_{V} det(I − q^{|V |}T F V̄ ; B)^{(−1)^{|V |}}.

This proves (4.13).

5. Meromorphic continuation and rationality

We prove Theorem 2.1, its family version and Theorem 2.3 in this section. We first prove Theorem 2.1. By Lemma 4.2, it suffices to prove the meromorphic continuation of det(I − T F_{B}) in the open disk ord_{q}(T) > −h(B). Recall that

h = h(B) = lim_{|u|→∞} inf \frac{ord_{q}(b_{u})}{log_{q} |u|}, F_{B} = (b_{qu−v})_{u,v∈Z^{n}}.

We can assume that h is finite. For a given ϵ > 0, there is an integer N_{ϵ} > 0 such that for all |u| > N_{ϵ},

(5.1)  ord_{q}(b_{u}) ≥ (h − \frac{ϵ}{2}) log_{q} |u|.

Define a non-negative weight function w(u) by

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

Let a_{u,v} be the twisted matrix q^{w(v)−w(u)}b_{qu−v} in M_{r×r}(R_{p}). Let F_{B}^{∗} be the twisted infinite Frobenius matrix (a_{u,v})_{u,v∈Z^{n}}. Clearly, the power (F_{B}^{∗})^{k} is well defined and has the same trace as F_{B}^{k} for each positive integer k. Thus det(I − T F_{B}^{∗}) is well defined and equal to det(I − T F_{B}). To finish the proof, by Proposition 3.6 we need to prove that F_{B}^{∗} ∈ M_{h−ϵ} for any ϵ > 0.

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

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

If |u| ≤ N_{ϵ}^{∗}, we have the trivial inequality ord_{q}(a_{u,v}) ≥ −w(u) ≥ −w(N_{ϵ}^{∗}). To prove the assertion that F_{B}^{∗} ∈ M_{h}, it suffices to prove the claim that ord_{q}(a_{u,v}) ≥ h − ϵ for all |u| > N_{ϵ}^{∗} and all v. Assume |u| > N_{ϵ}^{∗}. If v = 0 or qu, one checks that ord_{q}(a_{u,v}) ≥ w(qu) − w(u) = (h − ϵ/2) > h − 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 (5.1) and (5.2) we deduce that

ord_{q}(a_{u,v}) ≥ (h − \frac{ϵ}{2}) log_{q} \frac{|v|}{|u|}

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

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

≥ h − 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_{q}(a_{u,v}) ≥ (h − \frac{ϵ}{2}) log_{q} \frac{|v||qu − v|}{|u|}

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

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

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

≥ h − ϵ.

The claim is true and the proof of Theorem 2.1 is complete.

The proof shows that the meromorphic continuation is uniform if we have a uniform family B(X, y). We make this precise.

Definition 5.1. Let B(X, y) be a family of power series parameterized by y in some metric space Y:

B(X, y) = ∑_{u∈N^{n}} b_{u}(y)X^{u},

where b_{u}(y) is in M_{r×r}(R_{p}) for each y and b_{u}(y) goes to zero uniformly in u. Let α be a real number. The family B(X, y) is called uniformly α log-convergent if we have

lim_{|u|→∞} inf \frac{inf_{y} ord_{q}(b_{u}(y))}{log_{q} |u|} ≥ α.

The family B(X, y) is called continuous in y if each coefficient b_{u}(y) is a continuous map. This continuity is automatically uniform in u since b_{u}(y) goes to zero uniformly in y as |u| goes to infinity.

We have the following strengthened form of Theorems 1.1 and 2.1.

THEOREM 5.2. Let B(X, y) be a family of power series parameterized by y in some metric space Y. Assume that the family B(X, y) is continuous in y and uniformly α log-convergent. Then for any ϵ > 0, the zeroes and poles of the family of L-functions L(B(X, y)/G_{m}^{n}, T) (resp. L(B(X, y)/A^{n}, T)) flow continuously on the closed disk ord_{q}(T) ≥ −(α − ϵ).

Proof. We prove only the assertion for the L-function L(B(X, y)/G_{m}^{n}, T) on the torus as the proof for L(B(X, y)/A^{n}, T) is completely similar. Since B(X, y) is uniformly α log-convergent, the above proof of Theorem 2.1 shows that the twisted Frobenius matrix F_{B}^{∗} = (q^{w(v)−w(u)}b_{qu−v}(y)) is uniformly in M_{α−ϵ/2} for any ϵ > 0. By Proposition 3.9, the family of Fredholm determinants det(I − T F_{B(X,y)}) is uniformly analytic on the closed disk ord_{q}(T) > −(α − ϵ) for any ϵ > 0. Since B(X, y) is continuous, the Euler product definition of L-functions shows that the family of L-functions L(B(X, y)/G_{m}^{n}, T) is continuous in y. The trace formula (4.10) shows that the family of Fredholm determinants det(I − T F_{B(X,y)}) is also continuous in y. We have proved that det(I − T F_{B(X,y)}) is a continuous family of uniformly analytic functions on the closed disk ord_{q}(T) > −(α − ϵ). By Lemma 3.8, the zeroes of the family det(I − T F_{B(X,y)}) flow continuously on the closed disk ord_{q}(T) > −(α − ϵ). The theorem then follows by applying the trace formula (4.9). The proof is complete.

Now, we turn to proving Theorem 2.3 and Theorem 1.3. Write

B(X) = ∑_{|u|≤d,u∈N^{n}} b_{u}X^{u}, b_{u} ∈ M_{r×r}(R_{p}).

The infinite Frobenius matrix F_{B} is (b_{qu−v})_{u,v∈N^{n}}. The block entry b_{qu−v} is easily seen to be zero if |qu − v| > d since B(X) is a polynomial of degree d. We order the set N^{n} in such a way that |u| > |v| implies that u > v. Then the first \binom{n+k}{k} elements in N^{n} are just the lattice points u = (u_{1}, ⋯, u_{n}) ∈ N^{n} with u_{1} + ⋯ + u_{n} ≤ k. With this ordering, for u ≥ v and |u| > [d/(q − 1)], we have |qu − v| ≥ (q − 1)|u| > d and thus b_{qu−v} = 0. We have proved that the infinite matrix F_{B} has the block form

(5.4)  F_{B} = ( \begin{matrix}F_{1} & ∗ \\ 0 & F_{2} \end{matrix} ) ,

where F_{1} is the finite matrix (b_{qu−v}) with u and v run over the lattice points in N^{n} such that both |u| ≤ [d/(q − 1)] and |v| ≤ [d/(q − 1)], and F_{2} is the infinite matrix (b_{qu−v}) with u and v run over the lattice points in N^{n} such that both |u| > [d/(q−1)] and |v| > [d/(q−1)]. Furthermore, any block entry on or below the main diagonal of F_{2} is zero. This implies that det(I −T F_{B}) = det(I −T F_{1}) is a polynomial of degree at most

(5.5)  r\binom{n + [d/(q − 1)]}{[d/(q − 1)]} = r\binom{n + [d/(q − 1)]}{n}.

This and Lemma 4.2 together give Theorem 2.3.

If U is a subset of {1, ⋯, n}, recall that the F_{B}^{U} is the submatrix (b_{qu−v}) such that u_{i} ≥ 1 and v_{i} ≥ 1 for all i ∈ U. One computes that det(I − T F_{B}^{U}) is a polynomial of degree at most

(5.6)  r\binom{n + [d/(q − 1)] − |U|}{[d/(q − 1)] − |U|} = r\binom{n − |U| + [d/(q − 1)]}{n}.

This and Lemma 4.3 together give the total degree bound in Theorem 1.3.

Example. Let k be a positive integer and let B(X) = ∑_{|u|≤k(q−1)} b_{u}X^{u} be a polynomial in X = (X_{1}, ⋯, X_{n}) of degree d = k(q − 1) with coefficients b_{u} in M_{r×r}(R_{p}). Assume that k ≤ q − 2 and b_{u} = 0 if u is not divisible by (q − 1). Then for u ≠ v and |u|, |v| ≤ k, one has b_{qu−v} = 0 since qu − v ≡ u − v ≢ 0 (mod (q −1)). Thus, the upper left matrix F_{1} in (5.4) is of the diagonal form whose block matrices on the main diagonal are the r × r matrices b_{(q−1)u} with |u| ≤ k. It follows that

(5.7)  det(I − T F_{B}) = det(I − T F_{1}) = ∏_{|u|≤k} det(I − T b_{(q−1)u}).

If each b_{(q−1)u} in (5.7) is invertible, then det(I −T F_{B}) is a polynomial of degree exactly

r\binom{n + k}{n} = r\binom{n + [d/(q − 1)]}{n}

and for each subset U of {1, ⋯, n}, det(I − T F_{B}^{U}) is a polynomial of degree exactly

r\binom{n − |U| + k}{n} = r\binom{n − |U| + [d/(q − 1)]}{n}.

If each b_{(q−1)u} is invertible modulo p, then there cannot have any cancellation in (4.9) and (4.13). Thus, the bounds for the total degree of L-functions in Theorems 1.3 and 2.3 are sharp in this case.

6. Counterexamples

We now begin to prove Theorem 2.2. Let r = n = 1. In this case, a general B(X) is a p-adic analytic function of one variable. Write

B(X) = ∑_{k=0}^{∞} b_{k}X^{k}, b_{0} = 1, b_{k} ∈ R_{p}, lim_{k→∞} b_{k} = 0.

To construct best possible examples for the meromorphic continuation theorem, we further suppose that

(6.1)  b_{k} = 0 unless k is of the form q^{i} − 1.

Let F_{B} be the infinite Frobenius matrix associated with B, namely, F_{B} = (b_{qu−v}), where u and v run over the non-negative integers. Let F′_{B} be the submatrix of F_{B} consisting of those entries b_{qu−v} with u ≥ 1 and v ≥ 1. Then, the Frobenius matrix F_{B} has the following block form:

(6.2)  F_{B} = ( \begin{matrix}1 & ∗ \\ 0 & F′_{B} \end{matrix} ) .

Now,

(6.3)  det (I − T F_{B}) = (1 − T)det (I − T F′_{B}) = (1 − T) ∑_{k=0}^{∞} α_{k}T^{k},

where the α_{k} are the coefficients in the expansion of det (I − T F′_{B}). More precisely, α_{0} = 1 and for k ≥ 1,

(6.4)  α_{k} = (−1)^{k} ∑_{1≤u_{1}<···<u_{k}} ∑_{σ} sign(σ)b_{qu_{1}−σ(u_{1})} ⋯ b_{qu_{k}−σ(u_{k})},

where σ runs over all permutations of {u_{1}, ⋯, u_{k}}. To evaluate ord_{q}(α_{k}), we make use of the special form given in (6.1).

Let k be a positive integer. Let u_{1} < ⋯ < u_{k} be positive integers. For a permutation σ of the k-tuple {u_{1}, ⋯, u_{k}}, the corresponding term in (6.4) is non-zero only if

qu_{1} − σ (u_{1}) = q^{a(1)} − 1,
qu_{2} − σ (u_{2}) = q^{a(2)} − 1,

⋯ (6.5)

qu_{k} − σ (u_{k}) = q^{a(k)} − 1,

where the a(i) are non-negative integers. Fix such a term in (6.4). First, we suppose that σ is a k-cycle. By renumbering if necessary, we may suppose u_{i+1} = σ (u_{i}) for all i, where the indices are taken modulo k. The resulting u_{i} are then distinct positive integers, not necessarily in increasing order. Let

(6.6)  qu_{i} − σ (u_{i}) = qu_{i} − u_{i+1} = q^{a(i)} − 1.

Solving (6.6), one gets

u_{1} = \frac{1}{q^{k} − 1} (q^{k−1+a(1)} + q^{k−2+a(2)} + ⋯ + q^{a(k)} − q^{k−1} − q^{k−2} − ⋯ − 1) ,

⋯ (6.7)

u_{i} = \frac{1}{q^{k} − 1} (q^{k−1+a(i)} + q^{k−2+a(i+1)} + ⋯ + q^{a(i−1)} − q^{k−1} − q^{k−2} − ⋯ − 1) ,

⋯

with the restriction that the u_{i} are distinct positive integers. To estimate ord_{q}(α_{k}), the key step is to prove

LEMMA 6.1. Let a(i) be non-negative integers. Suppose that the u_{i} defined by (6.7) are distinct positive integers. Then

∑_{i=1}^{k} a(i) ≥ k.

Proof. Without loss of generality, we may assume that 0 ≤ a(i) < k for all 1 ≤ i ≤ k. If a(i) ≥ i for some i, we may write

q^{k−i+a(i)} = (q^{k} − 1) q^{a(i)−i} + q^{a(i)−i}.

Then,

(6.8)  u_{1} = ∑_{a(i)≥i} q^{a(i)−i} + \frac{1}{q^{k} − 1} (∑_{i=1}^{k} q^{b(i)} − q^{k−1} − q^{k−2} − ⋯ − 1) ,

where

b(i) = \{ \begin{matrix}a(i) − i, & if a(i) ≥ i \\ k − i + a(i), & if a(i) < i. \end{matrix}

Since a(i) < k for all i, we have

0 ≤ b(i) < k.

To prove Lemma 6.1, we first prove

LEMMA 6.2. Let b(i) be integers satisfying 0 ≤ b(i) < k for all 1 ≤ i ≤ k. If

(q^{k} − 1) |(∑_{i=1}^{k} q^{b(i)} − q^{k−1} − q^{k−2} − ⋯ − 1) ,

then the sequence b(i) is a permutation of the integers {0, 1, 2, ⋯, k − 1}.

Proof. For 0 ≤ j ≤ k − 1, let N(j) be the number of indices i such that b(i) = j. Then

∑_{i=1}^{k} q^{b(i)} = ∑_{j=0}^{k−1} N(j)q^{j}, ∑_{j=0}^{k−1} N(j) = k.

We first assume that N(j) ≥ q for at least one j. Carrying over all the q-adic digits whenever possible, we can then write

∑_{j=0}^{k−1} N(j)q^{j} ≡ ∑_{j=0}^{k−1} M(j)q^{j} (mod (q^{k} − 1)) ,

where 0 ≤ M(j) ≤ q − 1 and

(6.9)  0 < ∑_{j=0}^{k−1} M(j) < ∑_{j=0}^{k−1} N(j) = k.

The second inequality in (6.9) is true because we carried over the digits at least once. The first inequality in (6.9) is true because all we did was to carry over digits and we did not reduce the number (q − 1)q^{k−1} + ⋯ + (q − 1) to zero modulo (q^{k} − 1). The condition in Lemma 6.2 shows that

(6.10)  (q^{k} − 1) |∑_{j=0}^{k−1} (M(j) − 1) q^{j}, −1 ≤ M(j) − 1 ≤ q − 2.

If q > 2, equation (6.10) forces M(j) − 1 = 0 for all j, that is M(j) = 1 for all 0 ≤ j ≤ k − 1. In particular,

∑_{j=0}^{k−1} M(j) = k.

This is a contradiction to (6.9). If q = 2, equation (6.10) shows that either M(j) = 1 for all j or M(j) = 0 for all j. The first case is handled in the above. The second case contradicts with (6.9) because we now have ∑_{j} M(j) = 0.

If all N(j) ≤ q − 1, we then have

(q^{k} − 1) |∑_{j=0}^{k−1} (N(j) − 1) q^{j}, −1 ≤ N(j) − 1 ≤ q − 2, ∑_{j=1}^{k} N(j) = k.

This forces N(j) = 1 for all j. Thus, the b_{j} form a permutation of {0, 1, ⋯, k− 1}. Lemma 6.2 is proved.

We now return to the proof of Lemma 6.1. Using Lemma 6.2, we deduce

(6.11)  ∑_{i=1}^{k} b(i) = ∑_{i=0}^{k−1} i = \frac{(k − 1)k}{2} .

By the definition of the b(i), we also have

(6.12)  \begin{aligned}∑_{i=1}^{k} b(i) & = ∑_{a(i)≥i} (a(i) − i) + ∑_{a(i)<i} (a(i) − i + k) \\ & = ∑_{i=1}^{k} (a(i) − i) + ∑_{a(i)<i} k. \end{aligned}

Equations (6.11) and (6.12) together imply

(6.13)  \begin{aligned}∑_{i=1}^{k} a(i) & = \frac{(k − 1)k}{2} + \frac{k(k + 1)}{2} − k ∑_{a(i)<i} 1 \\ & = k(k − ∑_{a(i)<i} 1). \end{aligned}

We deduced (6.13) just starting from u_{1} in (6.8). Similarly, if we start with u_{j} in (6.7), then we will deduce that

(6.14)  ∑_{i=1}^{k} a(i) = k(k − ∑_{{a(i+j−1)<i \atop 1≤i≤k}} 1),

where we define a(k + i) = a(i).

Since u_{1} ≥ 1, by (6.7) there is some j such that a(j) ≥ 1 (i.e. not all a(i) will be zero). Fix one such j, then (6.14) shows

∑_{i=1}^{k} a(i) = k(k − ∑_{{a(i+j−1)<i \atop 1≤i≤k}} 1)

= k(k − ∑_{{a(i+j−1)<i \atop 2≤i≤k}} 1)

≥ k.

Lemma 6.1 is proved.

THEOREM 6.3. Let h be a positive integer and

B(X) = 1 + ∑_{k>0} q^{(hk+1)}c_{k} · X^{(q^{k}−1)}, c_{k} ∈ R_{p}.

Then

det (I − T F′_{B}) = 1 + ∑_{k=1}^{∞} α_{k}T^{k},

where for all k ≥ 1,

(6.15)  α_{k} ≡ (−1)q^{(hk+1)}c_{k} ( mod q^{hk+2}) .

Proof: We have b_{0} = 1, b_{q^{k}−1} = q^{(hk+1)}c_{k} for k > 0 and b_{k} = 0 otherwise. Recall that for k ≥ 1,

(6.16)  α_{k} = (−1)^{k} ∑_{1≤u_{1}<···<u_{k}} ∑_{σ} sign(σ)b_{qu_{1}−σ(u_{1})} ⋯ b_{qu_{k}−σ(u_{k})}.

If σ is a k-cycle and the corresponding term in (6.16) is non-zero, then

qu_{i} − σ(u_{i}) = q^{a(i)} − 1

for all i, where the a(i) are non-negative integers. Lemma 6.1 shows

(6.17)  \begin{matrix}ord_{q} (b_{qu_{1}−σ(u_{1})} ⋯ b_{qu_{k}−σ(u_{k})}) = & ∑_{{i=1 \atop a(i)≥1}}^{k} (ha(i) + 1) \\ ≥ & (∑_{i=1}^{k} ha(i)) + 1 \\ ≥ & h · k + 1. \end{matrix}

In order for the above term to have a non-trivial contribution in (6.15), both inequalities in (6.17) have to be equalities. Now, the first inequality is an equality if and only if only one a(j) ≥ 1 and all other a(i) = 0. The second

inequality is an equality if and only if ∑_{i=1}^{k} a(i) = k. Thus, both inequalities are

equalities if and only if there is an index j such that a(i) = 0 for all i ≠ j and a(j) = k.

Since u_{1} < u_{2} < ⋯ < u_{k}, we have qu_{k} − σ (u_{k}) > 0. This implies that j = k and

(6.18)  \{ \begin{matrix}qu_{k} − σ(u_{k}) = q^{k} − 1, \\ qu_{i} − σ(u_{i}) = 0, & for all 1 ≤ i ≤ k − 1. \end{matrix}

Solving this system and using the fact that the u_{i} are increasing, we deduce that 1 ≤ u_{1}, u_{2} = qu_{1}, u_{3} = q^{2}u_{1}, ⋯, u_{k−1} = q^{k−2}u_{1} and u_{k} = (q^{k} −1+u_{1})/q. That u_{k} is an integer implies u_{1} ≡ 1(modq). If k = 1, the equation u_{1} = (q − 1 + u_{1})/q gives u_{1} = 1. If k > 1, then u_{k} > u_{k−1}. That is, (q^{k} − 1 + u_{1}) > q^{k−1}u_{1}. This inequality together with the congruence u_{1} ≡ 1(modq) gives u_{1} = 1. Thus, for all k > 0, we have u_{1} = 1, u_{2} = q, ⋯, u_{k} = q^{k−1} and σ (u_{i}) = u_{i+1} for all i. The contribution of this term in (6.15) is given by

(−1)^{k}(−1)^{k−1}b_{q^{k}−1} = (−1)q^{hk+1}c_{k}.

If σ is not a k-cycle, we can apply Lemma 6.1 to each cycle contained in σ. Since σ contains at least two proper cycles, the first sum in (6.17) contains at least two terms. Thus, the first inequality in (6.17) is a strict inequality and it follows that

ord_{q} (b_{qu_{1}−σ(u_{1})} ⋯ b_{qu_{k}−σ(u_{k})}) ≥ hk + 2.

This proves that if σ is not a k-cycle, then its contribution in (6.15) is zero. The theorem is proved.

THEOREM 6.4. Let B(X) be as in Theorem 6.3. Assume that the power series

∑_{k=1}^{∞} c_{k}T^{k} modulo the maximal ideal of R_{p} is not rational in its residue field. Then, the L-function L(B/G_{m}, T) is p-adic meromorphic in the open disk ord_{q}(T) > −h but not meromorphic on the closed disk ord_{q}(T) ≥ −h.

Proof: The Dwork trace formula (Lemma 4.2) and (6.3) show that

(6.19)  L(B/G_{m}, T) = \frac{(1 − T)det(I − T F′_{B})}{(1 − qT)det(I − qT F′_{B})}.

By (6.15), the L-function L(B/G_{m}, T) is clearly meromorphic in the open disk ord_{q}(T) > −h. On the other hand, Theorem 6.3 shows that the following power series

C(T) = \frac{(−1) (det (I − q^{−h}T F′_{B}) − 1)}{q}

has p-adic integral coefficients satisfying the congruence

C(T) ≡ ∑_{k=1}^{∞} c_{k}T^{k}

modulo the maximal ideal of R_{p}. If L(B/G_{m}, T) (and hence det(I − T F′_{B}) by (6.19)) were meromorphic on the closed disk ord_{q}(T) ≥ −h, then C(T) would be meromorphic on the closed unit disk. By the Weierstrass factorization theorem, the power series C(t) modulo the maximal ideal of R_{p} must be a rational function. This is a contradiction. The theorem is proved.

7. A generalization to local fields

Let K be a complete normed field of any characteristic. Let X be a topological space with a continuous map F : X → X. Let B(X) be a map from X to M_{r×r}(K), not necessarily continuous. Assume that for each positive integer k, the number of fixed points of the iterated map F^{k} in X is finite. Then we can define the L-function as before. (7.1)

L(B/X, T) = ∏_{closed points x∈X} \frac{1}{det(I − T^{d(x)}B(F^{d(x)−1}(x)) ⋯ B(F (x))B(x))},

where a closed point means the orbit of a periodic point of F and d(x) denotes the length of the orbit. In the case when K is an archimedean field, the Lfunction in (7.1) is simply Ruelle’s dynamic zeta function which accounts for the number of periodic points with weights.

It is clear that the Dwork trace formula (Lemma 4.2) can be applied to study the L-function in (7.1) in some cases, particularly when K is a nonarchimedean field. For example, let K be a complete discrete valuation ring contained in some complete and algebraically closed non-archimedean field Ω. Let X be the affine space A^{n}(Ω) and F be the q-th power map, where q is a positive integer, not necessarily a prime power. The periodic points of F are just those points x = (x_{1}, ⋯, x_{n}) whose coordinates are either zero or (q^{k} −1)- th roots of unity for some positive integer k. The multiplicity of a (q^{k} − 1)-th root of unity is counted when (q^{k} − 1) is divisible by the characteristic of K. Then (4.4) always holds. Let B(X) be a power series ∑_{u∈N^{n}} b_{u}X^{u}, where b_{u} ∈ M_{r×r}(K) and b_{u} → 0. Then the additive form of the Dwork trace formula holds, see Lemma 4.1 whose proof depends only on (4.4). The multiplicative form also holds in characteristic zero. If K has non-zero characteristic, the multiplicative form still holds because one can lift the coefficients of the power series B(X) to characteristic zero. Let π be a uniformizer of K. Define

h(B) = lim_{|u|→∞} inf \frac{ord_{π}(b_{u})}{log_{q} |u|}.

Then, Theorem 1.1 and Theorem 1.3 are true in this more general setting with the disk ord_{q}(T) > −h(B) replaced by the disk ord_{π}(T) > −h(B). The counterexample in Theorem 1.2 also carries over if, in addition, the number q is not a unit in the ring of integers of K. In the case that q is a unit in K, the counterexample in Theorem 1.2 still carries over if one assumes that the power

series ∑_{k=1}^{∞} c_{k}T^{k}/ ∑_{k=1}^{∞} c_{k}(qT)^{k} modulo π is not rational in its residue field.

Remark. In the above example, we can take F to be the p-th power map (p is a prime number) and K to be the ℓ-adic integers Z_{ℓ} with π = ℓ, where ℓ is a prime number which may be different from p.

Finally, we describe in details the characteristic p results for the torus and affine space cases, which will be used in [28].

Let F_{q} be the finite field of q elements. Let A be the ring of regular functions of a smooth absolutely irreducible affine curve, possibly defined over a finite extension field of F_{q}. Let π be a prime of A. Let A_{π} be the completion of A at π. We also use π to denote a uniformizer of A_{π}.

Let n be a fixed positive integer. Let B(X) be an r × r-matrix whose entries are power series over A_{π}. Namely, we can write

(7.2)  B(X) = ∑_{u∈N^{n}} b_{u}X^{u}, b_{u} ∈ M_{r×r}(A_{π}),

where u runs over all lattice points in N^{n} and |u| = |u_{1}| + ⋯ + |u_{n}|. Define

(7.3)  h(B) = lim_{|u|→∞} inf \frac{ord_{π}(b_{u})}{log_{q} |u|}.

The L-functions L(B/G_{m}^{n}, T) and L(B/A^{n}, T) are defined by (2.1) and (1.1) with x = x̄ since we are in characteristic p and therefore p-adic lifting is not necessary. This is a power series with coefficients in A_{π}. Then, we have

THEOREM 7.1. Both L(B/G_{m}^{n}, T)^{(−1)^{n−1}} and L(B/A^{n}, T)^{(−1)^{n−1}} are π-

adic entire functions in the open disk ord_{π}(T) > −h(B).

THEOREM 7.2. Let r = n = 1. For a positive integer h, let

B(X) = 1 + ∑_{u>0} π^{hu+1}c_{u}X^{(q^{u}−1)}, c_{u} ∈ A_{π}.

Assume that the power series ∑_{u>0} c_{u}T^{u} modulo the maximal ideal of A_{π} is not a rational function over its residue field, then both L(B/G_{m}, T) and L(B/A, T) can not be extended to be π-adic meromorphic on the closed disk ord_{π}(T) ≥ −h.

THEOREM 7.3. Let B(X) = ∑ b_{u}X^{u} be a polynomial in X with coefficients in M_{r×r}(A_{π}) of degree d. Then the L-function L(B/G_{m}^{n}, T)^{(−1)^{n−1}} is a polynomial in T whose degree is at most

r\binom{n + [d/(q − 1)]}{n},

and the L-function L(B/A^{n}, T)^{(−1)^{n−1}} is a polynomial in T whose degree is at most

r\binom{[d/(q − 1)]}{n}.

Proof. Let F_{B} be the infinite Frobenius matrix (b_{qu−v}). As mentioned above, the additive trace formula (4.2) is always true. The multiplicative trace formula is also true since one can lift B(X) to characteristic zero. One just lifts the coefficients b_{u} in the power series expansion of B(X) to characteristic zero. Reducing Lemma 4.2 and Lemma 4.3 modulo p give us the characteristic p formulas:

(7.4)  L(B/G_{m}^{n}, T)^{(−1)^{n−1}} = det(I − T F_{B})

and

(7.5)  L(B/A^{n}, T)^{(−1)^{n−1}} = det(I − T F_{B}^{S}),

where S is the full set {1, ⋯, n}. The proofs in sections 5-6 carry over word for word with ord_{q} replaced by ord_{π}, etc. Thus, Theorems 7.1-7.2 follow. Theorem 7.3 follows from (5.5)-(5.6) and (7.4)-(7.5). The proof is complete.

Note that the conclusion in Theorem 7.1 is stronger than the conclusion for the p-adic case. It says that the L-functions raised to (−1)^{n−1}-th power is actually entire in the same disk, not just meromorphic. The example in section 5 shows that the degree bounds in Theorem 7.3 are attained in some cases. The following strengthened form of Theorem 7.1 is also valid. This is a direct analogue of Theorem 5.2.

Let B(X, y) be a family of power series parameterized by y in some metric space Y:

B(X, y) = ∑_{u∈N^{n}} b_{u}(y)X^{u},

where b_{u}(y) is in M_{r×r}(A_{π}) for each y and b_{u}(y) goes to zero uniformly in u. Let α be a real number. The family B(X, y) is called uniformly α log-convergent if we have

lim_{|u|→∞} inf \frac{inf_{y} ord_{π}(b_{u}(y))}{log_{q} |u|} ≥ α.

The family B(X, y) is called continuous in y if each coefficient b_{u}(y) is a continuous map. This continuity is automatically uniform in u since b_{u}(y) goes to zero uniformly in y as |u| goes to infinity.

THEOREM 7.4. Let B(X, y) be a family of power series parameterized by y in some metric space Y. Assume that the family B(X, y) is continuous in y and uniformly α log-convergent. Then for any ϵ > 0, the zeroes of the family of L-functions

L(B(X, y)/G_{m}^{n}, T)^{(−1)^{n−1}} (resp. L(B(X, y)/A^{n}, T)^{(−1)^{n−1}}) flow continuously on

the closed disk ord_{π}(T) ≥ −(α − ϵ).

THE PENNSYLVANIA STATE UNIVERSITY, UNIVERSITY PARK

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