On Families of Pure Slope L-Functions

Elmar Grosse-Klönne

Received: August 6, 2002 Revised: January 11, 2003

Communicated by Peter Schneider

Abstract. Let R be the ring of integers in a finite extension K of ℚ_{p}, let k be its residue field and let χ : π_{1}(X) → R^{×} = GL_{1}(R) be a ”geometric” rank one representation of the arithmetic fundamental group of a smooth affine k-scheme X. We show that the locally Kanalytic characters κ : R^{×} → ℂ_{p}^{×} are the ℂ_{p}-valued points of a K-rigid space W and that

L(κ ∘ χ, T) = ∏_{\overline{x}∈X} \frac{1}{1 − (κ ∘ χ)(F rob\overline{x})T^{deg(\overline{x})}} ,

viewed as a two variable function in T and κ, is meromorphic on 𝔸_{ℂ_{p}}^{1} × W. On the way we prove, based on a construction of Wan, a slope decomposition for ordinary overconvergent (finite rank) σmodules, in the Grothendieck group of nuclear σ-modules.

2000 Mathematics Subject Classification: Primary 14F30; Secondary 14G10, 14G13, 14G15, 14G22

Introduction

In a series of remarkable papers [14] [15] [16], Wan recently proved a long outstanding conjecture of Dwork on the p-adic meromorphic continuation of unit root L-functions arising from an ordinary family of algebraic varieties defined over a finite field k. We begin by illustrating his result by a concrete example. Fix n ≥ 0 and let Y be the affine n + 1-dimensional 𝔽_{p}-variety in 𝔸^{1} × 𝔾_{m}^{n+1} defined by

z^{p} − z = x_{0} + … + … x_{n}.

Define u : Y → 𝔾_{m} by sending (z, x_{0}, …, x_{n}) to x_{0}x_{1} ⋯ x_{n}. For r ≥ 1 and y ∈ 𝔽_{p^{r}}^{×} let Y_{y}/𝔽_{p^{r}} be the fibre of u above y. For m ≥ 1 let Y_{y}(𝔽_{p^{rm}}) be the set of 𝔽_{p^{rm}}-rational points and (Y_{y})_{0} the set of closed points of Y_{y}/𝔽_{p^{r}} (a closed point z is an orbit of an \overline{𝔽}_{p^{r}}-valued point under the p^{r}-th power Frobenius map σ_{p^{r}}; its degree deg_{r}(z) is the smallest positive integer d such that σ_{p^{r}}^{d} fixes the orbit pointwise). The zeta function of Y_{y}/𝔽_{p^{r}} is

Z(Y_{y}/𝔽_{p^{r}}, T) = exp( ∑_{m=1}^{∞} \frac{|Y_{y}(𝔽_{p^{rm}})|}{m}T^{m}) = ∏_{z∈(Y_{y})_{0}} \frac{1}{1 − T^{deg_{r}(z)}} .

On the other hand for a character Ψ : 𝔽_{p} → ℂ define the Kloosterman sum

K_{m}(y) = ∑_{x_{0}x_{1}···x_{n}=y^{x_{i}∈𝔽{× \atop p^{rm}}}} Ψ(Tr_{𝔽_{p^{rm}} /𝔽_{p}}(x_{0} + x_{1} + … + x_{n}))

and let L_{Ψ}(Y, T) be the series such that

T dlogL_{Ψ}(y, T) = ∑_{m=1}^{∞} K_{m}(y)T^{m}.

Then, as series,

∏_{Ψ} L_{Ψ}(Y, T) = Z(Y_{y}/𝔽_{p^{r}}, T),

hence to understand Z(Y_{y}/𝔽_{p^{r}}, T) we need to understand all the L_{Ψ}(y, T). Suppose Ψ is non-trivial. It is known that L_{Ψ}(y, T) is a polynomial of degree n + 1: there are algebraic integers α_{0}(y), …, α_{n}(y) such that

L_{Ψ}(y, T)^{(−1)^{n−1}} = (1 − α_{0}(y)T) ⋯ (1 − α_{n}(y)T).

These α_{i}(y) have complex absolute value p^{rn/2} and are ℓ-adic units for any prime ℓ ≠ p. We ask for their p-adic valuation and their variation with y. Embedding \overline{ℚ} → \overline{ℚ_{p}} we have α_{i}(y) ∈ ℚ_{p}(π) where π^{p−1} = −p. Sperber has shown that we may order the α_{i}(y) such that ord_{p}(α_{i}(y)) = i for any 0 ≤ i ≤ n. Fix such an i and for k ∈ ℤ consider the L-function

∏_{y∈(𝔾_{m})_{0}/𝔽_{p}} \frac{1}{1 − α_{i}^{k}(y)T^{deg_{1}(y)}}

(here deg_{1}(y) is the minimal r such that y ∈ 𝔽_{p^{r}}^{×}, and (𝔾_{m})_{0}/𝔽_{p} is the set of closed points of 𝔾_{m}/𝔽_{p} defined similarly as before). A priori this series defines a holomorphic function only on the open unit disk. Dwork conjectured and Wan proved that it actually extends to a meromorphic function on 𝔸_{ℂ_{p}}^{1}, and varies uniformly with k in some sense. Now let W be the rigid space of locally ℚ_{p}(π)-analytic characters of the group of units in the ring of integers of ℚ_{p}(π). In this paper we show that

L(T, κ) = ∏_{y∈(𝔾_{m})_{0}/𝔽_{p}} \frac{1}{1 − κ(α_{i}(y))T^{deg_{1}(y)}}

Documenta Mathematica 8 (2003) 1–42

On Families of Pure Slope L-Functions 3

defines a meromorphic function on 𝔸_{ℂ_{p}}^{1} ×W. Specializing κ ∈ W to the character r ↦ r^{k} for k ∈ ℤ we recover Wan’s result. The conceptual way to think of this example is in terms of σ-modules: 𝔽_{p} acts on Y via z ↦ z + a for a ∈ 𝔽_{p}. It induces an action of 𝔽_{p} on the relative n-th rigid cohomology ℝ^{n}u_{rig,∗}O_{Y} of u, and over ℚ_{p}(π) the latter splits up into its eigencomponents for the various characters of 𝔽_{p}. The Ψ-eigencomponent (ℝ^{n}u_{rig,∗}O_{Y})^{Ψ} is an overconvergent σ-module and L_{Ψ}(y, T)^{(−1)^{n−1}} is the characteristic polynomial of Frobenius acting on its fibre in y. Crucial is the slope decomposition of (ℝ^{n}u_{rig,∗}O_{Y})^{Ψ}: it means that for fixed i the α_{i}(y) vary rigid analytically with y in some sense. We are thus led to consider Dwork’s conjecture, i.e. Wan’s theorem, in the following general context. Let R be the ring of integers in a finite extension K of ℚ_{p}, let π be a uniformizer and k the residue field. Let X be a smooth affine k-scheme, let A be the coordinate ring of a lifting of X to a smooth affine weak formal R-scheme (so A is a wcfg-algebra) and let Â be the p-adic completion of A. Let σ be an Ralgebra endomorphism of A lifting the q-th power Frobenius endomorphism of X, where q = |k|. A finite rank σ-module over Â (resp. over A) is a finite rank free Â-module (resp. A-module) together with a σ-linear endomorphism ϕ. A finite rank σ-module over Â is called overconvergent if it arises by base change A → Â from a finite rank σ-module over A. Let the finite rank overconvergent σ-module Φ over Â be ordinary, in the strong sense that it admits a Frobenius stable filtration such that on the j-th graded piece we have: the Frobenius is divisible by π^{j} and multiplied with π^{−j} it defines a unit root σ-module Φ_{j}, i.e. a σ-module whose linearization is bijective. (Recall that unit root σ-modules over Â are the same as continuous representations of π_{1}(X) on finite rank free R-modules.) Although Φ is overconvergent, Φ_{j} will in general not be overconvergent; and this is what prevented Dwork from proving what is now Wan’s theorem: the L-function L(Φ_{j}, T) is meromorphic on 𝔸_{ℂ_{p}}^{1}. Moreover he proved the same for powers (=iterates of the σ-linear endomorphism) Φ_{j}^{k} of Φ_{j} and showed that in case Φ_{j} is of rank one the family {L(Φ_{j}^{k}, T)}_{k∈ℤ} varies uniformly with k ∈ ℤ in a certain sense. At the heart of Wan’s striking method lies his ”limiting σ-module” construction which allows him to reduce the analysis of the not necessarily overconvergent Φ_{j} to that of overconvergent σ-modules — at the cost of now working with overconvergent σ-modules of infinite rank, but which are nuclear. To the latter a generalization of the Monsky trace formula can be applied which expresses L(Φ_{j}^{k}, T) as an alternating sum of Fredholm determinants of completely continuous Dwork operators. The first aim of this paper is to further explore the significance of the limiting σ-module construction which we think to be relevant for the search of good p-adic coefficients on varieties in characteristic p. Following an argument of Coleman [4] we give a functoriality result for this construction. This is then used to prove (Theorem 7.2) a slope decomposition for ordinary overconvergent finite rank σ-modules, in the Grothendieck group Δ(Â) of nuclear σ-modules over Â. More precisely, we show that any Φ_{j} as above, not necessarily overconvergent, can be written, in Δ(Â), as a sum of virtual nuclear overconvergent σ-modules. (This is the global version of the decomposition of the corresponding L-function found by Wan.) Our second aim is to strengthen Wan’s uniform results on the family {L(Φ_{j}^{k}, T)}_{k∈ℤ} in case Φ_{j} is of rank one. More generally we replace Φ_{j} by the rank one unit root σ-module det(Φ_{j}) if Φ_{j} has rank > 1. Let det Φ_{j} be given by the action of α ∈ Â^{×} on a basis element. For \overline{x} ∈ X a closed point of degree f let x : Â → R_{f} be its Teichmüller lift, where R_{f} denotes the unramified extension of R of degree f. Then

α\overline{x} = x(ασ(α) … σ^{f−1}(α))

lies in R^{×}. We prove that for any locally K-analytic character κ : R^{×} → ℂ_{p}^{×} the twisted L-function

L(α, T, κ) = ∏_{\overline{x}∈X} \frac{1}{1 − κ(α\overline{x})T^{deg(\overline{x})}}

is p-adic meromorphic on 𝔸_{ℂ_{p}}^{1}, and varies rigid analytically with κ. More precisely, building on work of Schneider and Teitelbaum [13], we use Lubin-Tate theory to construct a smooth ℂ_{p}-rigid analytic variety W whose ℂ_{p}-valued points are in natural bijection with the set Hom_{K-an}(R^{×}, ℂ_{p}^{×}) of locally Kanalytic characters of R^{×}. Then our main theorem is:

Theorem 0.1. On the ℂ_{p}-rigid space 𝔸_{ℂ_{p}}^{1} × W there exists a meromorphic function L_{α} whose pullback to 𝔸_{ℂ_{p}}^{1} via 𝔸_{ℂ_{p}}^{1} → 𝔸_{ℂ_{p}}^{1} × W, t ↦ (t, κ) for any κ ∈ Hom_{K-an}(R^{×}, ℂ_{p}^{×}) = W(ℂ_{p}) is a continuation of L(α, T, κ).

The statement in the abstract above follows by the well known correspondence between representations of the fundamental group and unit-root σ-modules. The analytic variation of the L-series L(α, T, κ) with the weight κ makes it meaningful to vastly generalize the eigencurve theme studied by Coleman and Mazur [5] in connection with the Gouvêa-Mazur conjecture. Namely, we can ask for the divisor of the two variable meromorphic function L_{α} on 𝔸_{ℂ_{p}}^{1} × W. From a general principle in [3] we already get: for fixed λ ∈ ℝ_{>0}, the difference between the numbers of poles and zeros of L_{α} on the annulus |T | = λ is locally constant on W. We hope for better qualitative results if the σ-module over A giving rise to the σ-module Φ over Â carries an overconvergent integrable connection, i.e. is an overconvergent F -isocrystal on X in the sense of Berthelot. The eigencurve from [5] comes about in this context as follows: The Fredholm determinant of the U_{p}-operator acting on overconvergent p-adic modular forms is a product of certain power rank one unit root L-functions arising from the universal ordinary elliptic curve, see [3]. Also, again in the general case, the p-adic L-function on W which we get by specializing T = 1 in L_{α} should be of particular interest. The proof of Theorem 0.1 consists of two steps. First we prove (this is essentially Corollary 4.12) the meromorphic continuation to 𝔸_{ℂ_{p}}^{1} × W^{0} for a certain open subspace W^{0} of W which meets every component of W: the subspace of characters of the type κ(r) = r^{ℓ}u(r)^{x} for ℓ ∈ ℤ and small x ∈ ℂ_{p}, with u(r) denoting the one-unit part of r ∈ R^{×}. (In particular, W^{0} contains the characters r ↦ κ_{k}(r) = r^{k} for k ∈ ℤ; for these we have L(Φ_{j}^{k}, T) = L(α, T, κ_{k}).) For this we include det(Φ_{j}) in a family of nuclear σ-modules, parametrized by W^{0}: namely, the factorization into torsion part and one-unit part and then exponentiation with ℓ ∈ ℤ resp. with small x ∈ ℂ_{p} makes sense not just for R^{×}-elements but also for α, hence an analytic family of rank one unit root σmodules parametrized by W^{0}. In the Grothendieck group of W^{0}-parametrized families of nuclear σ-modules, we write this deformation family of det(Φ_{j}) as a sum of virtual families of nuclear overconvergent σ-modules. In each fibre κ ∈ W^{0} we thus obtain, by an infinite rank version of the Monsky trace formula, an expression of the L-function L(α, T, κ) as an alternating product of characteristic series of nuclear Dwork operators. While this is essentially an ”analytic family version” of Wan’s proof (at least if X = 𝔸^{n}), the second step, the extension to the whole space 𝔸_{ℂ_{p}}^{1} × W, needs a new argument. We use a certain integrality property (w.r.t. W) of the coefficients of (the logarithm of) L_{α} which we play out against the already known meromorphic continuation on 𝔸_{ℂ_{p}}^{1} × W^{0}. However, we are not able to extend the limiting modules from W^{0} to all of W; as a consequence, for κ ∈ W − W^{0} we have no interpretation of L(α, T, κ) as an alternating product of characteristic series of Dwork operators. Note that for K = ℚ_{p}, the locally K-analytic characters of R^{×} = ℤ_{p}^{×} are precisely the continuous ones; the space W^{0} in that case is the weight space considered in [3] while W is that of [5]. Now let us turn to some technical points. Wan develops his limiting σ-module construction and the Monsky trace formula for nuclear overconvergent infinite rank σ-modules only for the base scheme X = 𝔸^{n}. General base schemes X he embeds into 𝔸^{n} and treats (the pure graded pieces of) finite rank overconvergent σ-modules on X by lifting them with the help of Dwork’s F -crystal to σ-modules on 𝔸^{n} having the same L-functions. We work instead in the infinite rank setting on arbitrary X. Here we need to overcome certain technical difficulties in extending the finite rank Monsky trace formula to its infinite rank version. The characteristic series through which we want to express the L-function are those of certain Dwork operators ψ on spaces of overconvergent functions with non fixed radius of overconvergence. To get a hand on these ψ’s one needs to write these overconvergent function spaces as direct limits of appropriate affinoid algebras on which the restrictions of the ψ’s are completely continuous. Then statements on the ψ’s can be made if these affinoid algebras have a common system of orthogonal bases. Only for X = 𝔸^{n} we find such bases; but we show how one can pass to the limit also for general X. An important justification for proving the trace formula in this form (on general X, with function spaces with non fixed radius of overconvergence) is that in the future it will allow us to make full use of the overconvergent connection in case the σ-module over A giving rise to the σ-module Φ over Â underlies an overconvergent F -isocrystal on X (see above) — then the limiting module also carries an overconvergent connection. Deviating from [14] [15], instead of working with formally free nuclear σ-modules with fixed formal bases we work, for concreteness, with the infinite square matrices describing them. This is of course only a matter of language. A brief overview. In section 1 we show the existence of common orthogonal bases in overconvergent ideals which might be of some independent interest. In section 2 we define the L-functions and prove the trace formula. In section 3 we introduce the Grothendieck group of nuclear σ-modules (and their deformations). In section 4 we concentrate on the case where ϕ_{j} is the unit root part of ϕ and is of rank one: here we need the limiting module construction. In section 5 we introduce the weight space W, in section 6 we prove (an infinite rank version of) Theorem 0.1, and in section 7 (which logically could follow immediately after section 4) we give the overconvergent representation of Φ_{j}.

Acknowledgments: I wish to express my sincere thanks to Robert Coleman and Daqing Wan. Manifestly this work heavily builds on ideas of them, above all on Wan’s limiting module construction. Wan invited me to begin further elaborating his methods, and directed my attention to many interesting problems involved. Coleman asked me for the meromorphic continuation to the whole character space and provided me with some helpful notes [4]. In particular the important functoriality result 4.10 for the limiting module and the suggestion of varying it rigid analytically is due to him. Thanks also to Matthias Strauch for discussions on the weight space.

Notations: By |.| we denote an absolute value of K and by e ∈ ℕ the absolute ramification index of K. By ℂ_{p} we denote the completion of a fixed algebraic closure of K and by ord_{π} and ord_{p} the homomorphisms ℂ_{p}^{×} → ℚ with ord_{π}(π) = ord_{p}(p) = 1. For R-modules E with πE ≠ E we set

ord_{π}(x) := sup\{r ∈ ℚ; r = \frac{n}{m} for some n ∈ ℕ_{0}, m ∈ ℕ such that x^{m} ∈ π^{n}E\}

for x ∈ E. Similarly we define ord_{p} on such E. For n ∈ ℕ we write μ_{n} = {x ∈ ℂ_{p}; x^{n} = 1}. We let ℕ_{0} = ℤ_{≥0}. For an element g in a free polynomial ring A[X_{1}, …, X_{n}] over a ring A we denote by deg(g) its (total) degree.

1 Orthonormal bases of overconvergent ideals

In this preparatory section we determine explicit orthonormal K-bases of ideals in overconvergent K-Tate algebras T_{n}^{c} (1.5). Furthermore we recall the complete continuity of certain Dwork operators (1.7).

1.1 For c ∈ ℕ we let

T_{n}^{c} := \{ ∑_{α∈ℕ_{0}^{n}} b_{α}π^{[ \frac{|α|}{c}]}X^{α}; b_{α} ∈ K, lim_{|α|→∞} |b_{α}| = 0\}

where as usual |α| = ∑_{i=1}^{n} α_{i} for α = (α_{1}, …, α_{n}) ∈ ℕ_{0}^{n} and where [r] ∈ ℤ for a given r ∈ ℚ denotes the unique integer with [r] ≤ r < [r] + 1. This is the ring of power series in X_{1}, …, X_{n} with coefficients in K, convergent on the polydisk

\{x ∈ ℂ_{p}^{n}; ord_{π}(x_{i}) ≥ −\frac{1}{c} for all 1 ≤ i ≤ n\}.

We view T_{n}^{c} as a K-Banach module with the unique norm |.|_{c} for which \{π^{[ \frac{|α|}{c}]}X^{α}\}_{α∈ℕ_{0}^{n}} is an orthonormal basis (this norm is not power multiplicative). Suppose we are given elements g_{1}, …, g_{r} ∈ R[X_{1}, …, X_{n}] − πR[X_{1}, …, X_{n}]. Let \overline{g}_{j} ∈ k[X_{1}, …, X_{n}] be the reduction of g_{j}, let d_{j} = deg(\overline{g}_{j}) ≤ deg(g_{j}) be its degree.

Lemma 1.2. For each 1 ≤ j ≤ r and each c > max_{j} deg(g_{j}) we have

|π^{[ \frac{|α|+d_{j}}{c} ]}X^{α}g_{j}|_{c} = 1.

Proof: Write g_{j} = ∑_{β∈ℕ_{0}^{n}} b_{β}X^{β} with b_{β} ∈ K. There exists a β_{1} ∈ ℕ_{0}^{n} with |β_{1}| = d_{j} and |b_{β_{1}}| = 1. Hence

|π^{[ \frac{|α|+d_{j}}{c} ]}X^{α}b_{β_{1}}X^{β_{1}}|_{c} = |π^{[ \frac{|α|+|β_{1}|}{c}]}X^{α+β_{1}}|_{c} = 1.

Now let β ∈ ℕ_{0}^{n} be arbitrary, with b_{β} ≠ 0. If |β| > d_{j} then |b_{β}| ≤ |π|. Hence

|π^{[ \frac{|α|+d_{j}}{c} ]}X^{α}b_{β}X^{β}|_{c} ≤ |π^{[ \frac{|α|+d_{j}}{c} ]−[ \frac{|α|+|β|}{c}]+1}π^{[ \frac{|α|+|β|}{c}]}X^{α+β}|_{c}.

But [\frac{|α|+d_{j}}{c}] − [\frac{|α|+|β|}{c}] + 1 ≥ 0 because b_{β} ≠ 0, hence c > |β|. Thus,

|π^{[ \frac{|α|+d_{j}}{c} ]}X^{α}b_{β}X^{β}|_{c} ≤ 1.

On the other hand, if |β| ≤ d_{j}, then [\frac{|α|+d_{j}}{c}] ≥ [\frac{|α|+|β|}{c}] and |b_{β}| ≤ 1, and again we find

|π^{[ \frac{|α|+d_{j}}{c} ]}X^{α}b_{β}X^{β}|_{c} ≤ 1.

We are done.

1.3 The Tate algebra in n variables over K is the algebra

T_{n} := { ∑_{α∈ℕ_{0}^{n}} b_{α}X^{α}; b_{α} ∈ K, lim_{|α|→∞} |b_{α}| = 0}.

Let I_{∞} (resp. I_{c}) be the ideal in T_{n} (resp. in T_{n}^{c}) generated by g_{1}, …, g_{r}. As all ideals in T_{n}^{c}, the ideal I_{c} is closed in T_{n}^{c}. We view I_{c} as a K-Banach module with the norm |.|_{c} induced from T_{n}^{c}.

Lemma 1.4. If I_{∞} ⊂ T_{n} is a prime ideal, I_{∞} ≠ T_{n}, then I_{c} = I_{∞} ∩ T_{n}^{c} for c >> 0.

Proof: For c >> 0 also I_{c} is a prime ideal in T_{n}^{c}. The open immersion of K-rigid spaces Sp(T_{n}) → Sp(T_{n}^{c}) induces an open immersion V (I_{∞}) → V (I_{c}) of the respective zero sets of g_{1}, …, g_{r}. That I_{c} is prime means that V (I_{c}) is irreducible, and I_{∞} ≠ T_{n} means that V (I_{∞}) is non empty. Hence an element of I_{∞} ∩ T_{n}^{c}, since it vanishes on V (I_{∞}), necessarily also vanishes on V (I_{c}). By Hilbert’s Nullstellensatz ([2]) it is then an element of I_{c}.

Now we fix an integer c′ > max_{j} deg(g_{j}). By 1.2 we find a subset E of ℕ_{0}^{n} ×

\{1, …, r\} such that \{π^{[ \frac{|α|+d_{j}}{c′} ]}X^{α}g_{j}\}_{(α,j)∈E} is an orthonormal basis of I_{c′} over K.

Theorem 1.5. For integers c ≥ c′, the set \{π^{[ \frac{|α|+d_{j}}{c} ]}X^{α}g_{j}\}_{(α,j)∈E} is an orthonormal basis of I_{c} over K.

Proof: Let K^{c} be a finite extension of K containing a c-th root π^{\frac{1}{c}} and a c′-th root π^{\frac{1}{c′}} of π. The absolute value |.| extends to K^{c}. Any norm on a K-Banach module M extends uniquely to a K^{c}-Banach module norm on M ⊗_{K} K^{c}, and we keep the same name for it. It is enough to show that \{π^{[ \frac{|α|+d_{j}}{c} ]}X^{α}g_{j}\}_{(α,j)∈E} is an orthonormal basis of I_{c} ⊗_{K} K^{c} over K^{c}. Let |.|′_{c} be the supremum norm on T_{n}^{c} ⊗_{K} K^{c}. This is the norm for which \{X^{\frac{α}{c}} \}_{α∈ℕ_{0}^{n}} is an orthonormal basis over K^{c}. For j ∈ {1, …, r} write g_{j} = ∑_{β∈ℕ_{0}^{n}} b_{β}X^{β} with b_{β} ∈ K. Then, by a computation similar to that in 1.2 we find

|π^{\frac{|α|+d_{j}}{c}} X^{α}b_{β}X^{β}|′_{c} = 1 if |β| = d_{j} and |b_{β}| = 1,
|π^{\frac{|α|+d_{j}}{c}} X^{α}b_{β}X^{β}|′_{c} < 1 otherwise.

In particular it follows that |π^{\frac{|α|+d_{j}}{c}} X^{α}g_{j}|′_{c} = 1. Now a comparison of expan-

sions shows that \{π^{[ \frac{|α|+d_{j}}{c} ]}X^{α}g_{j}\}_{(α,j)∈E} is an orthonormal basis of I_{c} ⊗_{K} K^{c}

over K^{c} with respect to |.|_{c} if and only if \{π^{\frac{|α|+d_{j}}{c}} X^{α}g_{j}\}_{(α,j)∈E} is an orthonormal basis of I_{c} ⊗_{K} K^{c} over K^{c} with respect to |.|′_{c}. In particular it follows on

the one hand that we only need to show that \{π^{\frac{|α|+d_{j}}{c}} X^{α}g_{j}\}_{(α,j)∈E} is an orthonormal basis of I_{c} ⊗_{K} K^{c} over K^{c} with respect to |.|′_{c}, and on the other hand

it follows (applying the above with c′ instead of c) that \{π^{\frac{|α|+d_{j}}{c′}} X^{α}g_{j}\}_{(α,j)∈E} is an orthonormal basis of I_{c′} ⊗_{K} K^{c} over K^{c} with respect to |.|′_{c′}. Consider the isomorphism

T_{n}^{c} ⊗_{K} K^{c} ≅ T_{n}^{c′} ⊗_{K} K^{c}, π^{\frac{α}{c}} X^{α} ↦ π^{\frac{α}{c′}} X^{α}

which is isometric with respect to |.|′_{c} resp. |.|′_{c′}. It does not necessarily map I_{c} ⊗_{K} K^{c} to I_{c′} ⊗_{K} K^{c}. However, from our above computations of the values |π^{\frac{|α|+d_{j}}{c}} X^{α}b_{β}X^{β}|′_{c} it follows that this isomorphism identifies the reductions

of the elements of the set \{π^{\frac{|α|+d_{j}}{c}} X^{α}g_{j}\}_{(α,j)∈ℕ_{0}^{n}×\{1,...,r\}} with the reductions

of the elements of the set \{π^{\frac{|α|+d_{j}}{c′}} X^{α}g_{j}\}_{(α,j)∈ℕ_{0}^{n}×\{1,...,r\}} (here by reduction we mean reduction modulo elements of absolute value < 1). The K^{c}-vector subspaces spanned by these sets are dense in I_{c} ⊗_{K} K^{c} resp. in I_{c′} ⊗_{K} K^{c}. Since for a subset of |.| = 1 elements in an orthonormizable K^{c}-Banach module the property of being an orthonormal basis is equivalent to that of inducing an (algebraic) basis of the reduction, the theorem follows.

1.6 Let B_{K} be a reduced K-affinoid algebra, i.e. a quotient of a Tate algebra T_{m} over K (for some m), endowed with its supremum norm |.|_{sup}. Let

B = (B_{K})^{0} := {b ∈ B_{K}; |b|_{sup} ≤ 1}.

For positive integers m and c let

[m, c] := [m, c]_{R} := {z ∈ R[[X_{1}, …, X_{n}]];

z = ∑_{j=0}^{∞} π^{j}p_{j} with p_{j} ∈ R[X_{1}, …, X_{n}] and deg(p_{j}) ≤ m + cj}

and

[m, c]_{B} := [m, c]ˆ⊗_{R}B

(the π-adically completed tensor product). Note that for m, c_{1}, c_{2} ∈ ℕ with c_{1} < c_{2} we have [m, c_{1}]_{B} ⊂ T_{n}^{c_{2}} ˆ⊗_{K}B_{K} and also (∪_{m,c}[m, c]_{B}) ⊗_{R} K = ∪_{c}(T_{n}^{c} ˆ⊗_{K}B_{K}). Let

R[X_{1}, …, X_{n}]^{†} := R[X]^{†} := ⋃_{m,c}[m, c].

Fix a Frobenius endomorphism σ of R[X]^{†} lifting the q-th power Frobenius endomorphism of k[X]. Also fix a Dwork operator θ (with respect to σ) on R[X]^{†}, i.e. an R-module endomorphism with θ(σ(x)y) = xθ(y) for all x, y ∈ R[X]^{†}. By [8] 2.4 we have θ(T_{n}^{c}) ⊂ T_{n}^{c} for all c >> 0, thus we get a B_{K}-linear endomorphism θ ⊗ 1 on T_{n}^{c} ˆ⊗_{K}B_{K}.

Proposition 1.7. Let I be a countable set, m′, c′ positive integers and M = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I} an I × I-matrix with entries a_{i_{1},i_{2}} in [m′, c′]_{B}. Suppose that M is nuclear, i.e. that for each M > 0 there are only finitely many i_{2} ∈ I such that inf_{i_{1}} ord_{π}a_{i_{1},i_{2}} < M. For c >> 0 and β ∈ ℕ_{0}^{n} develop (θ ⊗ 1)(π^{[ \frac{|β|}{c}]}X^{β}a_{i_{1},i_{2}}) ∈ T_{n}^{c} ˆ⊗_{K}B_{K} in the orthonormal basis \{π^{[ \frac{|α|}{c}]}X^{α}\}_{α} of the B_{K}-Banach module T_{n}^{c} ˆ⊗_{K}B_{K} and let G_{{α,i_{1}}{β,i_{2}}}^{c} ∈ B_{K} for α ∈ ℕ_{0}^{n} be its coefficients:

(θ ⊗ 1)(π^{[ \frac{|β|}{c}]}X^{β}a_{i_{1},i_{2}}) = ∑_{α} G_{\{α,i_{1}\}\{β,i_{2}\}}^{c}π^{[ \frac{|α|}{c}]}X^{α}.

Then for all c >> 0 and all M > 0 there are only finitely many pairs (α, i_{2}) ∈ ℕ_{0}^{n} × I such that

inf_{β,i_{1}} ord_{π}G_{{α,i_{1}}{β,i_{2}}}^{c} < M.

Proof: For simplicity identify I with ℕ. By [8] 2.3 we find integers r and c_{0} such that (θ⊗1)([qm, qc]_{B}) ⊂ [m+r, c]_{B} for all c ≥ c_{0}, all m. Increasing c_{0} and r we may assume that a_{i_{1},i_{2}} ∈ [q(r − 1), c_{0}]_{B} for all i_{1}, i_{2}. Now let c be so large that for c′ = c − 1 we have qc′ ≥ c_{0}. Then one easily checks that X^{β}a_{i_{1}i_{2}} ∈ [q(r + [\frac{|β|}{q}]), qc′]_{B} for all β, i_{1}, i_{2}. Hence (θ ⊗ 1)(X^{β}a_{i_{1},i_{2}}) ∈ [r + [\frac{|β|}{q}], c′]. This means

|α| ≤ r + [\frac{|β|}{q}] + c′(ord_{π}(G_{\{α,i_{1}\}\{β,i_{2}\}}^{c}) + [\frac{|α|}{c}] − [\frac{|β|}{c}])

for all α, and thus

ord_{π}(G_{\{α,i_{1}\}\{β,i_{2}\}}^{c}) ≥ [\frac{|β|}{c}] − [\frac{|α|}{c}] + \frac{|α| − r − [\frac{|β|}{q}]}{c′}.

Here the right hand side tends to infinity as |α| tends to infinity, uniformly for all β — independently of i_{1} and i_{2} — because c/q ≤ c′ ≤ c. Now let M ∈ ℕ be given. By the above we find N′(M) ∈ ℕ such that for all α with |α| ≥ N′(M) we have ord_{π}(G_{{α,i_{1}}{β,i_{2}}}^{c}) ≥ M. Now fix α. We have

ord_{π}(G_{\{α,i_{1}\}\{β,i_{2}\}}^{c}) ≥ [\frac{|β|}{c}] − [\frac{|α|}{c}] + ord_{π}(θ ⊗ 1)(X^{β}a_{i_{1},i_{2}}).

By nuclearity of M the right hand side tends to zero as i_{2} tends to infinity, uniformly for all i_{1}, all β. In other words, there exists N(α, M) such that ord_{π}(G_{{α,i_{1}}{β,i_{2}}}^{c}) ≥ M for all i_{2} ≥ N(α, M), for all i_{1}, all β. Now set

N(M) = N′(M) + max{N(α, M); |α| < N′(M)}.

Then we find inf_{β,i_{1}} ord_{π}G_{{α,i_{1}}{β,i_{2}}}^{c} ≥ M whenever |α| + i_{2} ≥ N(M). We are done.

2 L-functions

This section introduces our basic setting. We define nuclear (overconvergent) matrices (which give rise to nuclear (overconvergent) σ-modules), their associated L-functions and Dwork operators and give the Monsky trace formula (2.13).

2.1 Let q ∈ ℕ be the number of elements of k, i.e. k = 𝔽_{q}. Let X = Spec(\overline{A}) be a smooth affine connected k-scheme of dimension d. So \overline{A} is a smooth k-algebra. By [6] it can be represented as \overline{A} = A/πA where

A = \frac{R[X_{1}, …, X_{n}]^{†}}{(g_{1}, …, g_{r})}

Documenta Mathematica 8 (2003) 1–42

with polynomials g_{j} ∈ R[X_{1}, …, X_{n}] − πR[X_{1}, …, X_{n}] such that A is R-flat. By [10] we can lift the q-th power Frobenius endomorphism of \overline{A} to an Ralgebra endomorphism σ of A. Then A, viewed as a σ(A)-module, is locally free of rank q^{d}. Shrinking X if necessary we may assume that A is a finite free σ(A)-module of rank q^{d}. As before, B_{K} denotes a reduced K-affinoid algebra, and B = (B_{K})^{0}.

2.2 Let I be a countable set. An I ×I-matrix M = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I} with entries in an R-module E with E ≠ πE is called nuclear if for each M > 0 there are only finitely many i_{2} such that inf_{i_{1}} ord_{π}(a_{i_{1},i_{2}}) < M (thus M is nuclear precisely if its transpose is the matrix of a completely continuous operator, or in the terminology of other authors (e.g. [8]): a compact operator). An I × I-matrix M = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I} with entries in Aˆ⊗_{R}B is called nuclear overconvergent if there exist positive integers m, c and a nuclear matrix I × I-matrix M̃ with entries in [m, c]_{B} which maps (coefficient-wise) to M under the canonical map

[m, c]_{B} ↪ R[X]^{†} ˆ⊗_{R}B → Aˆ⊗_{R}B.

Clearly, if M is nuclear overconvergent then it is nuclear. Example: Let B_{K} = K. Nuclear overconvergence implies that the matrix entries are in the subring A of its completion Aˆ⊗_{R}R = Â. Conversely, if I is finite, an I×I-matrix with entries in A is automatically nuclear overconvergent. Similarly, if I is finite, any I × I-matrix with entries in Â is automatically nuclear.

2.3 For nuclear matrices N = (c_{h_{1},h_{2}})_{h_{1},h_{2}∈H} and N′ = (d_{g_{1},g_{2}})_{g_{1},g_{2}∈G} with entries in Aˆ⊗_{R}B define the (G × H) × (G × H)-matrix

N ⊗ N′ := (e_{(h_{1},g_{1}),(h_{2},g_{2})})_{(h_{1},g_{1}),(h_{2},g_{2})∈(G×H)},
e_{(h_{1},g_{1}),(h_{2},g_{2})} := c_{h_{1},h_{2}}d_{g_{1},g_{2}}.

Now choose an ordering of the index set H. For k ∈ ℕ_{0} let ⋀^{k}(H) be the set of k-tuples (h_{1}, …, h_{k}) ∈ H^{k} with h_{1} < … < h_{k}. Define the ⋀^{k}(H) × ⋀^{k}(H)- matrix

⋀^{k}(N) := N^{∧k} := (f_{h⃗_{1},h⃗_{2}})_{h⃗_{1},h⃗_{2}∈⋀^{k}(H)},

f_{h⃗_{1},h⃗_{2}} = f_{(h_{11},...,h_{1k}),(h_{21},...,h_{2k})} := ∏_{i=1}^{k} c_{h_{1,i}h_{2,i}}.

It is straightforward to check that N ⊗ N′ and ⋀^{k}(N) are again nuclear, and even nuclear overconvergent if N and N′ are nuclear overconvergent.

2.4 We will use the term ”nuclear” also for another concept. Namely, suppose ψ is an operator on a vector space V over K. For g = g(X) ∈ K[X] let

F (g) := ∩_{n}g(ψ)^{n}V and N(g) := ∪_{n} ker g(ψ)^{n}.

Let us call a subset S of K[X] bounded away from 0 if there is an r ∈ ℚ such that g(a) ≠ 0 for all {a ∈ ℂ_{p}; ord_{p}(a) ≥ r}. We say ψ is nuclear if for any subset S of K[X] bounded away from 0 the following two conditions hold:

(i) F (g) ⊕ N(g) = V for all g ∈ S (ii) N(S) := ∑_{g∈S} N(g) is finite dimensional. (In particular, if g ∉ (X), we can take S = {g} and as a consequence of (ii) get N(g) = ker g(ψ)^{n} for some n.) Suppose ψ is nuclear. Then we can define P_{S}(X) = det(1−Xψ|_{N(S)}) for subsets S of K[X] bounded away from 0. These S from a directed set under inclusion, and in [8] it is shown that

P (X) := lim_{S} P_{S}(X)

(coefficient-wise convergence) exists in K[[X]]: the characteristic series of ψ.

2.5 Let (N_{c})_{c∈ℕ} be an inductive system of B_{K}-Banach modules with injective (but not necessarily isometric) transition maps ρ_{c,c′} : N_{c} → N_{c′} for c′ ≥ c. Suppose this system has a countable common orthogonal B_{K}-basis, i.e. there is a subset {q_{m}; m ∈ ℕ} of N_{1} such that for all c and m ∈ ℕ there are λ_{m,c} ∈ K^{×} such that {λ_{m,c}ρ_{1,c}(q_{m}); m ∈ ℕ} is an orthonormal B_{K}-basis of N_{c}. Let

N := lim_{c^{→}} N_{c}” = ” ⋃_{c} N_{c}

and let N′ ⊂ N be a B_{K}-submodule such that N′_{c} = N′ ∩ N_{c} is closed in N_{c} for all c. Endow N′_{c} with the norm induced from N_{c} and suppose that also the inductive system (N′_{c})_{c∈ℕ} has a countable common orthogonal B_{K}-basis. Let u be a B_{K}-linear endomorphism of N with u(N′) ⊂ N′ and restricting to a completely continuous endomorphism u : N_{c} → N_{c} for each c. In that situation we have:

Proposition 2.6. u induces a completely continuous B_{K}-endomorphism u of N′′_{c} = N_{c}/N′_{c} for each c, and det(1 − uT; N′′_{c}) is independent of c. If B_{K} = K, the induced endomorphism u of N′′ = N/N′ is nuclear in the sense of 2.4, and its characteristic series coincides with det(1 − uT; N′′_{c}) for each c.

Proof: From [3] A2.6.2 we get that u on N′_{c} and u on N′′_{c} are completely continuous (note that N′′_{c} is orthonormizable, as follows from [3] A1.2), and that

det(1 − uT; N_{c}) = det(1 − uT; N′_{c}) det(1 − uT; N′′_{c})

for each c. The assumption on the existence of common orthogonal bases implies (use [5] 4.3.2)

det(1 − uT; N_{c}) = det(1 − uT; N_{c′}), det(1 − uT; N′_{c}) = det(1 − uT; N′_{c′})

for all c, c′. Hence

det(1 − uT; N′′_{c}) = det(1 − uT; N′′_{c′}) for all c, c′. Also note that for c′ ≥ c the maps N′′_{c} → N′′_{c′} are injective. The additional assumptions in case B_{K} = K now follow from [8] Theorem 1.3 and Lemma 1.6.

2.7 Shrinking X if necessary we may assume that the module of (p-adically separated) differentials Ω_{A/R}^{1} is free over A. Fix a basis ω_{1}, …, ω_{d}. With respect to this basis, let D be the d×d-matrix of the σ-linear endomorphism of Ω_{A/R}^{1} which the R-algebra endomorphism σ of A induces. Then D^{∧k} = ⋀^{k}(D)

is the matrix of the σ-linear endomorphism of Ω_{A/R}^{k} = ⋀^{k}(Ω_{A/R}^{1}) which σ induces. Let θ = σ^{−1} ∘ Tr be the endomorphism of Ω_{A/R}^{d} constructed in [7] Theorem 8.5. It is a Dwork operator: we have θ(σ(a)y) = aθ(y) for all a ∈ A, y ∈ Ω_{A/R}^{d}. Denote also by θ the Dwork operator on A which we get by transport of structure from θ on Ω_{A/R}^{d} via the isomorphism A ≅ Ω_{A/R}^{d} which sends 1 ∈ A to our distinguished basis element ω_{1} ∧ … ∧ ω_{d} of Ω_{A/R}^{d}. For c ∈ ℕ define the subring A^{c} of A_{K} = A ⊗_{R} K as the image of

T_{n}^{c} ↪ R[X]^{†} ⊗_{R} K → A_{K}.

This is again a K-affinoid algebra, and we have

θ(A^{c}) ⊂ A^{c}

for c >> 0. To see this, choose an R-algebra endomorphism σ̃ of R[X]^{†} which lifts both σ on A and the q-th power Frobenius endomorphism on k[X]. With respect to this σ̃ choose a Dwork operator θ̃ on R[X]^{†} lifting θ on A (as in the beginning of the proof of [8] Theorem 2.3). Then apply [8] Lemma 2.4 which says θ̃(T_{n}^{c}) ⊂ T_{n}^{c}.

2.8 Let M = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I} be a nuclear overconvergent I×I-matrix with entries in Aˆ⊗_{R}B. For c ∈ ℕ let M̌_{I}^{c} be the A^{c} ˆ⊗_{K}B_{K}-Banach module for which the set of symbols {ě_{i}}_{i∈I} is an orthonormal basis. For c ≥ c′ we have the continuous inclusion of B_{K}-algebras A^{c′} ˆ⊗_{K}B_{K} ⊂ A^{c} ˆ⊗_{K}B_{K}, hence a continuous inclusion of B_{K}-modules M̌_{I}^{c′} ⊂ M̌_{I}^{c}. Since M is nuclear overconvergent we have a_{i_{1},i_{2}} ∈ A^{c} ˆ⊗_{K}B_{K} for all c >> 0, all i_{1}, i_{2}. We may thus define for all c >> 0 the B_{K}-linear endomorphism ψ = ψ[M] of M̌_{I}^{c} by

ψ(∑_{i_{1}∈I} b_{i_{1}} ě_{i_{1}}) = ∑_{i_{1}∈I} ∑_{i_{2}∈I}(θ ⊗ 1)(b_{i_{1}}a_{i_{1},i_{2}})ě_{i_{2}}

(b_{i_{1}} ∈ A^{c} ˆ⊗_{K}B_{K}). Clearly these endomorphisms extend each other for increasing c, hence we get an endomorphism ψ = ψ[M] on

M̌_{I} := ⋃_{c>>0} M̌_{I}^{c}.

2.9 Suppose B_{K} = K and I is finite, and M is the matrix of the σ-linear endomorphism ϕ acting on the basis {e_{i}}_{i∈I} of the free A-module M. Then we define ψ[M] as the Dwork operator

ψ[M] : Hom_{A}(M, Ω_{A/R}^{d}) → Hom_{A}(M, Ω_{A/R}^{d}), f ↦ θ ∘ f ∘ ϕ.

This definition is compatible with that in 2.8: Consider the canonical embedding

Hom_{A}(M, Ω_{A/R}^{d}) → Hom_{A}(M, Ω_{A/R}^{d}) ⊗_{R} K ≅^{w} M̌_{I}

where the inverse of the A_{K}-linear isomorphism w sends ě_{i} ∈ M̌_{I} to the homomorphism which maps e_{i} ∈ M to ω_{1} ∧ … ∧ ω_{d} and which maps e_{i′} for i′ ≠ i to 0. This embedding commutes with the operators ψ[M].

Theorem 2.10. For each c >> 0, the endomorphism ψ = ψ[M] on M̌_{I}^{c} is a completely continuous B_{K}-Banach module endomorphism. Its Fredholm determinant det(1 − ψT; M̌_{I}^{c}) is independent of c. Denote it by det(1 − ψT; M̌_{I}). If B_{K} = K, the endomorphism ψ = ψ[M] on M̌_{I} is nuclear in the sense of [8], and its characteristic series as defined in [8] coincides with det(1 − ψT; M̌_{I}).

Proof: Choose a lifting of M = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I} to a nuclear matrix (ã_{i_{1},i_{2}})_{i_{1},i_{2}∈I} with entries in [m, c]_{B}. Also choose a lifting of θ on A to a Dwork operator θ̃ on R[X]^{†} (with respect to a lifting of σ, as in 2.7). Let N_{I}^{c} be the T_{n}^{c} ˆ⊗_{K}B_{K}- Banach module for which the set of symbols {(ě_{i})}_{i∈I} is an orthonormal basis, and define the B_{K}-linear endomorphism ψ̃ of N_{I}^{c} by

ψ̃(∑_{i_{1}∈I} b̃_{i_{1}}(ě_{i_{1}})) = ∑_{i_{1}∈I} ∑_{i_{2}∈I}(θ̃⊗ 1)(b̃_{i_{1}} ã_{i_{1},i_{2}})(ě_{i_{2}})

(b̃_{i_{1}} ∈ T_{n}^{c} ˆ⊗_{K}B). An orthonormal basis of N_{I}^{c} as a B_{K}-Banach module is given by

\{π^{[ \frac{|α|}{c}]}X^{α}(ě_{i})\}_{α∈ℕ_{0}^{n},i∈I}.  (1)

By 1.7 the matrix for ψ̃ in this basis is completely continuous; that is, ψ̃ is completely continuous. If I_{c} ⊂ T_{n}^{c} and I_{∞} ⊂ T_{n} denote the respective ideals generated by the elements g_{1}, …, g_{r} from 2.1, then I_{∞} ∩ T_{n}^{c} is the kernel of T_{n}^{c} → A^{c}, so by 1.4 the sequences

0 → I_{c} → T_{n}^{c} → A^{c} → 0  (2)

are exact for c >> 0. Let H be the B_{K}-Banach module with orthonormal basis the set of symbols {h_{i}}_{i∈I}. From (2) we derive an exact sequence

0 → I_{c} ˆ⊗_{K}H → T_{n}^{c} ˆ⊗_{K}H → A^{c} ˆ⊗_{K}H → 0  (3)

(To see exactness of (3) on the right note that one of the equivalent norms on A^{c} is the residue norm for the surjective map of K-affinoid algebras T_{n}^{c} → A^{c} (this surjection even has a continuous K-linear section as the proof of [3] A2.6.2 shows)). We use the following isomorphisms of T_{n}^{c} ˆ⊗_{K}B_{K}-Banach modules (in (i)) resp. of A^{c} ˆ⊗_{K}B_{K}-Banach modules (in (ii)):

T_{n}^{c} ˆ⊗_{K}H = (T_{n}^{c} ˆ⊗_{K}B_{K})ˆ⊗_{B_{K}} H ≅ N_{I}^{c}, 1 ⊗ h_{i} ↦ (ě_{i})  (i)
A^{c} ˆ⊗_{K}H = (A^{c} ˆ⊗_{K}B_{K})ˆ⊗_{B_{K}} H ≅ M̌_{I}^{c}, 1 ⊗ h_{i} ↦ ě_{i}  (ii)

By 1.5 we find a subset E of ℕ_{0}^{n} ×\{1, …, r\} such that \{π^{[ \frac{|α|+d_{j}}{c} ]}X^{α}g_{j}\}_{(α,j)∈E} is an orthonormal basis of I_{c} over K for all c >> 0. For the B_{K}-Banach modules I_{c} ˆ⊗_{K}H = (I_{c} ˆ⊗_{K}B_{K})ˆ⊗_{B_{K}} H we therefore have the orthonormal basis

\{π^{[ \frac{|α|+d_{j}}{c} ]}X^{α}g_{j} ⊗ h_{i}\}_{(α,j)∈E,i∈I}.  (4)

It is clear that the systems of orthonormal bases (1) resp. (4) make up systems of common orthonormal bases when c increases. (This is why we took pains to prove 1.5; the present argument could be simplified if we could prove the existence of a common orthogonal basis for the system (M̌_{I}^{c})_{c>>0}.) Now let N_{I} = ∪_{c}N_{I}^{c}. From the exactness of the sequences (3) and from the injectivity of the maps M̌_{I}^{c} → M̌_{I}^{c′} for c ≤ c′ we get I_{c} ˆ⊗_{K}H = T_{n}^{c} ˆ⊗_{K}H ∩ Ker(N_{I} → M̌_{I}). Thus the theorem follows from 2.6.

Corollary 2.11.

∏_{r=0}^{d} det(1 − ψ[M ⊗ D^{∧r}]T; M̌_{I})^{(−1)^{r−1}}

is the quotient of entire power series in the variable T with coefficients in B_{K}; in other words, it is a meromorphic function on 𝔸_{K}^{1} ×_{Sp(K)} Sp(B_{K}).

2.12 Let B_{K} = K. We want to define the L-function of a nuclear matrix M = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I} (with entries in Â). For f ∈ ℕ define the f-fold σ-power M^{(σ)^{f}} of M to be the matrix product

M^{(σ)^{f}} := ((a_{i_{1},i_{2}})_{i_{1},i_{2}∈I})(σ(a_{i_{1},i_{2}})_{i_{1},i_{2}∈I}) … ((σ^{f−1}(a_{i_{1},i_{2}}))_{i_{1},i_{2}∈I}).

Let \overline{x} ∈ X be a geometric point of degree f over k, that is, a surjective k-algebra homomorphism \overline{A} → 𝔽_{q^{f}}. Let R_{f} be the unramified extension of R with residue field 𝔽_{q^{f}}, and let x : Â → R_{f} be the Teichmüller lifting of \overline{x} with respect to σ (the unique σ^{f}-invariant surjective R-algebra homomorphism lifting \overline{x}). By (quite severe) abuse of notation we write

M\overline{x} := x(M^{(σ)^{f}}),

the I ×I-matrix with R_{f}-entries obtained by applying x to the entries of M^{(σ)^{f}} — the ”fibre of M in \overline{x}”. The nuclearity condition implies that M\overline{x} is nuclear; equivalently, its transpose is a completely continuous matrix over R_{f} in the sense of [12]. It turns out that the Fredholm determinant det(1 − M\overline{x}T^{deg(\overline{x})}) has coefficients in R, not just in R_{f}. We define the R[[T]]-element

L(M, T) := ∏_{\overline{x}∈X} \frac{1}{det(1 − M\overline{x}T^{deg(\overline{x})})}.

It is trivially holomorphic on the open unit disk. Let T be the set of kvalued points \overline{x} : \overline{A} → k of X. For a completely continuous endomorphism ψ of an orthonormizable K-Banach module we denote by Tr_{K}(ψ) ∈ K its trace.

Theorem 2.13. Let M be a nuclear overconvergent matrix over Â.

(1) For each \overline{x} ∈ T the element

S\overline{x} := ∑_{0≤j≤d}(−1)^{j}Tr((D^{∧d−j})\overline{x})

is invertible in R. For 0 ≤ i ≤ d, we have

Tr_{K}(ψ[M ⊗ D^{∧i}]) = ∑_{\overline{x}∈T} \frac{Tr((D^{∧d−i})\overline{x})Tr(M\overline{x})}{S\overline{x}}.

(2)

L(M, T) = ∏_{r=0}^{d} det(1 − ψ[M ⊗ D^{∧r}]T; M̌_{I})^{(−1)^{r−1}}.

In particular, by 2.11, L(M, T) is meromorphic on 𝔸_{K}^{1}.

Proof: Let J ⊂ A be the ideal generated by all elements of the form a − σ(a) with a ∈ A. Then Spec(A/J) is a direct sum of copies of Spec(R), indexed by T: It is the direct sum of all Teichmüller lifts of elements in T (or rather, their restrictions from Â to A; cf. [8] Lemma 3.3). Let C(A, σ) be the category of finite (not necessarily projective) A-modules (M, ϕ) with a σ-linear endomorphism ϕ, let m(A, σ) be the free abelian group generated by the isomorphism classes of objects of C(A, σ), and let n(A, σ) be the subgroup of m(A, σ) generated by the following two types of elements. The first type is of the form (M, ϕ) − (M_{1}, ϕ_{1}) − (M_{2}, ϕ_{2}) where

0 → (M_{1}, ϕ_{1}) → (M, ϕ) → (M_{2}, ϕ_{2}) → 0

is an exact sequence in C(A, σ). The second type is of the form (M, ϕ_{1} + ϕ_{2}) − (M, ϕ_{1}) − (M, ϕ_{2}) for σ-linear operators ϕ_{1}, ϕ_{2} on the same M. Set K(A, σ) = m(A, σ)/n(A, σ). By the analogous procedure define the group K^{∗}(A, σ) associated with the category of finite A-modules with a Dwork operator relative to σ. (Here we follow the notation in [16]. The notation in [8] is the opposite one !). By [8], both K(A, σ) and K^{∗}(A, σ) are free A/J-modules of rank one. For a finite square matrix N over A we denote by Tr_{A/J}(N) ∈ A/J the trace of the matrix obtained by reducing modulo J the entries of N. Moreover, for such N we view ψ[N] always as a Dwork operator on a (finite) Amodule as in 2.9, i.e. we do not invert π. From [16] sect.3 it follows that ψ[D^{∧i}] can be identified with the standard Dwork operator ψ_{i} on Ω_{A/R}^{i} from [8]. By [8] sect.5 Cor.1 we have

[ψ[D^{∧d}]] ∑_{0≤j≤d}(−1)^{j}Tr_{A/J}(D^{∧d−j}) = [(A, id)]  (1)

in K^{∗}(A, σ), and ∑_{0≤j≤d}(−1)^{j}Tr_{A/J}(D^{∧d−j}) is invertible in A/J. By [8] Theorem 5.2 we also have

[ψ[D^{∧i}]] = Tr_{A/J}(D^{∧d−i})[ψ[D^{∧d}]]  (2)

in K^{∗}(A, σ). To prove the theorem suppose first that M is a finite square matrix. It then gives rise to an element [M] of K(A, σ). By [16] 10.8 we have

[M] = Tr_{A/J}(M)[(A, id)]

in K(A, σ). Application of the homomorphism of A/J-modules

λ_{d−i} : K(A, σ) → K^{∗}(A, σ)

of [16] p.42 gives

[ψ[M ⊗ D^{∧i}]] = Tr_{A/J}(M)[ψ[D^{∧i}]]  (3)

in K^{∗}(A, σ). From (1), (2), (3) we get

[ψ[M ⊗ D^{∧i}]] = \frac{Tr_{A/J}(M)Tr_{A/J}(D^{∧d−i})}{∑_{0≤j≤d}(−1)^{j}Tr_{A/J}(D^{∧d−j})}[(A, id)]

in K^{∗}(A, σ). Taking the R-trace proves (1) in case M is a finite square matrix. Then taking the alternating sum over 0 ≤ i ≤ d gives the additive formulation of (2) in case M is a finite square matrix (see also [16] Theorem 3.1). The case where the index set I for M is infinite follows by a limiting argument from the case where I is finite. We explain this for (2), leaving the easier (1) to the reader. Let P(I) be the set of finite subsets of I. For I′ ∈ P(I), the I′ × I′-sub-matrix M^{I′} = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I′} of M is again nuclear overconvergent. Hence, in view of the finite square matrix case it is enough to show

L(M, T) = lim_{I′∈P(I)} L(M^{I′}, T)  (1)

and for any 0 ≤ r ≤ d also

det(1 − ψ[M ⊗ D^{∧r}]T; M̌_{I}) = lim_{I′∈P(I)} det(1 − ψ[M^{I′} ⊗ D^{∧r}]T; M̌_{I}) (2)

(coefficient-wise convergence). For I′ ∈ P(I) define the I × I-matrix M[I′] = (a_{i_{1},i_{2}}^{I′})_{i_{1},i_{2}∈I} by a_{i_{1},i_{2}}^{I′} = a_{i_{1},i_{2}} if i_{2} ∈ I′ and a_{i_{1},i_{2}}^{I′} = 0 otherwise. For a geometric point \overline{x} ∈ X we may view the fibre matrices M\overline{x} resp. M[I′]\overline{x} for I′ ∈ P(I) as the transposed matrices of completely continuous operators λ\overline{x} resp. λ[I′]\overline{x} acting all on one single K-Banach space E\overline{x} with orthonormal basis indexed by I. And we may view the fibre matrix MI′ ; \overline{x} as the transposed matrix of the restriction of λ[I′]\overline{x} to a λ[I′]\overline{x}-stable subspace of E\overline{x}, spanned by a finite subset of our given orthonormal basis and containing λ[I′]\overline{x}(E\overline{x}). For the norm topology on the space L(E\overline{x}, E\overline{x}) of continuous K-linear endomorphisms of E\overline{x} we find, using the nuclearity of M, that lim_{I′} λ[I′]\overline{x} = λ\overline{x}. Hence it follows from [12] prop.7,c) that

det(1 − M\overline{x}T^{deg(\overline{x})}) = lim_{I′∈P(I)} det(1 − M[I′]\overline{x}T^{deg(\overline{x})}).

But by [12] prop.7,d) we have

det(1 − M[I′]\overline{x}T^{deg(\overline{x})}) = det(1 − MI′ ; \overline{x}T^{deg(\overline{x})}).

Together we get (1). The proof of (2) is similar: By the proof of 1.7 we have indeed

lim_{I′∈P(I)} ψ[M[I′] ⊗ D^{∧r}] = ψ[M ⊗ D^{∧r}]

in the space of continuous K-linear endomorphisms of M̌_{I}^{c}, so [12] prop.7,c) gives

det(1 − ψ[M ⊗ D^{∧r}]T; M̌_{I}^{c}) = lim_{I′∈P(I)} det(1 − ψ[M[I′] ⊗ D^{∧r}]T; M̌_{I}^{c}).

Now the ψ[M[I′] ⊗ D^{∧r}] do not have finite dimensional image in general, but clearly an obvious generalization of [12] prop.7,d) shows

det(1 − ψ[M[I′] ⊗ D^{∧r}]T; M̌_{I}^{c}) = det(1 − ψ[M^{I′} ⊗ D^{∧r}]T; M̌_{I}^{c})

for I′ ∈ P(I). We are done.

3 The Grothendieck group

In this section we introduce the Grothendieck group Δ(Aˆ⊗_{R}B) of nuclear σ-modules. It is useful since on the one hand, formation of the L-function of a given nuclear σ-module factors over this group, and on the other hand, many natural nuclear σ-modules which are not nuclear overconvergent can be represented in this group through nuclear overconvergent ones.

3.1 We will write σ also for the endomorphism σ ⊗ 1 of Aˆ⊗_{R}B = Âˆ⊗_{R}B. For ℓ = 1, 2 let M_{ℓ} be I_{ℓ} × I_{ℓ}-matrices with entries in Aˆ⊗_{R}B, for countable index sets I_{ℓ}. We say M_{1} is σ-similar to M_{2} over Aˆ⊗_{R}B if there exist a I_{1} × I_{2}-matrix S and a I_{2} × I_{1}-matrix S′, both with entries in Aˆ⊗_{R}B, such that SS′ (resp. S′S) is the identity I_{1} × I_{1} (resp. I_{2} × I_{2}) -matrix, and such that S′M_{1}S^{σ} = M_{2} (in particular it is required that all these matrix products converge coefficient-wise in Aˆ⊗_{R}B). Clearly, σ-similarity is an equivalence relation.

3.2 Let m(Aˆ⊗_{R}B) be the free abelian group generated by the σ-similarity classes of nuclear matrices (over arbitrary countable index sets) with entries in Aˆ⊗_{R}B. Let Δ(Aˆ⊗_{R}B) be the quotient of m(Aˆ⊗_{R}B) by the subgroup generated by all the elements [M] − [M′] − [M′′] for matrices M = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I}, M′ = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I′} and M′′ = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I′′} where I = I′ ⊔ I′′ is a partition of I such that a_{i_{1},i_{2}} = 0 for all pairs (i_{1}, i_{2}) ∈ I′ × I′′ (in other words, M is in block triangular form and M′, M′′ are the matrices on the block diagonal). Elements z ∈ Δ(Aˆ⊗_{R}B) can be written as z = [M_{+}] − [M_{−}] with nuclear matrices M_{+}, M_{−}. If {M_{n}}_{n∈ℕ} is a collection of nuclear matrices such that ord_{π}(M_{n}) → ∞ (where ord_{π}(M) = min_{i_{1},i_{2}}{ord_{π}a_{i_{1},i_{2}}} for a matrix M = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I}) and if {ν_{n}}_{n∈ℕ} are integers, then the infinite sum ∑_{n∈ℕ} ν_{n}[M_{n}] can be viewed as an element of Δ(Aˆ⊗_{R}B) as follows: Sorting the ν_{n} according to their signs means breaking up this sum into a positive and a negative summand, so we may assume ν_{n} ≥ 1 for all n. Replacing M_{n} by the block diagonal matrix diag(M_{n}, M_{n}, …, M_{n}) with ν_{n} copies of M_{n} we may assume ν_{n} = 1 for all n. Since all M_{n} are nuclear and ord_{π}(M_{n}) → ∞ the block diagonal matrix M = diag(M_{1}, M_{2}, M_{3}, …) is nuclear. It represents the desired element of Δ(Aˆ⊗_{R}B). Matrix tensor product (see 2.2) defines a multiplication in Δ(Aˆ⊗_{R}B): One checks that

([M_{1,+}] − [M_{1,−}]) ⊗ ([M_{2,+}] − [M_{2,−}])

= [M_{1,+} ⊗ M_{2,+}] − [M_{1,+} ⊗ M_{2,−}] − [M_{1,−} ⊗ M_{2,+}] + [M_{1,−} ⊗ M_{2,−}]

is independent of the chosen representations.

3.3 A more suggestive way to think of Δ(Aˆ⊗_{R}B) is the following. We say that a subset {e_{i}}_{i∈I} of an Aˆ⊗_{R}B-module M is a formal basis if there is an isomorphism of Aˆ⊗_{R}B-modules

{(d_{i})_{i∈I}; d_{i} ∈ Aˆ⊗_{R}B} ≅ M

mapping for any j ∈ I the sequence (d_{i})_{i} with d_{j} = 1 and d_{i} = 0 for i ≠ j to e_{j}. A nuclear σ-module over Aˆ⊗_{R}B is an Aˆ⊗_{R}B-module M together with a σ-linear endomorphism ϕ such that there exists a formal basis {e_{i}}_{i∈I} of M such that the action of ϕ on {e_{i}}_{i∈I} is described by a nuclear matrix M with entries in Aˆ⊗_{R}B, i.e. ϕe_{i} = Me_{i} if we think of e_{i} as the i-th column of the identity I × I matrix. We usually think of a nuclear σ-module over Aˆ⊗_{R}B as a family of nuclear σ-modules over A, parametrized by the rigid space Sp(B_{K}). In the above situation, if S is a (topologically) invertible I × I-matrix with entries in Aˆ⊗_{R}B, then S^{−1}MS^{σ} is the matrix of ϕ in the new formal basis consisting of the elements Se_{i} = e′_{i} of M (if now we think of e′_{i} as the i-th column of the identity I×I-matrix). Hence we can view Δ(Aˆ⊗_{R}B) as the Grothendieck group of nuclear σ-modules over Aˆ⊗_{R}B, i.e. as the quotient of the free abelian group generated by (isomorphism classes of) nuclear σ-modules over Aˆ⊗_{R}B, divided out by the relations [(M, ϕ)]−[(M′, ϕ′)]−[(M′′, ϕ′′)] coming from short exact sequences

0 → (M′, ϕ′) → (M, ϕ) → (M′′, ϕ′′) → 0

which are Aˆ⊗_{R}B-linearly (but not necessarily ϕ-equivariantly) split.

Proposition 3.4. Let B_{K} = K. Let x ∈ Δ(Â) be represented by a convergent series x = ∑_{ℓ∈ℕ} ν_{ℓ}[M_{ℓ}] with nuclear matrices M_{ℓ} over Â. Then the L-series

L(x, T) := ∏_{ℓ∈ℕ} L(M_{ℓ}, T)^{ν_{ℓ}}

is independent of the chosen representation of x. If all M_{ℓ} are nuclear overconvergent, then L(x, T) represents a meromorphic function on 𝔸_{K}^{1}.

Proof: One checks that σ-similar nuclear matrices over Â have the same Lfunction. Indeed, even the Euler factors at closed points of X are the same: they are given by Fredholm determinants of similar (in the ordinary sense) completely continuous matrices. Now let M, M′ and M′′ give rise to a typical relation [M] = [M′]+[M′′] as in our definition of Δ(Aˆ⊗_{R}B). Then one checks that

L(M, T) = L(M′, T)L(M′′, T),

again by comparing Euler factors. And finally it also follows from the Euler product definition that ord_{π}(1 − L(M_{ℓ}, T)) → ∞ if ord_{π}(M_{ℓ}) → ∞. Altogether we get the well definedness of L(x, T). If the M_{l} are nuclear overconvergent, then the L(M_{ℓ}, T) are meromorphic by 2.13 and we get the second assertion.

4 Resolution of unit root parts of rank one

In this section we describe a family version of the limiting module construction. Given a rank one unit root σ-module (M_{unit}, ϕ_{unit}) which is the unit root part of a (unit root ordinary) nuclear σ-module (M, ϕ) and such that ϕ_{unit} acts by a 1-unit a_{i_{0},i_{0}} ∈ Â on a basis element of M_{unit}, we choose an affinoid rigid subspace Sp(B_{K}) of 𝔸_{K}^{1} such that for each ℂ_{p}-valued point x ∈ Sp(B_{K}) ⊂ ℂ_{p} the exponentiation a_{i_{0},i_{0}}^{x} is well defined. Hence we get a rank one σ-module over Aˆ⊗_{R}B. We express its class in Δ(Âˆ⊗_{R}B) through a set (indexed by r ∈ ℤ) of nuclear σ-modules (B^{r}(M), B^{r}(ϕ)) over Aˆ⊗_{R}B which are overconvergent if (M, ϕ) is overconvergent, even if (M_{unit}, ϕ_{unit}) is not overconvergent. Later Sp(B_{K}) will be identified with the set of characters κ : U_{R}^{(1)} → ℂ_{p}^{×} of the type κ(u) = κ_{x}(u) = u^{x} for small x ∈ ℂ_{p}, where U_{R}^{(1)} denotes the group of 1-units in R. To obtain the optimal parameter space for the B^{r}(M) (i.e. the maximal region in ℂ_{p} of elements x for which κ_{x} occurs in the parameter space) one needs to go to the union of all these Sp(B_{K}). This K-rigid space is not affinoid any more; in the case K = ℚ_{p} it is the parameter space B^{∗} from [3]. We will however not pass to this limit here, since for an extension of the associated unit root L-function even to the whole character space we will have another method available in section 6.

Lemma 4.1. Let E be a p-adically separated and complete ring such that E → E ⊗ ℚ is injective and denote again by ord_{p} the natural extension of ord_{p} from E to E ⊗ ℚ.

(i) Let x ∈ E. If ord_{p}(x) > \frac{1}{p−1}, then ord_{p}(\frac{x^{n}}{n!} ) ≥ 0 for all n ≥ 0, and

exp(x) = ∑_{n≥0} \frac{x^{n}}{n!}

converges. (ii) Let x ∈ E. If ord_{p}(x) > 0, then ord_{p}(\frac{x^{n}}{n}) ≥ 0 for all n ≥ 1, and

log(1 + x) = (−1)^{n−1} ∑_{n≥1} \frac{x^{n}}{n}

converges. Moreover, if ord_{p}(x) > β ≥ \frac{1}{p−1}, then ord_{p}(log(1 + x)) > β; if ord_{p}(x) ≥ \frac{1}{p−1} \frac{1}{p^{b}} for some b ∈ ℕ_{0}, then ord_{p}(log(1 + x)) ≥ \frac{1}{p−1} − b.

Proof: Proceed as in [11], p.252, p.356.

4.2 Fix a countable non empty set I and an element i_{0} ∈ I, let I_{1} = I − {i_{0}}. Let M = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I} be a nuclear I × I-matrix over Â. It is called 1-normal if 1−a_{i_{0},i_{0}} ∈ πÂ and if a_{i_{1},i_{2}} ∈ πÂ for all (i_{1}, i_{2}) ≠ (i_{0}, i_{0}). It is called standard normal if a_{i_{1},i_{0}} = 0 for all i_{1} ∈ I_{1}, if a_{i_{0},i_{0}} is invertible in Â and if a_{i_{1},i_{2}} ∈ πÂ for all (i_{1}, i_{2}) ≠ (i_{0}, i_{0}). It is called standard 1-normal if it is both standard normal and 1-normal. That M is standard normal means that the associated σ-module (M, ϕ) has a unique ϕ-stable submodule of rank one on which ϕ acts on a basis element by multiplication with a unit in Â: the unit root part (M_{unit}, ϕ_{unit}) of (M, ϕ). In general, (M_{unit}, ϕ_{unit}) will not be overconvergent even if (M, ϕ) is overconvergent. The purpose of this section is to present another construction of σ-modules departing from (M, ϕ) which does preserve overconvergence and allows us to recapture (M_{unit}, ϕ_{unit}) in Δ(Â), and even certain of its twists.

4.3 For ν ∈ ℚ we define the ℂ_{p}-subsets

D^{≥ν} := {x ∈ ℂ_{p}; ord_{p}(x) ≥ ν}
D^{>ν} := {x ∈ ℂ_{p}; ord_{p}(x) > ν}.

We use these notations also for the natural underlying rigid spaces. Let B(ν)_{K} be the reduced K-affinoid algebra consisting of power series in the free variable V, with coefficients in K, convergent on D^{≥ν} (viewing V as the standard coordinate). Thus

B(ν)_{K} = { ∑_{α∈ℕ_{0}} c_{α}V^{α}; c_{α} ∈ K, lim_{α→∞}(ord_{p}(c_{α}) + να) = ∞}.

4.4 Fix ν ∈ ℚ and let

B := (B(ν)_{K})^{0} := { ∑_{α∈ℕ_{0}} c_{α}V^{α} ∈ B(ν)_{K}; ord_{p}(c_{α}) + να ≥ 0 for all α},

J := {q : I_{1} → ℕ_{0}; q(i) = 0 for almost all i ∈ I_{1}},

C := (Aˆ⊗_{R}B)^{J} = ∏_{J} Aˆ⊗_{R}B.

Define a multiplication in C as follows. Given β = (β_{q})_{q∈J} and β′ = (β′_{q})_{q∈J} in C, the component at q ∈ J of the product ββ′ is defined as

(ββ′)_{q} = ∑_{q_{1}+q_{2}=q^{(q_{1},q_{2})∈J^{2}}} β_{q_{1}}β′_{q_{2}}.

C is p-adically complete. For c ∈ ℕ_{0} we defined [0, c]_{B} in 1.6, and now we let

C_{c} := ([0, c]_{B})^{J} = ∏_{J}[0, c]_{B},

a complete subring of C. We view C as a Aˆ⊗_{R}B-algebra by means of the ring morphism h : Aˆ⊗_{R}B → C defined for y ∈ Aˆ⊗_{R}B by h(y)_{q} = y ∈ Aˆ⊗_{R}B if q ∈ J is the zero map I_{1} → ℕ_{0}, and by h(y)_{q} = 0 ∈ Aˆ⊗_{R}B for all other q ∈ J. In turn,

C ≅ Aˆ⊗_{R}B[[I_{1}]],

the free power series ring on the set I_{1} (viewed as a set of free variables).

4.5 Let μ : S_{1} → S_{2} be a homomorphism of arbitrary R-modules. With I, i_{0}, I_{1} and J from above we now define a homomorphism

λ(μ) : (S_{1})^{I} = ∏_{I} S_{1} → (S_{2})^{J} = ∏_{J} S_{2}.

Given a = (a_{i})_{i∈I} ∈ ∏_{I} S_{1}, the q-component λ(μ)(a)_{q} of λ(μ)(a), for q ∈ J, is defined as follows. If q ∈ J is the zero map I_{1} → ℕ_{0}, then λ(μ)(a)_{q} = μ(a_{i_{0}}) ∈ S_{2}. If there is a i ∈ I_{1} such that q(i) = 1 and q(i′) = 0 for all i′ ∈ I_{1} − {i}, then λ(μ)(a)_{q} = μ(a_{i}) ∈ S_{2} (for this i). For all other q ∈ J we let λ(μ)(a)_{q} = 0 ∈ S_{2}. Returning to the situation in 4.4, the natural inclusion τ : Â → Aˆ⊗_{R}B = Âˆ⊗_{R}B gives us an embedding of Â-modules

λ = λ(τ) : Â^{I} = ∏_{I} Â → C = ∏_{J} Aˆ⊗_{R}B.

It is clear that λ(([0, c]_{R})^{I}) ⊂ C_{c}.

4.6 Now let M = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I} be a nuclear and 1-normal I × I-matrix over Â. Then

μ := inf(\{ord_{p}(a_{i_{0},i_{0}} − 1)\} ∪ \{ord_{p}(a_{i_{1},i_{2}}); (i_{1}, i_{2}) ≠ (i_{0}, i_{0})\}) ≥ \frac{1}{e} > 0.

If μ > \frac{1}{p−1} choose ν ∈ ℚ such that ν > \frac{1}{p−1} − μ. If only μ ≥ \frac{1}{p^{b}} \frac{1}{p−1} for some b ∈ ℕ_{0} choose ν ∈ ℚ such that ν > b. With this ν define B and C as above. We view M as the set, indexed by i_{2} ∈ I, of its columns

a_{(i_{2})} := (a_{i_{1},i_{2}})_{i_{1}∈I} ∈ Â^{I}.

For each r ∈ ℤ we now define a J × J-matrix B^{r}(M) = (b_{q_{1},q_{2}}^{(r)})_{q_{1},q_{2}∈J} over Aˆ⊗_{R}B associated with M. To define B^{r}(M) it is enough to define the set, indexed by q_{2} ∈ J, of the columns

b_{(q_{2})}^{(r)} = (b_{q_{1},q_{2}}^{(r)})_{q_{1}∈J} ∈ ∏_{J} Aˆ⊗_{R}B = C

of B^{r}(M). Using the ring structure of C we define

b_{(q_{2})}^{(r)} := λ(a_{(i_{0})})^{V} λ(a_{(i_{0})})^{r} \frac{∏_{i∈I_{1}} λ(a_{(i)})^{q_{2}(i)}}{λ(a_{(i_{0})})^{|q_{2}|}} .

Here |q| = ∑_{i∈I_{1}} q(i) for q ∈ J, and λ(a_{(i_{0})})^{V} ∈ C is defined as

λ(a_{(i_{0})})^{V} := exp(V log(λ(a_{(i_{0})}))).

For this to make sense note that ord_{p}(λ(a_{(i_{0})}) − 1_{C}) ≥ μ > \frac{1}{p−1} (resp. ord_{p}(λ(a_{(i_{0})}) − 1_{C}) ≥ μ ≥ \frac{1}{p^{b}} \frac{1}{p−1}), hence ord_{p}(log(λ(a_{(i_{0})}))) ≥ μ (resp. ord_{p}(log(λ(a_{(i_{0})}))) ≥ \frac{1}{p−1} − b) by 4.1(ii). Thus V log(λ(a_{(i_{0})})) is, in view of our choice of ν, indeed an element of Aˆ⊗_{R}B, with ord_{p}(V log(λ(a_{(i_{0})}))) ≥ μ + ν > \frac{1}{p−1} (resp. ord_{p}(V log(λ(a_{(i_{0})}))) ≥ \frac{1}{p−1} − b + ν > \frac{1}{p−1}), so we can apply 4.1(i) to it. If the free variable V specializes to integer values, λ(a_{(i_{0})})^{V} specializes to the usual exponentiation by integers of the unit λ(a_{(i_{0})}) in C (just as we use usual exponentiation for the other factors in the above definition of b_{(q_{2})}^{(r)}). Let B_{−}^{r}(M) be the matrix obtained from B^{r}(M) by replacing V with −V (i.e. the matrix defined by the same recipe, but now using λ(a_{(i_{0})})^{−V} in place of λ(a_{(i_{0})})^{V} as the first factor of b_{(q_{2})}^{(r)}).

4.7 The particular choice of ν made in 4.6 will play no role in the sequel. However, there is some theoretical interest in taking ν as small as possible: the smaller ν, the larger D^{≥ν} which is the parameter space for our families of σ-modules defined by the matrices B^{r}(M) and B_{−}^{r}(M). The ultimate result 6.11 on the family of twisted unit root L-functions does not depend on the choice (in the prescribed range) of ν here: for 6.11 it is not important how far the family extends, we only need to extend it to D^{≥ν} for some ν < ∞. But we get trace formulas, which are important for a further qualitative study, only for those members of this family of twisted unit root L-functions whose parameters (=locally K-analytic characters) are in D^{≥ν}.

Proposition 4.8. The matrices B^{r}(M) and B_{−}^{r}(M) are nuclear. If M is nuclear overconvergent, then B^{r}(M) and B_{−}^{r}(M) are nuclear overconvergent.

Proof: Nuclearity: Given M > 0, we need to show ord_{π}(b_{(q_{2})}^{(r)}) > M for all but finitely many q_{2} ∈ J. It is clear that ord_{π}(λ(a_{(i_{0})})^{V} λ(a_{(i_{0})})^{m}) = 0 for all m ∈ ℤ, therefore we need to concentrate only on the factors ∏_{i∈I_{1}} λ(a_{(i)})^{q_{2}(i)}. By nuclearity of M we know that ord_{π}(λ(a_{(i)})) = ord_{π}(a_{(i)}) > M for all but finitely many i ∈ I_{1}. Therefore we need to concentrate only on those q_{2} with support inside this finite exceptional subset of I_{1}. Among these q_{2} we have |q_{2}| > M for all but finitely many q_{2}. But |q_{2}| > M (and ord_{π}(a_{(i)}) ≥ 1 for all i ∈ I_{1}) implies

ord_{π}( ∏_{i∈I_{1}} λ(a_{(i)})^{q_{2}(i)}) = ∑_{i∈I_{1}} q_{2}(i)ord_{π}(a_{(i)}) ≥ ∑_{i∈I_{1}} q_{2}(i) = |q_{2}| > M.

Nuclearity is established. Now assume M is nuclear overconvergent. Then it can be lifted to a nuclear, overconvergent and 1-normal matrix M̃ = (ã_{i_{1},i_{2}})_{i_{1},i_{2}∈I} with entries in R[X]^{†}. Then, perhaps increasing the c from our nuclearity condition, there is a c ∈ ℕ such that ã_{i_{1},i_{2}} ∈ [0, c]_{R} for all (i_{1}, i_{2}) ≠ (i_{0}, i_{0}), and also ã_{i_{0},i_{0}} − 1 ∈ [0, c]_{R}. Then all entries of B^{r}(M̃) are in [0, c]_{B}. Hence B^{r}(M̃) is nuclear and overconvergent. Clearly it is a lifting of B^{r}(M), so we are done.

4.9 Now let us look at the σ-module over Aˆ⊗_{R}B defined by the matrix B^{r}(M). By construction, this is the Aˆ⊗_{R}B-module C (which in fact even is a Aˆ⊗_{R}Balgebra), with the σ-linear endomorphism defined by B^{r}(M). We view it as an analytic family, parametrized by the rigid space Sp(B(ν)_{K}) = D^{≥ν}, of nuclear σ-modules over Â; its fibres at points ℤ ∩ D^{≥ν} are Wan’s ”limiting modules” [15]. Yet another description is due to Coleman [4], which we now present (in a slightly generalized form). It will be used in the proof of 4.10. The nuclear matrix M over Â is the matrix in a formal basis {e_{i}}_{i∈I} of a σ-linear endomorphism ϕ on a Â-module M. The element e = e_{i_{0}} ∈ M can also be viewed as an element of the symmetric Â-algebra SymÂ(M) defined by M, so it makes sense to adjoin its inverse to SymÂ(M). Let D be the subring of degree zero elements in SymÂ(M)[\frac{1}{e}]ˆ⊗_{R}B: the Aˆ⊗_{R}B-sub-algebra of SymÂ(M)[\frac{1}{e}]ˆ⊗_{R}B generated by all \frac{m}{e} for m ∈ M. Let I ⊂ D be the ideal generated by all elements \frac{m}{e} for m ∈ M with ϕ(m) ∈ πM, and let B^{r}(M) be the (π, I)-adic completion of D. For all α ∈ (Aˆ⊗_{R}B)^{×}, all m_{1}, m_{2} ∈ M, if we set e′ = αe + πm_{1}, we have

\frac{m_{2}}{e′} = \frac{m_{2}}{αe} ∑_{i=0}^{∞}(\frac{π}{α} \frac{m_{1}}{e})^{i}  (∗)

in B^{r}(M). By our assumptions on M we know ϕ(e)−e ∈ πM. Therefore there exists a unique σ-linear ring endomorphism ψ of B^{r}(M) with ψ(\frac{m}{e}) = \frac{ϕ(m)}{ϕ(e)} for all m ∈ M: Take (∗) as a definition, with e′ = ϕ(e), m_{2} = ϕ(m) and α = 1. Similarly as in 4.6 we can define, for integers r ∈ ℤ, the element

(\frac{ϕ(e)}{e})^{V +r} = exp(V log(\frac{ϕ(e)}{e}))(\frac{ϕ(e)}{e})^{r}

of B^{r}(M). We define the σ-linear endomorphism B^{r}(ϕ) of B^{r}(M) by

B^{r}(ϕ)(y) = (\frac{ϕ(e)}{e})^{V +r}ψ(y)

for y ∈ B^{r}(M). Clearly B^{r}(M) is the matrix of B^{r}(ϕ) acting on the formal basis

\{∏_{i∈I}(\frac{e_{i}}{e})^{q(i)}\}_{q∈J}

of B^{r}(M) over Aˆ⊗_{R}B. The σ-module defined by B_{−}^{r}(M) is described similarly.

Proposition 4.10. The σ-similarity classes (over Aˆ⊗_{R}B) of B^{r}(M) and B_{−}^{r}(M) depend only on the σ-similarity class (over Â) of M.

Proof: We prove this for B^{r}(M), the argument for B_{−}^{r}(M) is the same. It is enough to prove that B^{r}(M), as a Aˆ⊗_{R}B-module together with its σlinear endomorphism B^{r}(ϕ), depends only on the σ-module M. Let M′ = (a′_{i_{1},i_{2}})_{i_{1},i_{2}∈I} be another 1-normal nuclear matrix over Â which is σ-similar to M. We can view M′ as the matrix of the same σ-linear endomorphism ϕ on the same Â-module M, but in another formal basis {e′_{i}}_{i∈I}. For the element e′ = e′_{i_{0}} our assumptions imply ϕ(e′) − e′ ∈ πM. Therefore e′ and e both generate the unit root part modulo π of M, hence there is a α ∈ Â^{×} with e′ − αe ∈ πM. Observe that

αe ≡ e′ ≡ ϕ(e′) ≡ ϕ(αe) ≡ σ(a)ϕ(e) ≡ σ(α)e

modulo πM. Since R^{×} is the subgroup of Â^{×} fixed by σ we may and will assume α ∈ R^{×}. From (∗) in 4.9 it follows that for m ∈ M the element \frac{m}{e′} of the π-adic completion of SymÂ(M)[M^{−1}]ˆ⊗_{R}B actually lies in its subring B^{r}(M). By a symmetry argument we deduce that B^{r}(M) is the same when constructed with respect to e or with respect to e′. Moreover the endomorphism ψ on B^{r}(M) is the same when constructed with respect to e or with respect to e′: it is uniquely determined by its action on B^{r}(M) ∩ SymÂ(M)[M^{−1}]ˆ⊗_{R}B, where it is characterized by ψ(\frac{m_{1}}{m_{2}} ) = \frac{ϕ(m_{1})}{ϕ(m_{2})} for m_{1}, m_{2} ∈ M. Now let

B^{r}(ϕ)′(y) = (\frac{ϕ(e′)}{e′})^{V +r}ψ(y)

for y ∈ B^{r}(M). The needed Aˆ⊗_{R}B-linear endomorphism λ_{r} of B^{r}(M) satisfying λ_{r} ∘ B^{r}(ϕ)′ = B^{r}(ϕ) ∘ λ_{r} we now define to be the multiplication with (\frac{e′}{αe})^{V +r} ∈ B^{r}(M) (by now obviously defined). Here we use that α ∈ R^{×}.

Now suppose M = (a_{i_{1},i_{2}})_{i_{1},i_{2}∈I} is even a standard 1-normal nuclear Â-matrix. Define M_{unit} := a_{i_{0},i_{0}} ∈ Â and (M_{unit})^{V} = exp(V log(M_{unit})) ∈ Aˆ⊗_{R}B as in 4.6.

Theorem 4.11. For s ∈ ℤ we have the following equalities in Δ(Aˆ⊗_{R}B):

[(M_{unit})^{s}(M_{unit})^{V}] = ⨁_{r≥1}(−1)^{r−1}r[B^{s−r}(M) ⊗ ⋀^{r}(M)]

[(M_{unit})^{s}(M_{unit})^{−V}] = ⨁_{r≥1}(−1)^{r−1}r[B_{−}^{s−r}(M) ⊗ ⋀^{r}(M)].

Proof: We prove the first equality, the second is proved similarly. First note that our assumptions imply that π^{r−1} divides ⋀^{r}(M), so the right hand side converges. Since M is standard 1-normal we have B^{s−r}(M) = (M_{unit})^{s}B^{−r}(M) so we may assume s = 0. Let M′′ = (a′′_{i_{1},i_{2}})_{i_{1},i_{2}∈I} be the Â-matrix with a′′_{i_{1},i_{2}} = a_{i_{1},i_{2}} for all (i_{1}, i_{2}) ∈ (I_{1} × I_{1}) ∪ {(i_{0}, i_{0})}, and a′′_{i_{1},i_{2}} = 0 for the other (i_{1}, i_{2}). Since M is standard 1-normal we see that [B^{−r}(M)⊗⋀^{r}(M)] = [B^{−r}(M′′)⊗⋀^{r}(M′′)] in view of the relations divided out in the definition of Δ(Aˆ⊗_{R}B). Hence we may assume M = M′′. Suppose that i_{0} is minimal in the ordering of I (which we tacitly chose to define ⋀^{r}(I) and ⋀^{r}(M), see 2.2). For r ≥ 1 let M_{r} be the Aˆ⊗_{R}B-module (Aˆ⊗_{R}B)^{(J×⋀^{r}(I))}. It has the formal basis (e_{(q,i⃗)})_{q∈J,i⃗∈⋀^{r}(I)}, where e_{(q,i⃗)} is the (q, i⃗)-th column of the identity (J × ⋀^{r}(I)) × (J × ⋀^{r}(I))-matrix. The matrix B^{−r}(M) ⊗ ⋀^{r}(M) describes the action of a σ-linear endomorphism ϕ_{r} of M_{r} on this basis. Actually we will need r copies of (M_{r}, ϕ_{r}) and its formal basis (e_{(q,i⃗)})_{q∈J,i⃗∈⋀^{r}(I)}: We denote them by (M_{r}^{(ℓ)}, ϕ_{r}^{(ℓ)}) and (e_{(q,i⃗)}^{(ℓ)})_{(q,i⃗)} for 1 ≤ ℓ ≤ r. We get the σ-module

(M_{r}^{•}, ϕ_{r}^{•}) = ⊕_{1≤ℓ≤r}(M_{r}^{(ℓ)}, ϕ_{r}^{(ℓ)})

with formal basis

H_{r} = (e_{(q,i⃗)}^{(ℓ)})_{q∈J,i⃗∈⋀^{r}(I),ℓ∈{1,...,r}}.

Define Aˆ⊗_{R}B-linear maps

α_{r}^{(ℓ)} : M_{r}^{(ℓ)} → M_{r+1}^{(ℓ)}
β_{r}^{(ℓ)} : M_{r}^{(ℓ)} → M_{r+1}^{(ℓ+1)}

as follows. For i⃗ = (i_{1}, …, i_{r}) ∈ ⋀^{r}(I) with i_{1} < … < i_{r}, and another i ∈ I, let τ(i⃗, i) = max({t ≤ r; i_{t} < i} ∪ {0}), and if in addition i ≠ i_{τ(i⃗,i)+1} let

[i⃗, i] = (i_{1}, …, i_{τ(i⃗,i)}, i, i_{τ(i⃗,i)+1}, …, i_{r}) ∈ ⋀^{r+1}(I).

For q ∈ J and i ∈ I_{1} with q(i) ≠ 0 define q^{i−} ∈ J by q^{i−}(i′) = q(i′) for i′ ∈ I_{1} − {i}, and q^{i−}(i) = q(i) − 1. Now set

α_{r}^{(ℓ)}(e_{(q,i⃗)}^{(ℓ)}) = e_{(q,[i⃗,i_{0}])}^{(ℓ)}

if i_{1} ≠ i_{0}, and set α_{r}^{(ℓ)}(e_{(q,i⃗)}^{(ℓ)}) = 0 if i_{1} = i_{0}. Set

β_{r}^{(ℓ)}(e_{(q,i⃗)}^{(ℓ)}) = ∑_{t}(−1)^{τ(i⃗,i_{t})}e_{(q^{i_{t}−},[i⃗,i_{t}])}^{(ℓ+1)}

where the sum runs through all 1 ≤ t ≤ r with i_{t} ≠ i_{0}, with i_{t} ≠ i_{τ(i⃗,i_{t})+1} and with q(i_{t}) ≠ 0. One checks that ϕ_{r+1}^{(ℓ)} ∘ α_{r}^{(ℓ)} = α_{r}^{(ℓ)} ∘ ϕ_{r}^{(ℓ)} (use the standard 1-normality of M), and that ϕ_{r+1}^{(ℓ+1)} ∘ β_{r}^{(ℓ)} = β_{r}^{(ℓ)} ∘ ϕ_{r}^{(ℓ)} (use M = M′′). Hence for

ψ_{r}^{•} = ⊕_{1≤ℓ≤r}(α_{r}^{(ℓ)} ⊕ β_{r}^{(ℓ)}) : M_{r}^{•} → M_{r+1}^{•}

we have ϕ_{r+1}^{•} ∘ ψ_{r}^{•} = ψ_{r}^{•} ∘ ϕ_{r}^{•}. Also note that ϕ_{1}^{•} = ϕ_{1}^{(1)} on M_{1}^{•} = M_{1}^{(1)} restricts on the rank one Aˆ⊗_{R}B-submodule M_{0}^{•} spanned by the basis element e_{(0,i_{0})}^{(1)} ∈ J × I = J × ⋀^{1}(I) to a σ-linear endomorphism ϕ_{0} with matrix (M_{unit})^{V}. Let ψ_{0}^{•} : M_{0}^{•} → M_{1}^{•} be the inclusion and consider

0 → M_{0}^{•} →^{ψ_{0}^{•}} M_{1}^{•} →^{ψ_{1}^{•}} M_{2}^{•} →^{ψ_{2}^{•}} ….  (∗)

We saw that this sequence is equivariant for the σ-linear endomorphisms ϕ_{r}^{•} which are described by matrices as occur in the statement of the theorem, so it remains to show that (∗) is split exact; more precisely, that for each r there are disjoint subsets G_{r}^{1} and G_{r}^{2} of M_{r}^{•} with the following properties: ψ_{r}^{•} induces a bijection of sets G_{r}^{2} ≅ G_{r+1}^{1}, and the union G_{r}^{1} ∪ G_{r}^{2} is a formal basis for M_{r}^{•} (transforming under an invertible matrix to the formal basis H_{r}). We let G_{0}^{1} = ∅, G_{0}^{2} = H_{0} = {e_{(0,i_{0})}^{(1)}}. For r ≥ 1 we let

G_{r}^{1} = {ψ_{r−1}^{•}(h); h ∈ H_{r−1}}.

We let G_{r}^{2} be the subset of H_{r} consisting of those e_{(q,i⃗)}^{(ℓ)} with ℓ, q and i⃗ = (i_{1}, …, i_{r}) ∈ ⋀^{r}(I) satisfying one of the following conditions: either

[ℓ = 1 and ((i_{1} ≠ i_{0}) or (i_{1} = i_{0} and ∃(i ∈ I_{1} − {i_{2}, …, i_{r}}) : q(i) ≠ 0))]

or

[ℓ ≠ 1 and ((i_{1} ≠ i_{0} and ∀(1 ≤ k ≤ r)∃(i ∈ I_{1} − {i_{k}}) : q(i) ≠ 0)
or (i_{1} = i_{0} and ∃(i ∈ I_{1} − {i_{2}, …, i_{r}}) : q(i) ≠ 0))].

The desired properties are formally verified, the proof is complete.

Corollary 4.12. Suppose our M is also overconvergent nuclear. Then for each s ∈ ℤ the series

∏_{\overline{x}∈X} \frac{1}{det(1 − (M_{unit})s ; \overline{x}(M_{unit})y ; \overline{x}T^{deg(\overline{x})})}

defines a meromorphic function in the variables T and y on 𝔸_{ℂ_{p}}^{1} × D^{≥ν}, specializing for y ∈ D^{≥ν}(K) to L(M_{unit}^{s+y}, T).

Proof: The series is trivially holomorphic on D^{>0} × D^{≥ν}. We claim that it is equal to

∏_{r≥1}(∏_{i=0}^{d} det(1 − ψ[B^{s−r}(M) ⊗ ⋀^{r}(M) ⊗ D^{∧i}]T)^{(−1)^{i−1}})^{(−1)^{r−1}r}

which clearly extends as desired. It suffices to prove equality at all specializations V = y at K-rational points y ∈ D^{≥ν}(K) (since these y are Zariski dense in D^{≥ν}). But for such y both series coincide with

∏_{r≥1} L(B^{s−r}(M)|_{V =y} ⊗ ⋀^{r}(M), T)^{(−1)^{r−1}r} :

For the series in the statement of 4.12 this follows from 3.4 and 4.11, for the first series written in this proof this follows from 2.13.

5 Weight space W

In this section we describe a K-rigid analytic space W whose set of ℂ_{p}- valued points can be identified with the set of locally K-analytic characters κ : R^{×} → ℂ_{p} occuring in Theorem 0.1.

5.1 For a K-analytic group manifold G (see [1]) we denote by Hom_{K-an}(G, ℂ_{p}^{×}) the group of locally K-analytic characters G → ℂ_{p}^{×}: characters which locally on G can be expanded into power series in dim_{K}(G)-many variables. If K = ℚ_{p} these are precisely the continuous characters G → ℂ_{p}^{×}. The examples relevant

for us are G = R, G = R^{×}, G = U_{R}^{(1)} and G = \overline{U}_{R}^{(1)}, where we write

U_{R}^{(1)} := 1 + πR and \overline{U}_{R}^{(1)} := \frac{U_{R}^{(1)}}{(U_{R}^{(1)})tors} .

To extinguish any confusion, although in these examples G even carries a natural structure of K-rigid group variety, the definition of Hom_{K-an}(G, ℂ_{p}^{×}) does not refer to this (indeed more ”rigid”) structure: local K-analycity of a character κ requires only that κ, as a ℂ_{p}^{×} ⊂ ℂ_{p} valued function on G, can be expanded into convergent power series on each member of some open covering of G — an open covering in the naive sense, not necessarily admissible in the sense of rigid geometry.

5.2 Let G = G_{π} be the Lubin-Tate formal group over R corresponding to our chosen uniformizer π ∈ R (see [9]). For x ∈ R denote by [x] ∈ U.R[[U]] the formal power series which defines the multiplication with x in the formal Rmodule G. The R-module Hom_{O_{ℂp}} (G ˆ⊗O_{ℂ_{p}}, 𝔾_{m,O_{ℂp}}) is free of rank one. Fix a generator with corresponding power series F (Z) ∈ Z.O_{ℂ_{p}}[[Z]]. Substitution yields power series F ([x]) ∈ U.O_{ℂ_{p}}[[U]] for x ∈ R. By [13] we have a group isomorphism

D^{>0} ⟶^{≅} Hom_{K-an}(R, ℂ_{p}^{×})

z ↦ [x ↦ 1 + F ([x])(z)].

Here D^{>0} carries the group structure defined by G. Let m ∈ ℤ_{≥−1} be minimal such that π^{m} log(U_{R}^{(1)}) ⊂ R. Since (U_{R}^{(1)})tors = Ker(log) we have a well defined injective homomorphism of K-analytic group varieties

\overline{U}_{R}^{(1)} ⟶^{θ} R, u ↦ π^{m} log(u) = θ(u)

inducing a homomorphism

Hom_{K-an}(R, ℂ_{p}^{×}) ⟶^{δ} Hom_{K-an}(\overline{U}_{R}^{(1)}, ℂ_{p}^{×}).

Note that Coker(θ) is finite. Thus Ker(δ) is finite, and on the other hand δ is surjective (since ℂ_{p}^{×} is divisible). In other words, Hom_{K-an}(\overline{U}_{R}^{(1)}, ℂ_{p}^{×}) is the quotient of Hom_{K-an}(R, ℂ_{p}^{×}) by a finite subgroup Δ ⊂ Hom_{K-an}(R, ℂ_{p}^{×}). The formal group law G defines a structure of ℂ_{p}-rigid analytic group variety on D^{>0} (with its standard coordinate U). By means of the above isomorphism we view Hom_{K-an}(R, ℂ_{p}^{×}) as its group of ℂ_{p}-valued points. Accordingly, we

view Hom_{K-an}(\overline{U}_{R}^{(1)}, ℂ_{p}^{×}) as the group (D^{>0}/Δ)(ℂ_{p}) of ℂ_{p}-valued points of the ℂ_{p}-rigid group variety D^{>0}/Δ. Let

U_{R}^{(1)} ←^{u} R^{×} →^{v} μ_{q−1}

be the natural projections. We have (U_{R}^{(1)})tors = μ_{p^{a}} for some a ≥ 0. Let

F = {0, …, p^{a} − 1} × {0, …, q − 2}.

For each (s, t) ∈ F let W_{(s,t)} be a copy of the ℂ_{p}-rigid space D^{>0}/Δ, and let

W := ∐_{(s,t)∈F} W_{(s,t)}.

For an element ω ∈ W_{(s,t)}(ℂ_{p}) ⊂ W(ℂ_{p}) we define the character

κ_{ω} : R^{×} → ℂ_{p}, r ↦ u(r)^{s}ω̃(u(r))v(r)^{t} =: κ_{ω}(r)

where ω̃ ∈ Hom_{K-an}(U_{R}^{(1)}, ℂ_{p}^{×}) is the image of ω under the natural map

W_{(s,t)}(ℂ_{p}) ≅ (D^{>0}/Δ)(ℂ_{p}) ≅ Hom_{K-an}(\overline{U}_{R}^{(1)}, ℂ_{p}^{×}) → Hom_{K-an}(U_{R}^{(1)}, ℂ_{p}^{×}).

Since R^{×} = μ_{q−1} × U_{R}^{(1)} we get:

Proposition 5.3. The assignment ω ↦ κ_{ω} defines a bijection

W(ℂ_{p}) ≅ Hom_{K-an}(R^{×}, ℂ_{p}^{×}).

Thus Hom_{K-an}(R^{×}, ℂ_{p}^{×}) can be viewed as the set of ℂ_{p}-valued points of the ℂ_{p}-rigid variety W.

Lemma 5.4. For ν ∈ ℚ, ν > \frac{m}{e} + \frac{1}{p−1}, there exists an open embedding of ℂ_{p}-rigid varieties ι : D^{≥ν} → D^{>0} such that for all x ∈ R and all y ∈ D^{≥ν} we have

1 + F ([x])(ι(y)) = exp(π^{−m}xy).

Proof: Let log_{G} be the logarithm of G. Write F (Z) = Ω.Z +… ∈ Z.O_{ℂ_{p}}[[Z]]. Then we have the identity of formal power series (cf. [13] sect.4)

1 + F ([x]) = exp(Ω log_{G}([x]))

in O_{ℂ_{p}}[[U]]. But log_{G}([x]) = x. log_{G}(U) by [9] 8.6 Lemma 2, therefore it is enough to find ι with

log_{G}(ι(y)) = π^{−m}Ω^{−1}y.

By [9] 8.6 Lemma 4 the power series inverse to log_{G} defines an open embedding exp_{G} : D^{≥β} → D^{>0} for β > \frac{1}{e(q−1)}. Thus ι(y) = exp_{G}(π^{−m}Ω^{−1}y) is appropriate; it is well defined on D^{≥ν} because we have ord_{p}(Ω) = \frac{1}{p−1} − \frac{1}{e(q−1)} by [13], hence ν − \frac{m}{e} − ord_{p}(Ω) > \frac{1}{e(q−1)}.

5.5 Example: Consider the case K = ℚ_{p}, π = p. Then

G = 𝔾_{m},  log_{G}(Z) = log(1 + Z),  m = −1

[x] = (1 + U)^{x} − 1 = ∑_{n≥1} \binom{x}{n}U^{n} (x ∈ ℤ_{p}).

We may choose F (Z) = Z, and for ν > \frac{2−p}{p−1} the associated embedding

ι : D^{≥ν} → D^{>0}; y ↦ ι(y) = exp(py) − 1

is an isomorphism ι : D^{≥ν} ≅ D^{≥ν+1} ⊂ D^{>0}.

6 Meromorphic continuation of unit root L-functions

In this section we prove (the infinite rank version of) Theorem 0.1. Let us give a sketch. For simplicity suppose that α ∈ Â is a matrix of the ordinary unit root part of some nuclear overconvergent σ-module M over A (in the general case, α splits into two factors each of which is of this more special type and can ”essentially” be treated separately). An appropriate multiplicative decomposition of α (see 6.3) allows us to assume that α is a 1-unit. Then the results of section 4, together with the trace formula 2.13 already show meromorphy of L_{α} on 𝔸 × W^{0} for some open subspace W^{0} ⊂ W meeting each component of W: this is essentially what we proved in 4.12. More precisely we get a decomposition of L_{α} into holomorphic functions on 𝔸 × W^{0} which are Fredholm determinants det(ψ) of certain completely continuous operators ψ arising from limiting modules. We express the coefficients of the logarithms of these det(ψ) through the traces Tr(ψ^{f}) of iterates ψ^{f} of these ψ. Then we repeat the limiting module construction in each fibre \overline{x} ∈ X and prove its commutation with its global counterpart. Together with the trace formula 2.13 and the description of the embedding W^{0} → W given in 5.4 this can be used to show that all the functions Tr(ψ^{f}), a priori living on W^{0}, extend to functions on W, bounded by 1. By the general principle 6.1 below this implies the theorem.

Lemma 6.1. For m ∈ ℕ let g_{m}(U) ∈ O_{ℂ_{p}}[[U_{1}, …, U_{g}]]. Suppose there exists a τ > 0 such that

f(T, U) = exp(− ∑_{m=1}^{∞} \frac{g_{m}(U)}{m}T^{m}) ∈ ℂ_{p}[[T, U_{1}, …, U_{g}]]

converges on 𝔸_{ℂ_{p}}^{1} × (D^{≥τ})^{g}, where T resp. U_{1}, …, U_{g} are the standard coordinates on 𝔸_{ℂ_{p}}^{1}, resp. on (D^{≥τ})^{g}. Then f(T, U) converges on all of 𝔸_{ℂ_{p}}^{1}×(D^{>0})^{g}.

Proof: We reduce the convergence of f at a given point x ∈ 𝔸_{ℂ_{p}}^{1} × (D^{>0})^{g} to the convergence of f at regions – chosen in dependence on x – of 𝔸_{ℂ_{p}}^{1} × (D^{≥τ})^{g} with possibly much larger T -coordinates than the T -coordinate of x. For m ≥ 1 let

I_{m} = {i = (i_{1}, …, i_{m}) ∈ (ℕ_{0})^{m}; i_{1} + 2i_{2} + … + mi_{m} = m}.

We may write

f(T, U) = 1 + ∑_{m=1}^{∞} α_{m}(U)T^{m}

α_{m}(U) = ∑_{i∈I_{m}}(−1)^{i_{1}+...+i_{m}} \frac{g_{1}(U)^{i_{1}} … g_{m}(U)^{i_{m}}}{i_{1}! … i_{m}!1^{i_{1}}2^{i_{2}} … m^{i_{m}}} = ∑_{ℓ∈(ℕ_{0})^{g}} β_{m,ℓ}U^{ℓ}

β_{m,ℓ} = ∑_{i∈I_{m}}(−1)^{i_{1}+...+i_{m}} \frac{γ_{m,ℓ,(i_{1},...,i_{m})}}{i_{1}! … i_{m}!1^{i_{1}}2^{i_{2}} … m^{i_{m}}}

for certain γ_{m,ℓ,(i_{1},...,i_{m})} ∈ O_{ℂ_{p}}. We have the estimate

ord_{p}(β_{m,l}) ≥ min_{i∈I_{m}} −ord_{p}(i_{1}! … i_{m}!1^{i_{1}}2^{i_{2}} … m^{i_{m}})

≥ min_{i∈I_{m}} − ∑_{j=1}^{m}(ord_{p}(i_{j}!) + i_{j}(\frac{j}{p}))

≥ min_{i∈I_{m}} − ∑_{j=1}^{m}(i_{j} + i_{j}(\frac{j}{p}))

≥ min_{i∈I_{m}} −2 ∑_{j=1}^{m} ji_{j} = −2m.

Now let (t, u_{1}, …, u_{g}) ∈ 𝔸_{ℂ_{p}}^{1} × (D^{>0})^{g} be given. Set

0 < ρ = min\{1, \frac{ord_{p}(u_{1})}{τ}, …, \frac{ord_{p}(u_{g})}{τ}\} ≤ 1

λ = \frac{ord_{p}(t) − 2(1 − ρ)}{ρ}.

Documenta Mathematica 8 (2003) 1–42

Then we find

ord_{p}(β_{m,ℓ}u^{ℓ}t^{m}) ≥ −2m(1 − ρ) + ρord_{p}(β_{m,ℓ}) + ρ|ℓ|τ + ρmλ + 2m(1 − ρ)
= ρ(ord_{p}(β_{m,l}) + |ℓ|τ + mλ)

and this term tends to infinity as |ℓ| + m tends to infinity since by hypothesis f converges at the points (t̃, ũ_{1}, …, ũ_{g}) with ord_{p}(t̃) ≥ λ and ord_{p}(ũ_{i}) ≥ τ. The lemma follows.

Now let I and i_{0} ∈ I be as in 4.2. In particular we can talk about 1-normality and standard normality of I × I-matrices.

Lemma 6.2. Suppose the nuclear I ×I-matrix M with entries in Â is 1-normal. Then M is σ-similar to a standard 1-normal nuclear I × I-matrix.

Proof: In case A = R[X]^{†}, this is the translation of [15] Lemma 6.5 into matrix terminology. But the proof works for general A.

Lemma 6.3. Let N be a nuclear overconvergent I × I-matrix over A which is σ-similar to a standard normal nuclear I × I-matrix over Â. Then there exist a ξ ∈ A and a nuclear overconvergent 1-normal I × I matrix M over A, both unique up to σ-similarity, such that

(i) the 1 × 1-matrix ξ^{q−1} is σ-similar to 1 ∈ A, and (ii) ξM is σ-similar to N.

Proof: For the existence see Wan [16] (there I is finite, but at this point this is not important). For the uniqueness (which by the way we do not need in the sequel) we follow Coleman [4]. Let ξ′ and M′ be another such pair. Then ξ′ = aξ for some a ∈ A^{×}, hence a^{q−1} = \frac{σ(b)}{b} for some b ∈ A^{×} by hypothesis (i) for ξ and ξ′. On the other hand, from hypothesis (ii) for M and M′ it follows that M′ and \frac{1}{a}M are σ-similar, and by 1-normality of M and M′ this implies a = \frac{dσ(c)}{c} for some c, d ∈ A^{×} with d − 1 ∈ πA. Thus for e = \frac{b}{c^{q−1}} we have d^{q−1} = \frac{σ(e)}{e}. In particular (σ(e) − e) ∈ πA, hence e ∈ R + πA, so we may assume in addition e − 1 ∈ πA. For (the unique) f ∈ A with f^{q−1} = e and f − 1 ∈ πA we then see d = \frac{σ(f)}{f}. Thus a = \frac{σ(ef)}{ef} and it follows that ξ is σ-similar to σ′, and M to M′. We are done.

6.4 Let M be a standard 1-normal nuclear I × I-matrix over Â. Define I_{1} = I − \{i_{0}\} and J as in 4.4. Let \overline{x} be a closed point of X of degree f and write M\overline{x} = (a\overline{x} ; i_{1},i_{2})_{i_{1},i_{2}∈I} for the fibre matrix M\overline{x} with entries a\overline{x} ; i_{1},i_{2} in R_{f} as defined in 2.12. We denote its i_{2}-column for i_{2} ∈ I by

a\overline{x} ; (i_{2}) := (a\overline{x} ; i_{1},i_{2})_{i_{1}∈I} ∈ ∏_{I} R_{f}.

Let

η = 1 + F ([π^{m} log(a\overline{x} ; i_{0},i_{0})]) ∈ O_{ℂ_{p}}[[U]]

with F and m as in 5.2. For r ∈ ℤ we now define a nuclear J × J-matrix B̃^{r}(M\overline{x}) = (b_{q_{1},q_{2}}^{(r),\overline{x}})_{q_{1},q_{2}∈J} with entries in O_{ℂ_{p}}[[U]]. It is enough to give the columns

b_{(q_{2})}^{(r),\overline{x}} := (b_{q_{1},q_{2}}^{(r),\overline{x}})_{q_{1}∈J} ∈ ∏_{j∈J} O_{ℂ_{p}}[[U]],

indexed by q_{2} ∈ J, of B̃^{r}(M\overline{x}). The natural embedding ρ : R_{f} → O_{ℂ_{p}}[[U]] defines a map

λ = λ(ρ) : ∏_{I} R_{f} → ∏_{J} O_{ℂ_{p}}[[U]]

as explained in 4.5. We will also need the O_{ℂ_{p}}[[U]]-algebra structure on ∏_{J} O_{ℂ_{p}}[[U]] analogous to that on C in 4.4. Namely, the one we get from the natural identification

∏_{J} O_{ℂ_{p}}[[U]] ≅ O_{ℂ_{p}}[[U]][[I_{1}]],

the formal power series ring over O_{ℂ_{p}}[[U]] on the set I_{1} (viewed as a set of free variables). Using this O_{ℂ_{p}}[[U]]-algebra structure we set

b_{(q_{2})}^{(r),\overline{x}} := ηλ(a\overline{x} ; (i_{0}))^{r} \frac{∏_{i∈I_{1}} λ(a\overline{x} ; (i))^{q_{2}(i)}}{λ(a\overline{x} ; (i_{0}))^{|q_{2}|}} .

Note that λ(a\overline{x} ; (i_{0})) = a\overline{x} ; i_{0},i_{0} in the O_{ℂ_{p}}[[U]]-algebra ∏_{J} O_{ℂ_{p}}[[U]] since M is

standard normal. Let B̃_{−}^{r}(M\overline{x}) be the matrix defined by the same recipe, but now using η^{−1} in place of η.

6.5 Let ξ ∈ A be a unit, let (s, t) ∈ F, let r_{1}, r_{2} ∈ ℕ, for ℓ = 1 and ℓ = 2 let I^{(ℓ)} be a countable index set, i_{0}^{(ℓ)} ∈ I^{(ℓ)} an element and M_{ℓ} a standard 1-normal (with respect to i_{0}^{(ℓ)}) nuclear I^{(ℓ)} ×I^{(ℓ)}-matrix over Â. Arguing as in 4.8, where we proved that the matrices B^{r}(M) are nuclear, we see that the trace

g_{\overline{x},ξ,M_{1},M_{2}}^{r_{1},r_{2},s,t}(U) := Tr(ξt ; \overline{x}B̃^{s−r_{1}}(M_{1,\overline{x}}) ⊗ ⋀^{r_{1}}(M_{1})\overline{x} ⊗ B̃_{−}^{−s−r_{2}}(M_{2,\overline{x}}) ⊗ ⋀^{r_{2}}(M_{2})\overline{x})

(the fibre ξ\overline{x} ∈ R is defined as in 2.12 by viewing ξ as a 1 × 1-matrix) is a well defined element in O_{ℂ_{p}}[[U]], i.e. the infinite sum of diagonal elements of this tensor product matrix converges in O_{ℂ_{p}}[[U]]. We may view it as a function on D^{>0}. Let ν ∈ ℚ satisfy both 5.4 and the condition from 4.6 for both M_{1} and M_{2} so that we may form the matrices B^{s−r_{1}}(M_{1}) and B_{−}^{−s−r_{2}}(M_{2}) over Aˆ⊗_{R}B with B = (B(ν)_{K})^{0}. Recall the embedding ι : D^{≥ν} → D^{>0} from 5.4 and that we view the free variable V as standard coordinate on the source D^{≥ν}, and the free variable U as standard coordinate on the target D^{>0} of ι.

For a matrix N with coefficients in Aˆ⊗_{R}B and for y ∈ D^{≥ν} we denote by N |_{V =y} the matrix with entries in Â obtained from N by specializing elements a ⊗ V^{n} ∈ Aˆ⊗_{R}B (for n ∈ ℕ_{0}) to a ⊗ y^{n} ∈ Â.

Lemma 6.6. For K-rational points y ∈ D^{≥ν} we have

g_{\overline{x},ξ,M_{1},M_{2}}^{r_{1},r_{2},s,t}(ι(y)) =

Tr(((ξ^{t}B^{s−r_{1}}(M_{1}) ⊗ ⋀^{r_{1}}(M_{1}) ⊗ B_{−}^{−s−r_{2}}(M_{2}) ⊗ ⋀^{r_{2}}(M_{2}))|_{V =y})\overline{x})

Proof: Taking \overline{x}-fibres commutes with ⊗, thus

ξt ; \overline{x}(B^{s−r_{1}}(M_{1})|_{V =y})\overline{x} ⊗ ⋀^{r_{1}}(M_{1})\overline{x} ⊗ (B_{−}^{−s−r_{2}}(M_{2})|_{V =y})\overline{x} ⊗ ⋀^{r_{2}}(M_{2})\overline{x}

= ((ξ^{t}B^{s−r_{1}}(M_{1}) ⊗ ⋀^{r_{1}}(M_{1}) ⊗ B_{−}^{−s−r_{2}}(M_{2}) ⊗ ⋀^{r_{2}}(M_{2}))|_{V =y})\overline{x}.

Therefore it suffices to show

(B^{r}(M)|_{V =y})\overline{x} = B̃^{r}(M\overline{x})|_{U=ι(y)} and (B_{−}^{r}(M)|_{V =y})\overline{x} = B̃_{−}^{r}(M\overline{x})|_{U=ι(y)},

for standard 1-normal nuclear matrices M over Â and r ∈ ℤ. This is essentially a statement on commutation of the two operations ”M ↦ B^{r}(M)” and ”taking the f-fold σ-power of a square matrix”. In our situation this holds since M is standard normal, as we will now explain. For such M we keep the notation from 6.4. From 5.4 it follows that B̃^{r}(M\overline{x})|_{U=ι(y)} is the matrix constructed by the same recipe as B̃^{r}(M\overline{x}), but using

(a\overline{x} ; i_{0},i_{0})^{y} = exp(y log(a\overline{x} ; i_{0},i_{0})) ∈ R_{f}

in place of η. Observe that a\overline{x} ; i_{0},i_{0} = (a_{i_{0},i_{0}})\overline{x} where (a_{i_{0},i_{0}})\overline{x} is defined as in 2.12 by viewing the (i_{0}, i_{0})-entry a_{i_{0},i_{0}} of M as a 1 × 1-matrix — this is because M is standard. In particular we see (a\overline{x} ; i_{0},i_{0})^{y} = ((a_{i_{0},i_{0}})\overline{x})^{y} ∈ R. Let (M, ϕ) be the σ-module over Â such that the action of ϕ on a formal basis {e_{i}}_{i∈I} of M is given by M. As in 4.9 consider the Â-algebra D = SymÂ(M)[\frac{1}{e_{i_{0}}} ]^{0} of degree zero elements in SymÂ(M)[\frac{1}{e_{i_{0}}} ]. Let B^{r}(M) be its completion as in 4.9. Denote by ψ the natural σ-linear ring endomorphism of B^{r}(M) defined by ϕ, as in 4.9. Then B^{r}(M)|_{V =y} is the matrix of the σ-linear endomorphism ψ_{y+r} = B^{r}(ϕ)|_{V =y} = ((a_{i_{0},i_{0}})\overline{x})^{y+r}ψ (use that M is standard). Hence (B^{r}(M)|_{V =y})\overline{x} is the matrix of the R_{f}-linear endomorphism (ψ_{y+r}^{f})\overline{x} which the f-fold iterate ψ_{y+r}^{f} of ψ_{y+r} induces on the fibre B^{r}(M)\overline{x} = B^{r}(M) ⊗Â R_{f} (formed with respect to the Teichmüller lift x : Â → R_{f} of \overline{x}). On the other hand we can view B^{r}(M)\overline{x} as the completion (analogously to 4.9) of Sym_{R_{f}} (M\overline{x})[\frac{1}{e_{i_{0}}} ]^{0} (with M\overline{x} = M ⊗Â R_{f}). Then B̃^{r}(M\overline{x})|_{U=ι(y)} is the matrix of the R_{f}-linear endomorphism ((a_{i_{0},i_{0}})\overline{x})^{y+r}ψ_{f,\overline{x}} of B^{r}(M)\overline{x} where ψ_{f,\overline{x}} is the R_{f}-linear ring endomorphism of B^{r}(M)\overline{x} induced by the endomorphism which the f-fold iterate ϕ^{f} of ϕ induces on M\overline{x}. Thus it remains to show (ψ_{y+r}^{f})\overline{x} = ((a_{i_{0},i_{0}})\overline{x})^{y+r}ψ_{f,\overline{x}}. Now we clearly have (ψ_{y+r}^{f})\overline{x} = ((a_{i_{0},i_{0}})\overline{x})^{y+r}(ψ^{f})\overline{x} with (ψ^{f})\overline{x} the fibre of ψ^{f} in B^{r}(M)\overline{x}. Therefore we conclude using the functoriality (ψ^{f})\overline{x} = ψ_{f,\overline{x}} of the (σ-linear) functor SymÂ(?).

6.7 For f ∈ ℕ let T_{f} be the set of all closed points of X of degree f. Let A_{f} = A ⊗_{R} R_{f}. Note that the f-fold σ-power (D^{∧i})^{(σ)^{f}} (as defined in 2.12) is the matrix describing the endomorphism which the R_{f}-algebra endomorphism σ^{f} ⊗ 1 of A_{f} induces on Ω_{A_{f} /R_{f}}^{i} = Ω_{A/R}^{i} ⊗_{R} R_{f}. Therefore we may apply 2.13 to the situation obtained by base change ⊗_{R}R_{f}, with σ^{f} ⊗ 1 ∈ End(A_{f}) replacing σ ∈ End(A). We get that

S\overline{x} := ∑_{0≤i≤d}(−1)^{i}Tr((D^{∧i})\overline{x})

for \overline{x} ∈ T_{f} is invertible in R_{f}. For 0 ≤ j ≤ d we may define

h_{f,ξ,M_{1},M_{2}}^{r_{1},r_{2},j,s,t}(U) := ∑_{\overline{x}∈T_{f}} \frac{Tr((D^{∧d−j})\overline{x})}{S\overline{x}}g_{\overline{x},ξ,M_{1},M_{2}}^{r_{1},r_{2},s,t} ∈ O_{ℂ_{p}}[[U]],

D_{ξ,M_{1},M_{2}}^{r_{1},r_{2},j,s,t}(T, U) := exp(− ∑_{f=1}^{∞} \frac{h_{f,ξ,M_{1},M_{2}}^{r_{1},r_{2},j,s,t}(U)}{f}T^{f}) ∈ ℂ_{p}[[T, U]].

Theorem 6.8. If M_{1} and M_{2} are σ-similar to 1-normal nuclear overconvergent matrices over A, then D_{ξ,M_{1},M_{2}}^{r_{1},r_{2},j,s,t}(T, U) defines a holomorphic function on 𝔸_{ℂ_{p}}^{1} ×D^{>0}. There exists a nuclear overconvergent matrix N over Aˆ⊗_{R}B which is σ-similar to

ξ^{t}B^{s−r_{1}}(M_{1}) ⊗ ⋀^{r_{1}}(M_{1}) ⊗ B_{−}^{−s−r_{2}}(M_{2}) ⊗ ⋀^{r_{2}}(M_{2}),

and for K-rational points y ∈ D^{≥ν} we have

D_{ξ,M_{1},M_{2}}^{r_{1},r_{2},j,s,t}(T, ι(y)) = det(1 − ψ[N |_{V =y} ⊗ D^{∧j}]T).

Proof: The existence of N follows from 4.8 and 4.10. Next let us make a general remark. For a nuclear overconvergent matrix M over A we defined the completely continuous operator ψ[M] = ψ_{A}[M] in 2.8 relative to the Frobenius endomorphism σ on A. Now consider the f-fold σ-power M^{(σ)^{f}} of M from 2.12 and view it as a matrix over A_{f} = A ⊗_{R} R_{f}. As such we define the K_{f} = R_{f} ⊗ ℚ-linear completely continuous operator ψ_{A_{f}} [M^{(σ)^{f}}] relative to the Frobenius endomorphism σ^{f} on A_{f}. One finds

ψ_{A_{f}} [M^{(σ)^{f}}] = ψ_{A}[M]^{f} ⊗_{K} K_{f}.

We apply this to M = N |_{V =y} ⊗ D^{∧j} and obtain

Tr_{K}(ψ_{A}[N |_{V =y} ⊗ D^{∧j}]^{f}) = Tr_{K_{f}} (ψ_{A}[N |_{V =y} ⊗ D^{∧j}]^{f} ⊗_{K} K_{f})

= Tr_{K_{f}} (ψ_{A^{f}} [(N |_{V =y} ⊗ D^{∧j})^{(σ)^{f}}])

= Tr_{K_{f}} (ψ_{A^{f}} [(N |_{V =y})^{(σ)^{f}} ⊗ (D^{∧j})^{(σ)^{f}}])

= ∑_{\overline{x}∈T_{f}} \frac{Tr((D^{∧d−j})\overline{x})Tr((N |_{V =y})\overline{x})}{S\overline{x}}.

where for the last equality we applied 2.13. But

Tr((N |_{V =y})\overline{x}) = Tr(((ξ^{t}B^{s−r_{1}}(M_{1})⊗⋀^{r_{1}}(M_{1})⊗B_{−}^{−s−r_{2}}(M_{2})⊗⋀^{r_{2}}(M_{2}))|_{V =y})\overline{x})

which by 6.6 is equal to g_{\overline{x},ξ,M_{1},M_{2}}^{r_{1},r_{2},s,t}(ι(y)). Thus the stated formula is proven since its right hand side may be written as

exp(− ∑_{f=1}^{∞} \frac{Tr_{K}(ψ[N |_{V =y} ⊗ D^{∧j}]^{f})}{f}T^{f}).

Furthermore the points ι(y) for K-rational points y ∈ D^{≥ν} are Zariski dense in ι(D^{≥ν}), therefore we get the equality of holomorphic functions

D_{ξ,M_{1},M_{2}}^{r_{1},r_{2},j,s,t}(T, U) = det(1 − ψ[N ⊗ D^{∧j}]T)

on D^{>0} × ι(D^{≥ν}), where in the function on the right hand side we substitute V by ι^{−1}(U). But the right hand side extends to a holomorphic function on 𝔸_{ℂ_{p}}^{1} × ι(D^{≥ν}), since ψ[N ⊗ D^{∧j}] is completely continuous by 2.10. The definition of D_{ξ,M_{1},M_{2}}^{r_{1},r_{2},j,s,t}(T, U) and 6.1 now show the holomorphy on all of 𝔸_{ℂ_{p}}^{1} × D^{>0}, completing the proof.

6.9 Let α ∈ Â be a unit. For closed points \overline{x} ∈ X define α\overline{x} ∈ R as in 2.12 by viewing α as a 1 × 1-matrix. For κ ∈ Hom_{K-an}(R^{×}, ℂ_{p}^{×}) we ask for the twisted L-function

L(α, T, κ) := ∏_{\overline{x}∈X} \frac{1}{1 − κ(α\overline{x})T^{deg(\overline{x})}} .

It can be written as a power series with coefficients in O_{ℂ_{p}}, hence is trivially holomorphic on D^{>0} (in the variable T).

6.10 We say that α ∈ Â is ordinary geometric if there exists a nuclear G × Gmatrix H = (h_{g_{1},g_{2}})_{g_{1},g_{2}∈G} over Â, a non negative integer j ∈ ℕ_{0} and a nested sequence of (j + 1) finite subsets G_{0} ⊂ G_{1} ⊂ … ⊂ G_{j} of the (countable) index set G such that:

(i) H is σ-similar to a nuclear overconvergent matrix over A.

(ii) h_{g_{1},g_{2}} = 0 whenever there is a 0 ≤ ℓ ≤ j with g_{2} ∈ G_{ℓ} and g_{1} ∉ G_{ℓ}. Thus, H is in block triangular form. (iii) π^{ℓ+1} divides h_{g_{1},g_{2}} whenever g_{2} ∉ G_{ℓ}, for all 0 ≤ ℓ ≤ j. (iv) For all 0 ≤ ℓ ≤ j the element

H_{ℓ} := π^{− ∑_{i=1}^{ℓ} i(c_{i}−c_{i−1})} det((h_{g_{1},g_{2}})_{g_{1},g_{2}∈G_{ℓ}})

of Â is a unit, where we set c_{ℓ} = |G_{ℓ}|. Set H_{−1} = 1.

(v) We have α = H_{j}/H_{j−1} = π^{−j(c_{j}−c_{j−1})} det((h_{g_{1},g_{2}})_{g_{1},g_{2}∈(G_{j}−G_{j−1})}). The meaning of this definition is that α is the determinant of the pure slope j part (as a unit root σ-module) of a nuclear σ-module over A which is ordinary up to slope j and overconvergent (but neither the pure slope j part itself nor its determinant need to be overconvergent). See [16] for details on the Hodge-Newton decomposition by slopes.

Theorem 6.11. Suppose α is ordinary geometric. Then there exists a meromorphic function L_{α} on the ℂ_{p}-rigid space 𝔸_{ℂ_{p}}^{1} × W whose pullback to 𝔸_{ℂ_{p}}^{1} via

𝔸_{ℂ_{p}}^{1} → 𝔸_{ℂ_{p}}^{1} × W, t ↦ (t, κ)

for any κ ∈ Hom_{K-an}(R^{×}, ℂ_{p}^{×}) = W(ℂ_{p}) is a continuation of L(α, T, κ).

Proof: We treat every component W_{(s,t)} of W separately, so let us fix (s, t) ∈ F. Keeping the notation from 6.10 we begin with some definitions. For 0 ≤ ℓ ≤ j let I^{(ℓ)} be the index set of the nuclear matrix ⋀^{c_{ℓ}}(H). Our assumptions on H imply that ⋀^{c_{ℓ}}(H) is standard normal with respect to some i_{0}^{(ℓ)} ∈ I_{ℓ}. Moreover it is σ-similar to a nuclear overconvergent I^{(ℓ)} × I^{(ℓ)} matrix. Thus we may apply 6.3 to get a ξ_{ℓ} ∈ A and a 1-normal (with respect to i_{0}^{(ℓ)}) nuclear overconvergent I^{(ℓ)} × I^{(ℓ)}-matrix M_{ℓ} over A such that ξ_{ℓ}^{q−1} is σ-similar to 1 ∈ A, and ξ_{ℓ}M_{ℓ} is σ-similar to ⋀^{c_{ℓ}}(H). By 6.2 there is a standard 1-normal (with respect to i_{0}^{(ℓ)}) nuclear I^{(ℓ)} × I^{(ℓ)}-matrix M′_{ℓ} over Â which is σ-similar to M_{ℓ}. Let M′_{ℓ,unit} ∈ Â be the (i_{0}^{(ℓ)}, i_{0}^{(ℓ)})-entry of M′_{ℓ}. This is a 1-unit. Then

ξ_{ℓ}M′_{ℓ,unit} ∈ Â is σ-similar to the (i_{0}^{(ℓ)}, i_{0}^{(ℓ)})-entry of ⋀^{c_{ℓ}}(H) which we denote by a_{ℓ}. We will need these definitions for ℓ = j − 1 and ℓ = j if j > 0. If j = 0 we set ξ_{−1} = M_{−1} = M′_{−1} = M′_{−1,unit} = a_{−1} = 1 ∈ R. Our definitions imply α = a_{j}/a_{j−1}, thus if we set ξ = ξ_{j}/ξ_{j−1} and μ = M′_{j,unit}/M′_{j−1,unit} we find that α is σ-similar to ξμ. Let

H(T, U) = ∏_{\overline{x}∈X} \frac{1}{1 − ξt ; \overline{x}μs ; \overline{x}(1 + F ([π^{m} log(μ\overline{x})]))T^{deg \overline{x}}} .

This is a holomorphic function on D^{>0} × D^{>0} where we view T (resp. U) as coordinate for the first (resp. second) factor D^{>0}. Recall from 5.2 the finite étale covering of rigid spaces D^{>0} → W_{(s,t)} which on ℂ_{p}-valued points is given by

D^{>0} → W_{(s,t)}(ℂ_{p}) ≅ Hom_{K-an}(\overline{U}_{R}^{(1)}, ℂ_{p}^{×})

z ⟼ [\overline{w} ↦ 1 + F ([π^{m} log(\overline{w})])(z)].

We see that for any w ∈ U_{R}^{(1)} the holomorphic function 1 + F ([π^{m} log(w)])(U) in the variable U on D^{>0} descends to W_{(s,t)}. Thus our H(T, U) descends to a holomorphic function L_{α,(s,t)} on D^{>0} × W_{(s,t)}. Moreover, for κ ∈ W_{(s,t)}(ℂ_{p}) ⊂ W(ℂ_{p}) = Hom_{K-an}(R^{×}, ℂ_{p}^{×}) the pullback of L_{α,(s,t)} via

D^{>0} → D^{>0} × W_{(s,t)}, t ↦ (t, κ)

is L(α, T, κ): this is immediate since ξ\overline{x}μ\overline{x} = α\overline{x} is the decomposition of α\overline{x} ∈ R^{×} according to R^{×} = μ_{q−1} × U_{R}^{(1)}, for any \overline{x} ∈ X. These considerations also show that for K-rational points y ∈ D^{≥ν} we have

H(T, ι(y)) = L(ξ^{t}μ^{s}μ^{y}, T)  (1)

with ι : D^{≥ν} → D^{>0} from 5.4. To show that L_{α,(s,t)} is meromorphic on 𝔸_{ℂ_{p}}^{1} × W_{(s,t)} it is enough to show that H(T, U) is meromorphic on 𝔸_{ℂ_{p}}^{1} × D^{>0}. Consider the ℂ_{p}[[T, U]]-element

\underline{H}(T, U) = ∏_{r_{1},r_{2}≥1}(∏_{i=0}^{d} D_{ξ,M′_{j},M′_{j−1}}^{r_{1},r_{2},i,s,t}(T, U)^{(−1)^{i−1}})^{(−1)^{r_{1}+r_{2}} r_{1}r_{2}}.

By 6.8 each factor D_{ξ,M′_{j},M′_{j−1}}^{r_{1},r_{2},i,s,t}(T, U) is holomorphic on 𝔸_{ℂ_{p}}^{1} × D^{>0}. Moreover,

since ⋀^{r_{ℓ}}(M′_{ℓ}) is divisible by π^{r_{ℓ}−1} it also follows from 6.8 that ord_{π}(1 − D_{ξ,M′_{j},M′_{j−1}}^{r_{1},r_{2},i,s,t}(T, U)) tends to infinity as r_{1} + r_{2} tends to infinity (if the index set G is finite then the above product is even finite). Therefore \underline{H}(T, U) is meromorphic on 𝔸_{ℂ_{p}}^{1} × D^{>0}. We claim

\underline{H}(T, U) = H(T, U)

as meromorphic functions on D^{>0} × D^{>0}. As in 6.8 it is enough to check this on all subsets D^{>0} × ι(y) ⊂ D^{>0} × D^{>0} for K-rational points y ∈ D^{≥ν}. From 4.11 we get the following equalities in the Grothendieck group Δ(Â):

[ξ^{t}(M′_{j,unit})^{s}(M′_{j,unit})^{y}] = ⨁_{r≥1}(−1)^{r−1}r[ξ^{t}B^{s−r}(M′_{j})|_{V =y} ⊗ ⋀^{r}(M′_{j})]

[(M′_{j−1,unit})^{−s}(M′_{j−1,unit})^{−y}] = ⨁_{r≥1}(−1)^{r−1}r[B_{−}^{−s−r}(M′_{j−1})|_{V =y}⊗⋀^{r}(M′_{j−1})]

(with the notation |_{V =y} explained in 6.5 still in force: V is the standard coordinate on D^{≥ν}). Together

[ξ^{t}μ^{s}μ^{y}] = [\frac{ξ^{t}(M′_{j,unit})^{s}(M′_{j,unit})^{y}}{(M′_{j−1,unit})^{s}(M′_{j−1,unit})^{y}}] =

Documenta Mathematica 8 (2003) 1–42

⨁_{r_{1},r_{2}≥1}(−1)^{r_{1}+r_{2}}r_{1}r_{2}[ξ^{t}(B^{s−r_{1}}(M′_{j})⊗⋀^{r_{1}} M′_{j} ⊗B_{−}^{−s−r_{2}}(M′_{j−1})⊗⋀^{r_{2}} M′_{j−1})|_{V =y}].

Combining with 6.8 and the trace formula 2.13 we get

\underline{H}(T, ι(y)) = L(ξ^{t}μ^{s}μ^{y}, T).  (2)

Comparing (1) and (2) completes the proof.

7 Higher rank

7.1 A finite rank σ-module (M, ϕ) over Â is called ordinary if it admits a separated and exhausting ϕ-stable filtration by free sub-Â-modules

0 = M_{0} ⊂ M_{1} ⊂ M_{2} ⊂ …

of M with free quotients, such that each quotient (M_{i}/M_{i+1}, ϕ) is of the form (U_{i}, π^{i}.ϕ_{i}) where (U_{i}, ϕ_{i}) is a unit root σ-module; that is, Â_{σ} ⊗ U_{i} → U_{i}, a ⊗ u ↦ a.ϕ_{i}(u) is bijective. The (U_{i}, ϕ_{i}) are called the graded pieces of (M, ϕ), and (U_{0}, ϕ_{0}) is called the unit root part of (M, ϕ), also denoted by ϕ_{unit}.

Theorem 7.2. (Hodge-Newton slope decomposition for overconvergent σmodules) Let M be the matrix, in some basis, of a graded piece of an ordinary and overconvergent finite rank σ-module (M, ϕ) over Â. Then there exists a convergent series representation

[M] = ∑_{r≥1} ±[C_{r}]

in Δ(Â) with nuclear overconvergent matrices C_{r} over A.

Proof: (1) By induction on m we prove that for each m ∈ ℕ_{0} there exist finite index sets J_{m}^{1}, J_{m}^{2}, ordinary overconvergent σ-modules α_{t} and β_{t} of finite rank for each t ∈ J_{m}^{1} ∪ J_{m}^{2}, with (β_{t})_{unit} of rank one, and integers m_{t} ≥ m for each t ∈ J_{m}^{2}, such that

[M] = ( ∑_{t∈J_{m}^{2}} ±[π^{m_{t}}(α_{t})_{unit} ⊗ (β_{t})_{unit}^{−1}]) + ( ∑_{t∈J_{m}^{1}} ±[α_{t} ⊗ (β_{t})_{unit}^{−1}]) (∗)

in Δ(Â). Here, by abuse of notation, we identify a finite rank σ-module with the σ-similarity class of matrices it corresponds to. For m = 0 one has [M] = [α_{unit} ⊗ β_{unit}^{−1}] for some α, β, by [16] 6.2. Now let us pass from m to m + 1. Fix t ∈ J_{m}^{2}. Let (α_{t,t′})_{t′∈T_{t}} be the set of higher graded pieces of α_{t} (i.e. (α_{t})_{unit} omitted). By [16] 6.2 again, there exist for each t′ ∈ T_{t} ordinary overconvergent finite rank σ-modules α_{t,t′} and β̃_{t,t′}, with (β̃_{t,t′})_{unit} of rank one, such that [α_{t,t′}] = [(α_{t,t′})_{unit} ⊗ (β̃_{t,t′})_{unit}^{−1}]. Thus

[(α_{t})_{unit} ⊗ (β_{t})_{unit}^{−1}] = [α_{t} ⊗ (β_{t})_{unit}^{−1}] − ∑_{t∈T_{t}}[π^{m_{t′}} (α_{t,t′}) ⊗ (β_{t})_{unit}^{−1}]

= [α_{t} ⊗ (β_{t})_{unit}^{−1}] − ∑_{t∈T_{t}}[π^{m_{t′}} (α_{t,t′})_{unit} ⊗ (β_{t} ⊗ β̃_{t,t′})_{unit}^{−1}]

with integers m_{t′} ≥ 1 (the higher slopes of α_{t}). Inserting this into the formula given by induction hypothesis for m gives the formula for m + 1.

(2) To get the desired convergent series representation for M it is now enough to express, for t ∈ J_{m}^{1}, the terms (β_{t})_{unit}^{−1} in (∗) through overconvergent matrices. This is achieved by factoring β_{t} according to 6.3 and applying 4.11 (with V = 0 and s = −1 there) to the 1-normal overconvergent factor of β_{t}.

7.3 As a corollary of Theorem 7.2 (and 3.4) we recover Wan’s result: that L(M, T) is a meromorphic function on 𝔸^{1}.

References

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Elmar Grosse-Klönne

Mathematisches Institut der Universität Münster Einsteinstrasse 62 48149 Münster klonne@math.uni-muenster.de