On Families of Pure Slope L-Functions

Let R be the ring of integers in a finite extension K of Q p , let k be its residue field and let X: 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 K-analytic characters κ: R × → C × p are the Cp-valued points of a K-rigid space W and that L(κoX,T)=Π weX 1 1-(κoX)(Frob x )Tdeg(x), viewed as a two variable function in T and κ, is meromorphic on A 1 C p × W. On the way we prove, based on a construction of Wan, a slope decomposition for ordinary overconvergent (finite rank) σ-modules, in the Grothendieck group of nuclear σ-modules.


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 n+1 m 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 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 d p r fixes the orbit pointwise).The zeta function of Y y /F p r is On the other hand for a character Ψ : F p → C define the Kloosterman sum and let L Ψ (Y, T ) be the series such that Then, as series, 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 -adic units for any prime = p.We ask for their p-adic valuation and their variation with y.
Embedding Q → 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 (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 1  Cp , 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∈(Gm)0/Fp Cp × 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 A 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 A (resp. over A) is a finite rank free A-module (resp.A-module) together with a σ-linear endomorphism φ.A finite rank σ-module over A is called overconvergent if it arises by base change A → A from a finite rank σ-module over A. Let the finite rank overconvergent σ-module Φ over A 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 1  Cp .Moreover he proved the same for powers (=iterates of the σ-linear endomorphism) Φ k j of Φ j and showed that in case Φ j is of rank one the family {L(Φ k j , 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-modulesat 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 k j , 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 ∆( A) of nuclear σ-modules over A. More precisely, we show that any Φ j as above, not necessarily overcon-vergent, can be written, in ∆( A), 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 k j , 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 ∈ A × on a basis element.For x ∈ X a closed point of degree f let x : A → R f be its Teichmüller lift, where R f denotes the unramified extension of R of degree f .Then lies in R × .We prove that for any locally K-analytic character κ : Cp , 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: 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 1 Cp × 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 A 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 open subspace W 0 of W which meets every component of W: the subspace of characters of the type κ(r) = r u(r) x for ∈ 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 k j , 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 ∈ 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 1 Cp × 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 1 Cp × 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 A 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 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.

Orthonormal bases of overconvergent ideals
In this preparatory section we determine explicit orthonormal K-bases of ideals in overconvergent K-Tate algebras T c n (1.5).Furthermore we recall the complete continuity of certain Dwork operators (1.7).
1.1 For c ∈ N we let where as usual |α| = n i=1 α i for α = (α 1 , . . ., α n ) ∈ N n 0 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 We view T c n as a K-Banach module with the unique norm |.| c for which {π [ |α| c ] X α } α∈N n 0 is an orthonormal basis (this norm is not power multiplicative).Suppose we are given elements g be the reduction of g j , let d j = deg(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 We are done.

1.3
The Tate algebra in n variables over K is the algebra n , 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 . This is the norm for which {X α c } α∈N n 0 is an orthonormal basis over K c .For j ∈ {1, . . ., r} write g j = β∈N n 0 b β X β with b β ∈ K.Then, by a computation similar to that in 1.2 we find In particular it follows that |π In particular it follows on the one hand that we only need to show that {π and on the other hand it follows (applying the above with c instead of c) that {π X α g j } (α,j)∈N n 0 ×{1,...,r} with the reductions of the elements of the set {π |α|+d j c X α g j } (α,j)∈N n 0 ×{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 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 For positive integers m and c let (the π-adically completed tensor product).Note that for m, c 1 , c 2 ∈ N with Suppose that M is nuclear, i.e. that for each M > 0 there are only finitely many Then for all c >> 0 and all M > 0 there are only finitely many pairs 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 i1,i2 ∈ [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 for all α, and thus 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 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 c {α,i1}{β,i2} ) ≥ M for all i 2 ≥ N (α, M ), for all i 1 , all β.Now set Then we find inf β,i1 ord π G c {α,i1}{β,i2} ≥ M whenever |α| + i 2 ≥ N (M ).We are done.

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(A) be a smooth affine connected k-scheme of dimension d.So A is a smooth k-algebra.By [6] it can be represented as A = A/πA where By [10] we can lift the q-th power Frobenius endomorphism of 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 i1,i2 ) i1,i2∈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 i1 ord π (a i1,i2 ) < 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 i1,i2 ) i1,i2∈I with entries in A ⊗ R B is called nuclear overconvergent if there exist positive integers m, c and a nuclear matrix 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 Similarly, if I is finite, any I × I-matrix with entries in A is automatically nuclear.Now choose an ordering of the index set H.
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 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 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 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: 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 1 A/R 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 1  A/R which the R-algebra endomorphism σ of A induces.Then is the matrix of the σ-linear This is again a K-affinoid algebra, and we have for c >> 0. To see this, choose an R-algebra endomorphism σ of R[X] † which lifts both σ on A and the q-th power Frobenius endomorphism on k[X].With respect to this σ choose a Dwork operator θ 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 θ(T c n ) ⊂ T c n .
2.8 Let M = (a i1,i2 ) i1,i2∈I be a nuclear overconvergent I ×I-matrix with entries in A ⊗ R B. For c ∈ N let M c I be the A c ⊗ 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 We may thus define for all c >> 0 the 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 This definition is compatible with that in 2.8: Consider the canonical embedding where the inverse of the A K -linear isomorphism w sends ěi ∈ MI 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 on MI is nuclear in the sense of [8], and its characteristic series as defined in [8] coincides with det(1 − ψT ; MI ).
Proof: Choose a lifting of M = (a i1,i2 ) i1,i2∈I to a nuclear matrix ( a i1,i2 ) i1,i2∈I with entries in [m, c] B .Also choose a lifting of θ on A to a Dwork operator θ on R[X] † (with respect to a lifting of σ, as in 2.7).Let N c I be the T c n ⊗ 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 ψ of N c I by ).An orthonormal basis of N c I as a B K -Banach module is given by {π By 1.7 the matrix for ψ in this basis is completely continuous; that is, ψ is completely continuous.If I c ⊂ T c n and I ∞ ⊂ T n denote the respective ideals generated by the elements g 1 , . . ., g r from 2.1, then 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 (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 c n → 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 c n ⊗ K B K -Banach modules (in (i)) resp. of A c ⊗ K B K -Banach modules (in (ii)): By 1.5 we find a subset E of N n 0 ×{1, . . ., r} such that {π [ |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 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 ( M c I ) c>>0 .)Now let From the exactness of the sequences (3) and from the injectivity of the maps M c I → M c I for c ≤ c we get Thus the theorem follows from 2.6. 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 1 K × Sp(K) Sp(B K ).
2.12 Let B K = K.We want to define the L-function of a nuclear matrix M = (a i1,i2 ) i1,i2∈I (with entries in A).For f ∈ N define the f -fold σ-power M (σ) f of M to be the matrix product Let x ∈ X be a geometric point of degree f over k, that is, a surjective k-algebra homomorphism A → F q f .Let R f be the unramified extension of R with residue field F q f , and let x : A → R f be the Teichmüller lifting of x with respect to σ (the unique σ f -invariant surjective R-algebra homomorphism lifting x).By (quite severe) abuse of notation we write the I ×I-matrix with R f -entries obtained by applying x to the entries of M (σ) f -the "fibre of M in x".The nuclearity condition implies that M 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 . It is trivially holomorphic on the open unit disk.Let T be the set of kvalued points x : 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.
(1) For each x ∈ T the element (2) In particular, by 2.11, L(M, T ) is meromorphic on A 1 K .
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 A 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) 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 i A/R from [8].By [8] sect.5 Cor.1 we have 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 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 of [16] p.42 gives in K * (A, σ).From ( 1), ( 2), (3) we get 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. x acting all on one single K-Banach space E x with orthonormal basis indexed by I.And we may view the fibre matrix M I x as the transposed matrix of the restriction of λ[I ] x to a λ[I ] x -stable subspace of E x , spanned by a finite subset of our given orthonormal basis and containing λ[I ] x (E x ).For the norm topology on the space L(E x , E x ) of continuous K-linear endomorphisms of E x we find, using the nuclearity of M, that lim I λ[I ] x = λ x .Hence it follows from [12] prop.7,c) that But by [12] prop.7,d)we have ).
Together we get (1).The proof of ( 2) is similar: By the proof of 1.7 we have indeed lim in the space of continuous K-linear endomorphisms of M c I , so [12] prop.for I ∈ P(I).We are done.

The Grothendieck group
In this section we introduce the Grothendieck group ∆(A ⊗ 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 For = 1, 2 let M be I × I -matrices with entries in A ⊗ R B, for countable index sets I .We say M 1 is σ-similar to M 2 over A ⊗ R B if there exist a I 1 × I 2 -matrix S and a I 2 × I 1 -matrix S , both with entries in A ⊗ 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 A ⊗ R B).Clearly, σ-similarity is an equivalence relation.

Let
) 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 ∆(A ⊗ 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 ∆(A ⊗ R B). Matrix tensor product (see 2.2) defines a multiplication in ∆(A ⊗ R B): One checks that is independent of the chosen representations.

A more suggestive way to think of ∆(
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 A ⊗ R B is an A ⊗ R B-module M together with a σ-linear endomorphism φ such that there exists a formal basis {e i } i∈I of M Documenta Mathematica 8 (2003)  such that the action of φ on {e i } i∈I is described by a nuclear matrix M with entries in A ⊗ 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 A ⊗ 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 A ⊗ 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 ∆(A ⊗ R B) as the Grothendieck group of nuclear σ-modules over A ⊗ R B, i.e. as the quotient of the free abelian group generated by (isomorphism classes of) nuclear σ-modules over Altogether we get the well definedness of L(x, T ).If the M l are nuclear overconvergent, then the L(M , T ) are meromorphic by 2.13 and we get the second assertion.

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 i0,i0 ∈ A on a basis element of M unit , we choose an affinoid rigid subspace Sp(B K ) of A 1 K such that for each C p -valued point x ∈ Sp(B K ) ⊂ C p the exponentiation a x i0,i0 is well defined.Hence we get a rank one σ-module over A ⊗ R B. We express its class in ∆( A ⊗ R B) through a set (indexed by r ∈ Z) of nuclear σ-modules (B r (M ), B r (φ)) over A ⊗ 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 : 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 [11], p.252, p.356.

Fix a countable non empty set I and an element
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 A: 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 ∆( A), and even certain of its twists.

4.3
For ν ∈ Q we define the C p -subsets 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 for all α}, J := {q : I 1 → N 0 ; q(i) = 0 for almost all i ∈ I 1 }, 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 C is p-adically complete.For c ∈ N 0 we defined [0, c] B in 1.6, and now we let is the zero map I 1 → N 0 , and by h(y) q = 0 ∈ A ⊗ R B for all other q ∈ J.In turn, the free power series ring on the set I 1 (viewed as a set of free variables).
Returning to the situation in 4.4, the natural inclusion τ : gives us an embedding of A-modules λ = λ(τ ) : It is clear that λ(([0, c] R ) I ) ⊂ C c .

4.6
Now let M = (a i1,i2 ) i1,i2∈I be a nuclear and 1-normal I × I-matrix over A. Then With this ν define B and C as above.We view M as the set, indexed by i 2 ∈ I, of its columns a (i2) := (a i1,i2 ) i1∈I ∈ A I .
For each r ∈ Z we now define a J × J-matrix B r (M) = (b (r) q1,q2 ) q1,q2∈J over A ⊗ 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 Here |q| = i∈I1 q(i) for q ∈ J, and λ(a (i0) ) V ∈ C is defined as λ(a (i0) ) V := exp(V log(λ(a (i0) ))).
For this to make sense note that ord p (λ(a B r (M), so we are done.
4.9 Now let us look at the σ-module over A ⊗ R B defined by the matrix B r (M).By construction, this is the A ⊗ R B-module C (which in fact even is a A ⊗ 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 A; 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 A is the matrix in a formal basis {e i } i∈I of a σ-linear endomorphism φ on a A-module M .The element e = e i0 ∈ M can also be viewed as an element of the symmetric A-algebra Sym A (M ) defined by M , so it makes sense to adjoin its inverse to Sym A (M ).Let D be the of degree zero elements in Sym Elmar Grosse-Klönne 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 1 r and G 2 r of M • r with the following properties: r+1 , and the union G 1 r ∪ G 2 r is a formal basis for M • r (transforming under an invertible matrix to the formal basis H r ).We let (0,i0) }.For r ≥ 1 we let We let G 2 r be the subset of H r consisting of those e ( ) (q, ı) with , q and ı = (i 1 , . . ., i r ) ∈ r (I) satisfying one of the following conditions: either [ = 1 and ((i 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 defines a meromorphic function in the variables T and y on A 1 Cp × D ≥ν , specializing for y ∈ D ≥ν (K) to L(M s+y unit , T ).Proof: The series is trivially holomorphic on D >0 × D ≥ν .We claim that it is equal to 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 [9] 8.6 Lemma 2, therefore it is enough to find ι with log G (ι(y)) = π −m Ω −1 y.

Example: Consider the case
We may choose F (Z) = Z, and for ν > 2−p p−1 the associated embedding

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 ∈ A 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 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.
Suppose there exists a τ > 0 such that 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 A. 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 = 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 1 a M are σ-similar, and by 1-normality of M and M this implies a = dσ(c) c for some c, d ∈ A × with d − 1 ∈ πA.Thus for e = b c q−1 we have d q−1 = 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 = σ(f ) f .Thus a = 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 A. Define I 1 = I − {i 0 } and J as in 4.4.Let x be a closed point of X of degree f and write M x = (a x i1,i2 ) i1,i2∈I for the fibre matrix M x with entries a x i1,i2 in R f as defined in 2.12.We denote its i 2 -column for i 2 ∈ I by a x (i2) := (a x i1,i2 ) i1∈I ∈ I R f .
We apply this to M = N | V =y ⊗ D ∧j and obtain Tr((D ∧d−j ) x )Tr((N | V =y ) x ) S x .
where for the last equality we applied 2. which by 6.6 is equal to g r1,r2,s,t x,ξ,M1,M2 (ι(y)).Thus the stated formula is proven since its right hand side may be written as Cp × D >0 , completing the proof.
6.9 Let α ∈ A be a unit.For closed points x ∈ X define α 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) := It can be written as a power series with coefficients in O Cp , hence is trivially holomorphic on D >0 (in the variable T ).

6.10
We say that α ∈ A is ordinary geometric if there exists a nuclear G × Gmatrix H = (h g1,g2 ) g1,g2∈G over A, 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.

1 1 −
κ(α i (y))T deg 1 (y) Documenta Mathematica 8 (2003) 1-42 defines a meromorphic function on A 1 However, from our above computations of the values |π |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 |α|+d j c

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 ) Documenta Mathematica 8 (2003) 1-42 in [7] Theorem 8.5.It is a Dwork operator: we have θ(σ(a)y) = aθ(y) for all a ∈ A, y ∈ Ω d A/R .Denote also by θ the Dwork operator on A which we get by transport of structure from θ on Ω d A/R via the isomorphism Clearly these endomorphisms extend each other for increasing c, hence we get an endomorphism ψ = ψ[M] on MI := c>>0 M c I .Documenta Mathematica 8 (2003) 1-42

1 )
For I ∈ P(I), the I × I -sub-matrix M I = (a i1,i2 ) i1,i2∈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 ) (Elmar Grosse-Klönne and for any 0 ≤ r ≤ d also det(1 − ψ[M ⊗ D ∧r ]T ; MI ) = lim I ∈P(I) det(1 − ψ[M I ⊗ D ∧r ]T ; MI ) (2) (coefficient-wise convergence).For I ∈ P(I) define the I × I-matrix M[I ] = (a I i1,i2 ) i1,i2∈I by a I i1,i2 = a i1,i2 if i 2 ∈ I and a I i1,i2 = 0 otherwise.For a geometric point x ∈ X we may view the fibre matrices M x resp.M[I ] x for I ∈ P(I) as the transposed matrices of completely continuous operators λ x resp.λ[I ] 7,c) gives det(1 − ψ[M ⊗ D ∧r ]T ; M c I ) = lim I ∈P(I) det(1 − ψ[M[I ] ⊗ D ∧r ]T ; M c I ).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 ; M c I ) = det(1 − ψ[M I ⊗ D ∧r ]T ; M c I ) m(A ⊗ R B) be the free abelian group generated by the σ-similarity classes of nuclear matrices (over arbitrary countable index sets) with entries inA ⊗ R B. Let ∆(A ⊗ R B) be the quotient of m(A ⊗ R B) by the subgroup generated by all the elements [M] − [M ] − [M ] for matrices M = (a i1,i2 ) i1,i2∈I , M = (a i1,i2) i1,i2∈I and M = (a i1,i2 ) i1,i2∈I where I = I I is a partition of I such that a i1,i2 = 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 ∈ ∆(A ⊗ R B) can be written as z be represented by a convergent series x = ∈N ν [M ] with nuclear matrices M over A. Then the L-series L(x, T ) := ∈N L(M , T ) ν is independent of the chosen representation of x.If all M are nuclear overconvergent, then L(x, T ) represents a meromorphic function on A 1 K .Proof: One checks that σ-similar nuclear matrices over A 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 ∆(A ⊗ 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

Proposition 4 . 10 .
B generated by all m e for m ∈ M .Let I ⊂ D be the ideal generated by all elements 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 ∈ (A ⊗ R B) × , all m 1 , m 2 ∈ M , if we set e = αe + πm 1 , we have 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( m e ) = 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 ( φ(e) e ) V +r = exp(V log( B r (M ).We define the σ-linear endomorphism B r (φ) of B r (M ) by B r (φ)(y) = ( 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 ( e i e ) q(i) } q∈J of B r (M ) over A ⊗ R B. The σ-module defined by B r − (M) is described similarly.The σ-similarity classes (over A ⊗ R B) of B r (M) and B r − (M) depend only on the σ-similarity class (over A) of M.