Fractional differentiability overnon-Archimedean �elds

Enno Nagel

These note accompany my talk about non-Archimedean fractional differ-entiability in the realms of the p-adic Langlands program held on the 28thof november 2011 at the “Séminaire Automorphe” in Paris.

Contents

0 Prerequisites 2

1 Motivation 2p-adic Langlands for crystalline Galois representations 2The crystalline case 2

2 Our object of study 3Setup 3The universal unitary lattice inside I(χ) 4The universal unitary lattice of the E[P]-module I(χ)(N) 5Gluing the universal unitary lattice from its open cells 7

3 The simplest example 7The universal unitary completion of the compact open cell 7

4 De�nition of R-times differentiable functions for R ∈ R≥0 10Comparison with the Archimedean situation 10Conceptual approach 11Schikhof’s observation 12

5 Investigation of Cr -functions 13Fundamental properties 13Other more explicit descriptions 14

1

References 16

0 Prerequisites

Given a non-Archimedeanly valued complete field F we denote by oF its ring of integers.If its valuation v : F→ R ∪ {∞} is discret, we assume πF be a an element of positivevaluation generating v(F∗). If its residue field κF := oF/πFoF is of characteristic p > 0,we will assume v(p) = 1 and define a multiplicative valuation by |x | = p−v(x ) forx ∈ F.

1 Motivation

p-adic Langlands for crystalline Galois representations

Let F be a p-adic number field, E a sufficiently large p-adic number field serving as thefield of coefficients and G = GLn(F).Definition 1.1. By ap-adic Galois representationwemean a continuous representationof the absolute Galois group Gal(F/F) of a finite extension F over a finite dimensionalE-vector space.

Definition 1.2. By a unitary G-Banach space representation we mean an E-Banachspace with a continuous action ofG such that its topology can be defined by aG-invariantnorm.

Conjecture 1.3. The p-adic Langlands program:

{n-dimensional p-adic Galois-rep’s}↔ {unitary G-Banach space rep’s}

The crystalline case

Example 1.4. Let F = Qp . A filtered ϕ-module V over E is an n-dimensional E-vectorspace with

• an automorphism ϕ,

• and a decreasing, exhaustive and separated filtration . . . ⊇ Vn ⊇ Vn+1 ⊃ . . .indexed over n ∈ Z.

Definition. The filtration type τττ of a filtered ϕ-module V is defined as the sequence ofn integers τ1 ≤ . . . ≤ τn of the filtration’s jumps where an entry’s multiplicity is givenby the dimension of the associated nonzero graded piece.

Breuil and Schneider in [BS07] then associate a locally algebraic representationπ(ϕ) ⊗E ρ(τ) to the datum (ϕ,τττ) as follows:

2

• The construction of π(ϕ): We view the semisimple part ϕss of the endomorphism ϕas the semisimplen-dimensional unramified representation σ : W(F)→ FrobZ →GLn(E) of theWeil group ofGal(Qp/Qp) given by Frob 7→ ϕss. Then by (amodifiedversion of) the classical local Langlands correspondence σ corresponds to a smoothunramified principal series representation π = π(ϕ) of G.

• The construction of ρ(τττ): The n-tuple of increasing integers τττ corresponds (almostcanonically) to a dominant character ψ : T → E∗ of the subgroup of diagonalmatrices T ⊆ G. Then ψ in turn uniquely determines the irreducible rationalG-representation ρ(τττ) of highest weight ψ.

Conjecture 1.5 (See [BS07]). There is a G-invariant norm on π(ϕ) ⊗E ρ(τττ) if and onlyif the filtered ϕ-module V is admissible.

Wewill make neither the definition of an admissible filtered ϕ-module nor the followingdefinition of a crystalline p-adic Galois representation precise. Their importance stemsfrom the following theorem.

Theorem 1.6 (Colmez, Fontaine). We have an equivalence of categories

{admissible filtered ϕ-modules} ↔ {crystalline p-adic Galois representations}⋂full subcategory

⋂full subcategory

{filtered ϕ-modules} {p-adic Galois representations}.Remark. Up to this point we have only made use of the data given by the filtrationtype but not the actual filtration itself. Ideally one would like the family of crystallineGalois representations given by the possible admissible filtrations of V to correspond todifferent completions of π ⊗E ρ by G-invariant norms, but there is no precise conjectureyet.

2 Our object of study

Setup

Let F be a p-adic number field, E a sufficiently large p-adic number field serving as thefield of coefficients and G = GLn(F). Let T ⊆ G be the subgroup of diagonal matriceswith entries in F∗ and T0 ⊆ T its maximal compact open subgroup with entries in o∗F.Let P respectively P be the subgroup of higher respectively lower triangular matrices andN respectively N its subgroup of higher respectively lower triangular matrices whosediagonal entries are all 1. Then N0 ⊂ N denotes the subgroup of matrices with entriesin oF above the diagonal.

3

• Let V be a G-representation. A vector v in V is smooth respectively algebraicrespectively locally algebraic if its orbit map

ov : G→ V

д 7→ д · v

is a locally constant respectively rational respectively locally rational map.

• Let χ : T → E∗ be a character which we extend to P by the natural projectionP→ T. Then we define the G-representation

I(χ) := IndGPχ := {f : G→ E : f (pд) = χ(p) · f (д) for all p ∈ P,д ∈ G}

with the action of G given by right translation.

• We denote by I(χ)lc, I(χ)alg and I(χ)la the G-subrepresentation given by all smooth,algebraic and locally algebraic vectors inside I(χ) respectively. By the action ofG through right-translation these are given by all locally constant respectivelyrational respectively locally rational functions inside I(χ).

Let θ : T → E∗ be an unramified character, that is, it is trivial on T0 and hencefactorizing over T/T0 = (F/o∗F)n = Zn and let ψ : T → E∗ be an algebraic character,that is, its given by a rational function in the entries of T and hence of the form

ψ :*..,

t1. . .

tn

+//-7→ ta11 · · · t

ann

for some a1, . . . ,an ∈ Z.Then θ gives rise to a smooth principal series representation π(θ) = I(θ)lc and if ψ is

dominant (that is, a1 ≤ . . . ≤ an holds) it gives rise to the unique irreducible rationalrepresentation ρ(ψ) = I(ψ)alg of highest dominant weight ψ. Then π(θ)⊗Eρ(ψ) = I(χ)lawith χ = θψ.In the following we will write shorthand I(χ) for I(χ)la.

The universal unitary lattice inside I(χ)Recall that in the p-adic Langlands program one wants to link continuous Galoisrepresentations with unitary G-Banach space representations. One expects the rep-resentation I(χ) with respect to a unitary completion to correspond to a crystallineGalois-representation. The smooth part is obtained by the local Langlands correspon-dence whereas the algebraic part corresponds to the filtration type of associatedϕ-module.

4

Definition. A lattice L in an E-vector space V is an oE-submodule such that for anyv ∈ V, there is λ ∈ E∗ such that λv ∈ L.

Remark. Recall that we have a correspondence between non-Archimedean norms andlattices on an vector space V over a non-Archimedean field E as follows: Given a norm‖·‖, we take L to be its unit ball L = {v ∈ V : ‖v ‖ ≤ 1}. Given a lattice L, it has acorresponding norm ‖·‖L on V given by ‖v ‖L := inf{|λ| : λ ∈ E∗ with λv ∈ L}.Recall that we call a G-representation on a E-Banach space V unitary if the topology

of V may be defined by a G-invariant norm. Given a G-representation V there are anonzero G-invariant seminorm on V if and only if its universal unitary completion asdefined below is nonzero.

Definition. Let V be a G-representation. Then the unitary E-Banach space representa-tion V is the universal unitary completion of V if any E[G]-linear map V → W into aunitary E-Banach space representation W factors uniquely over V.

Remark. It is given by the completion with respect to the smallest G-stable lattice.

Here and in the following we will not distinguish between the commensurabilityclass of a lattice and the lattice itself. Correspondingly for the equivalence class of anorm and the norm itself.Let G be compact. Then given any G-representation V we can equip it by a G-invariant

norm by choosing any one of them ‖·‖0 on V and putting ‖v ‖ = sup{‖д · v ‖0 : д ∈ G}.This should make the following proposition plausible.

Proposition 2.1. The universal unitary lattice of the locally algebraic G-representationI(χ) is given by any lattice finitely generated as an oE[P]-module.

Proof: Follows from the following fulfilled conditions:

• That G is locally profinite and its Iwasawa decomposition G = PK

• That V locally finite dimensional (that is, for each vector exists a compact opensubgroup K ⊆ G such that the E-vector subspace generated by Kv is finitedimensional) and finitely generated as an E[P]-module. �

The universal unitary lattice of the E[P]-module I(χ)(N)We remark that we have a well-defined notion of support inside F := P\G for functionsin I(χ) as P acts on the left through multiplication with invertible scalars on I(χ). Then we

can view N as an open subset inF via the image of the open immersion N ⊆ Gcan.→ P\G.

Thus we can define I(χ)(N) to be the functions inside I(χ) whose support lies in theopen subset N ⊆ F. It is an E[P]-module.

5

Proposition 2.2. We have a P-equivariant injection

I(θ)(N) ⊗ Uψ∼→ Clc

cpt(N,E) ⊗E Uψ ↪→ Clccpt(N,E) ⊗E Calg(N,E) ∼→ Cla

cpt(N,E).Proof: Here the injectivity of the first isomorphism is clear and its surjectivity stemsfrom the fact that there is a system of neighborhoods of unity given by compact opensubgroups admitting an Iwahori factorization. The injectivity of the last map comes fromthe fact that by [Bor91, Theorem 21.20(i)] and the Taylor expansion, any polynomialfunction on N is in characteristic 0 uniquely determined on an open subset in N. Thesurjectivity holds by compactness of support. �

Definition 2.3. We denote the image of this injection by Cψ−lacpt (N,E). It can be described

by

Cψ−lacpt (N,E) = {f : N→ E of cpt. supp. : For all n ∈ N exists open U 3 n

in N such that f|U = p|U for some p ∈ IndGP(ψ)alg}.

Remark. Because N is as an algebraic variety a product of copies of A1 = Spec(K[X]),the function space Calg(N,E) consists of locally polynomial functions.

Corollary 2.4. We conclude that there is an isomorphism of E[P]-modules

I(χ)(N) ∼→ Cψ−lacpt (N,E),

Here Cψ−lacpt (N,E) is endowed with the P-action by f n = (·n) for n ∈ N and f t = χ(t) · f (·t )

for t ∈ T.

Proposition 2.5. Let I(ψ)alg be the unique irreducible algebraic G-representation of highestweight ψ and denote by u its unique — up to a scalar — vector fixed by N. Then

Cψ−lacpt (N,E) = E[P] · 1N0 ⊗ u .

Corollary 2.6. Let T+ be the dominant submonoid of T given by

T+ := {*..,

t1. . .

td

+//-: |t1 | ≥ . . . ≥ |td |} ⊆ T.

Then we find the universal unitary lattice L of the E[P]-module Cψ−lacpt (N,E) to be free if

and only if |χ(t)| ≤ 1 for all t ∈ T+.

Proof: Exhibit a nonzero norm ‖·‖ := ‖·‖Crrr ≤ ‖·‖L attached to differentiable functionsof multiple variables. For this, it suffices for the norm to satisfy the following twoconditions:

(i) It is invariant under translation by N.

(ii) There is a constant C > 0 such that ‖1tN0 ⊗ u‖ ≤ C · 1/|θψ(t)| for all t ∈ T. �

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Gluing the universal unitary lattice from its open cells

Let W be the Weyl group of G with respect to the choice of P and T whose system ofrepresentatives is given by all monomial matrices in G. We have a well-defined operationof W on characters through conjugation.One can show that im δP/δ

wP ⊆ p2Z and conclude that we have a well-defined

unramified character (δP/δwP )1/2 : T→ E∗.

Definition. Put θw := θw (δP/δwP )1/2. We call a character θ : P→ E∗ regular if θw = θonly if w = 1.

Let from now on θ : T→ E∗ be regular.

Corollary 2.7. For w ∈ W put χw := θwψ. Then there are nonzero morphisms ofG-representations Tχw : I(χw )→ I(χ) for all w ∈ W.

Proof: By tensoring the intertwining operators on the smooth part with the identitymorphism on the algebraic one. �

Proposition 2.8. We haveI(χ) =

∑w ∈W

Tχw (I(χ)(N)).

Corollary 2.9. The universal unitary lattice of I(χ) is given by

L =∑w ∈W

oE[P] · Tχw (1N0 ⊗ u).

Proof: By Proposition 2.8 and Proposition 2.5. �

3 The simplest example

The universal unitary completion of the compact open cell

Let F = Qp and n = 2 so that G = GL2(Qp). Let ψ = ψ1 ⊗ ψ2 : T → E∗ be thedominant algebraic character given by the characters ψ1 = ·

l+k and ψ2 = ·l on Q∗p with

l + k ≥ l ∈ Z and θ = θ1 ⊗ θ2 : T → E∗ the unramified character. Put χ = θψ. Thenidentifying N = Qp , we obtain the following description.

Proposition.

I(χ)(N) = Cla≤k (Qp ,E) := {f : Qp → E : f loc. pol. of degree ≤ k with cpt. supp.}with the P-action given by f t = χ(t)f (d/a · _) for all t = � a

d�∈ T and f n = f (· + n)

for n ∈ N.

7

Proof: We remark that in this case the unique irreducible algebraic representationI(ψ)alg of highest weight ψ has a basis given by k-fold products whose factors consist ofthe coordinate functions in the upper row and the determinant function. Restrictingthese functions to N leaves the monomial functions

�1 11

�,�1 X1

�, . . . ,

�1 Xk

1

�. �

Let r := v(χ(� p 1

�)). The universal unitary lattice inside I(χ)(N) is then generated by1Zpx

k . One can convince oneself that for the existence of a P-invariant norm on I(χ)(N)necessarily r ≥ 0 and |χ(Z)| = 1. We are therefore looking for the greatest norm ‖·‖ onCla≤k (Qp ,E) such that

(i) There is a constant C > 0 such that ‖1pnZpxk ‖ ≤ C · p(r−k )n for all n ∈ Z.

(ii) It is invariant under translation.

Definition. A function f : Zp → E is continuously differentiable if

f ]1[(x ,y) := f (x) − f (y)x − y

with x ,y ∈ Zp distinct

extends to a continuous function f [1] : Zp × Zp → E. We denote by C1(Zp ,E) theK-Banach space of all continuously differentiable functions with respect to the normgiven by ‖ f ‖C1 := max{‖ f ‖sup, ‖ f [1]‖sup}.Proposition 3.1. Let r = 1 and k > 0. The completion of Cla≤k (Zp ,E) with respect tothe restriction of ‖·‖ is given by C1(Zp ,E). (And if k = 0 it is given by C1

0(Zp ,E) = {f ∈C1(Zp ,E) : f ′ = 0})Proof: Let L be the universal unitary lattice and let LC1 := Cla≤k ∩B≤1(0) withB≤1(0) := {f ∈ C1(Zp ,E) : ‖ f ‖ ≤ 1} the unit ball inside C1(Zp ,E). We have tocheck that lim

←−−L/πnEE ⊗oE E = lim

←−−LC1/πnELC1 ⊗oE E.

By definition ‖·‖C1 is translation invariant and it is quickly checked that

‖1pnZpxk ‖C1 ≤ |

pkn − 0

pn − pn−1| = C · p(1−k )n

with C := p−1 > 0 for all n ∈ N. Therefore LC1 ⊇ L.The van der Put basis

{e0n := 1n+pl (n)Zp : n ∈ N} ∪ {e1n := 1pl (n)Zpx(· − n) : n ∈ N}with l(0) = 0 and l(n) = blogp(n)c for n > 0 is an orthogonal basis of C1(Zp ,E) with

‖e0n ‖ = pl (n) and ‖e1n ‖ = 1.

In particular B := {pl (n)e0n : n ∈ N} ∪ {e1n : n ∈ N} is a basis of the oE-moduleLC1 ∩ Cla≤1(Zp ,E). Since B ⊆ L∩ Cla≤1(Zp ,E) we find therefore LC1 ∩ Cla≤1(Zp ,E) =

8

L ∩ Cla≤1(Zp ,E). We note that since Cla≤1(Zp ,E) ⊆ C1(Zp ,E) is dense and the norm‖·‖C1 on Cla≤1(Zp ,E) coincides with the one induced by LC1 and LC1 ∩ Cla≤1(Zp ,E),the completion by LC1∩Cla≤1(Zp ,E) is given by C1(Zp ,E). We therefore have inclusions

LC1 ∩ Cla≤1(Zp ,E) ⊆ L ⊆ LC1

corresponding to the following morphisms between their completions

C1(Zp ,E) ↪→ lim←−−n

L/πnEL ⊗oE E ↪→ C1(Zp ,E).

Then it is a general fact that the completion in the middle is also isomorphic to C1(Zp ,E),seen as follows: Let ϕ be right-hand side morphism. Because its image is dense itsuffices to see that it is closed as well. We have ‖·‖C1 ≤ ‖·‖L ≤ ‖·‖C1 . Therefore it is atopological isomorphism onto its image. Since lim

←−−nL/πnEL ⊗oE E is complete its image

im ϕ ⊆ C1(Zp ,E) is complete as well and hence closed. �

Remark. Because I(χ) = I(χ)(N0) ⊕ I(χ)(N1w) with N1 =� 1 pZp0 1

�and w =

�0 11 0

�

and I(χ)(N1w) = I(χ)(N0) = Cla≤k (Zp ,E), we find I(χ) to be a sum of two copiesof Cla≤k (Zp ,E). Therefore Lu ⊇ LCr := Lu ∩ Cla≤k (Zp ,E) ⊕ Lu ∩ Cla≤k (Zp ,E). Bycompleting with respect to these lattices in I(χ) we obtain a mapping

I(χ)Cr := Cr (Zp ,E) ⊕ Cr (Zp ,E)→ I(χ).It is surjective as the lattices’ surrounding vector spaces coincide. If r = 0 it is anisomorphism. Let r > 0. Its kernel L is given by the closed E[P]-module generated bythe functions in I(χ)Cr restricting to 1,x , . . . ,xν on the dense subset N ↪→ P\G withν = dre − 1 < r . This is seen as follows: We firstly have to convince ourselves thatindeed {1,x , . . . ,xν} ⊂ I(χ)Cr . Let F ∈ I(χ)Cr . Put f := F|N : N→ E and f w : Nw → Ewith f w := f (·w) its translate. Then identifying their domains by N = Qp = Nw wecompute that f w|Q∗p is given in terms of f by

f w (x) = θ1/θ2(x)(−x)k f (1/x).This map is locally constant onQ∗p and for f = 1, . . . ,xν can be shown to extend to a Cr -function at 0. (Use the criterion by the Taylor polynomial Formula in [Nag11][ChapterI].) Therefore 1, . . . ,xν ∈ I(χ)Cr .Let i ∈ {0, . . . ,ν}. Let t = � p−n

1

�. Then 1Zpx

i t = χ(t)pni1p−nZpx i = λn1p−nZpx i ∈Lu with λ := pi/α and, since i ≤ ν < r , we have |λ| > 1. Thus we find (fn)n∈N withfn := 1p−nZpx

i ∈ Lu , converging to f := x i in I(χ)Cr , to be a zero sequence in I(χ)and thus 1, . . . ,xν ∈ L.On the other hand, we note that by the relation x i = 0 for i = 0, . . . ,ν in I(χ), we

obtain I(χ)(N0) 3 1Zpxi = f ∈ I(χ)(N1w). Because I(χ)(N0) is unitary under T+N0

9

with T+ = {t = � ad

�∈ T: |a | ≤ |d |} whereas I(χ)(N1w) is unitary under T−N∗1

−1 withT− = {t = � a

d�∈ T : |a | ≥ |d |} and N∗1 = N1 − {1}, we find 1Zp , . . . ,1Zpx

ν to beunitary unter TN = T+T−N0N∗1

−1. As the E[P]-module generated by 1Zp , . . . ,1Zpxν

is dense inside I(χ), we obtain that I(χ)Cr /L is unitary under P = TN. ThereforeI(χ)Cr /L → I(χ) is an isomorphism, the inverse mapping given by the universalproperty of I(χ).

4 De�nition of R-times differentiable functions for R ∈ R≥0

Comparison with the Archimedean situation

We want to show how the classic notion of differentiability over the real numberscompares to the one over non-Archimedeanly valued fields. We will then introduce theclassical approach to iterated differentiability over non-Archimedan vector spaces dueto Schikhof.Let K be a complete valued field, X ⊆ K an open subset and f : X→ K some function.

Recall the function

f ]1[(x ,y) := f (x) − f (y)x − y

for all distinct x ,y ∈ X.

Proposition 4.1. Let K = R. Then f ′ is continuous if and only if f ]1[ extends to acontinuous functions f [1] : X × X→ K.

Proof: A direct consequence of the mean value theorem. �

In general, a lot of fundamental facts in differential calculus over R depends on theintermediate value theorem.The intermediate value theorem implies the

(i) Mean value theorem, which in turn implies

(a) the Local invertibility of functions with nonvanishing differential

(b) the fact that a function of many variables is continuously differentiable ifand only if it is partially continuously differentiable.

(ii) The fundamental theorem of calculus, which in turn implies

(a) the Completeness of C1(X,R).To compensate for the lack of the intermediate value theorem over complete non-

Archimedeanly valued fields — which in fact renders all of these last consequences withthe common Archimedean definition of differentiability false (With the exception of 1b),which I am not sure about for the moment) - we turn Proposition 4.1 into a definition.

10

Definition. A function f : X→ K is continuously differentiable if

f [1](x ,y) := f (x) − f (y)x − y

with x ,y ∈ X distinct

extends to a continuous function f [1] : X × X→ K.

Remark. The space of continuously differentiable functions is then as before endowedwith a natural norm. In fact by this definition all of the above final consequences areeasily shown to remain true.

Problem: Already the first differential quotient f [1] of a one-variable function is afunction of two variables. (In contrast to the situation over an Archimedean field.)Therefore in order to iterate this definition with the aim to obtain a notion of higherdifferentiability for one-variable functions, we must already have established thedefinition of differentiability for a function of many variables.

Conceptual approach

Let V = Kd and E be a K-Banach space. Let e1, . . . ,ed be the canonical basis of V.

Definition. Let X ⊆ V be open. Define for all x + h,x ∈ X with h ∈ K∗d the function

f ]1[ : (x + h,x) 7→ A ∈ HomK(V,E)by

Ahk · ek = f (x + h1e1 + · · · + hk−1 · ek−1 + hk · ek ) − f (x + h1e1 + · · · + hk−1 · ek−1)for k = 1, . . . ,d. Then f is a C1-function if and only if f ]1[ extends to a continuousfunction f [1] : X × X→ E.

Example. Let f : X′ × X′′ → K with X′,X′′ ⊂ K open. Then

f ]1[ = (f ]1,0[, f ]0,1[)with

f ]1,0[(x + h,x) = f (x ′ + h′,x ′′) − f (x)h′

and f ]0,1[(x + h,x) = f (x ′,x ′′ + h′′) − f (x)h′′

for x + h,x ∈ X′ × X′′ with h = (h′,h′′) ∈ K∗ × K∗.

Then we can obtain a notion of ν-fold differentiability for ν ≥ 0 as follows: Letf ∈ C1(X,E) and let us regard the function f [1]. Its domain X × X is again a K-vectorspace with canonical choice of basis and its range HomK(V,E) again K-Banach space. We

11

can therefore iterate this definition by applying it to f [1]. That is, we define f ∈ C2(X,E)if and only if f [1 exists and

f ]2[ = (f [1])]1[ : (X[1])]1[ → HomK(HomK(V,E),E)extends to a continuous function f [2] on X[2]. (Here X]1[ = {(x +h,x) ∈ X2 : h ∈ K∗d}.)We aim to give a definition of r -fold differentiability. Write r = ν+ρ ≥ 0 with ν ∈ N andρ ∈ [0,1[. Then we define ρ-fold differentiability by a strengthened Lipschitz-continuitycondition as follows.

Definition 4.2. Let A ⊆ X and f : A→ E. Then f is Cρ at a point a ∈ X if for all ε > 0there exits Uε 3 a such that

‖ f (x) − f (y)‖ ≤ ε · ‖x − y‖ρ for all x ,y ∈ Uε ∩ A.

Then we can define r -fold differentiability by demanding the ν-th iterated differencequotient f ]ν[ to be Cρ everywhere.

Schikhof’s observation

This definition can also be given point-wise and more concisely by taking into accountthe symmetry properties of the difference quotients as observed by Schikhof. We willsee that for a symmetric function we are hence brought down to checking solely partialdifferentiability in its first coordinate, reducing an exponential growth of parameters toa linear one.

Remark 4.3. Let X ⊆ Kd be open and let F: X × X→ E be a symmetric function. ThenF ∈ C1 if A := F]1,0[(x + h,x ;y) ∈ HomK(V,E) defined for all x ′ + h,x ′ ∈ X,x ′′ ∈ Ywith h ∈ K∗d by

A ·hkek := f (x ′+h1e1+ · · ·+hk−1ek−1+hkek ,x′′)− f (x ′+h1e1+ · · ·+hk−1ek−1,x

′′)extends to a continuous function F[1,0] : (X × X) × X→ E.

The following is in fact the definition of two-fold differentiability as employedin [Sch84].

Example. Let f ∈ C1(X,E) and put F = f [1]. Then F ∈ C1(X × X,E) if and only if f ]2[defined by

f ]2[(x ,y; z) = f [1](x ,z) − f [1](y,z)x − y

for all distinct x ,z ∈ X andy ∈ X extends to a continuous function f [2] : (X×X)×X→ E.

We will now give the explicit definition of an r -times differentiable function for thecase of two variables, f : X × Y → E with X,Y ⊆ K open.

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Definition. • Let X ⊆ K open. Then we define X[k ] = X{0, ...,k} and X]k [ := {x ∈X{0, ...,k} : xi ,x j pairwise distinct } ⊆ X[k].

• We define f [i,j] : X]i[ × Y]j[ → E recursively by

f ]i,j[(x0,x1,x2, . . . ,xi ;y0, . . . ,yj ):=

f ]i−1,j[(x0,x2, . . . ,xi ;y0, . . . ,yj ) − f ]i,j[(x1,x2, . . . ,xi ;y0, . . . ,yj )x0 − x1

=f ]i,j−1[(x0, . . . ,xi ;y0,y2, . . . ,yj ) − f ]i,j[(x0, . . . ,xi ;y1,y2, . . . ,yj )

y0 − y1.

• Then we say that f is Cr at a point (a,b) ∈ X × Y if f ]i,j[ is Cρ at (~a; ~b) =(a, . . . ,a;b, . . . ,b) ∈ X[i] × X[j] for all i, j with i + j = ν.

• We say that f is a Cr -function if f is Cr everywhere and put Cr (X × Y,E) ={Cr -functions f : X × Y → E}.

5 Investigation of Cr -functions

Fundamental properties

Proposition 5.1. Let X ⊆ Kd and Y ⊆ Ke be open and f ∈ Cr (X,Y),д ∈ Cr (Y,E). Thenд ◦ f ∈ Cr (X,E) if r ≥ 1. (If r < 1, then the same conclusion holds provided eitherfunction is Locally Lipschitz-continuous.)

Proof: By the conceptual introduction in the previous chapter, this follows from the factthat the composed function’s difference quotient is given by the composition of thelinear morphisms given by their difference quotients, that is, a matrix product. SinceCρ-functions are closed under pointwise products, the proposition follows. �

As a Corollary we find that there is a well behaved notion of a Cr -manifold.

Corollary 5.2. Let r ≥ 1. Let M be a topological manifold with a Cr -atlas (that is, acovering by charts whose transition maps are given by Cr -functions). Then there is amaximal Cr -atlas of M and a function f : M → E is a Cr -function with respect to themaximal Cr -atlas if and only if it is a Cr -function with respect to any Cr -atlas.

Proof: Follows from the following two facts:

(i) The definition is pointwise, in particular local.

(ii) The Cr -functions are closed under composition.

�

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Lemma 5.3. We find f ∈ Cr (X × Y,E) if and only if f ]i,j[ : X]i[ × Y]j[ → E extends to aCρ-function on X[i] × X[j] for all i, j with i + j = ν.

Remark 5.4. By induction, using the fact that if A ⊆ X is dense and f is a Cρ-function(that is, Cρ everywhere), then it extends to a Cρ-function F: X→ E.

Corollary 5.5. There is a natural topology on the space of Cr -functions which renders itto a complete locally convex K-algebra.

Proof: By the supremum norm of the partial difference quotients f [i,j] over compactsubsets. �

Remark 5.6. For the following propositions to be meaningful, we note Cla(X,E) ⊆C∞(X,E) := ∩r ≥0 Cr (X,E).Proposition 5.7. The locally polynomial functions of degree ≤ ν are dense inside Cr (X,E).Corollary 5.8. The polynomial functions are dense inside Cr (X,E).Definition 5.9. Define Dr (X,E) as the dual of Cr (X,E). Let D(X,E) = lim

−−→Dr (X,E)

with the transition maps Dr (X,E) ↪→ Ds (X,E) for r ≤ s given by restriction (andinjective by the previous propositions). It is given by all distributions µ : C∞(X,E)→ Kextending onto some Cr (X,E) for some r ≥ 0.

Proposition 5.10. Let X be a group with C∞-multiplication so that D(X,E) is a K-algebraby the convolution product equipped with a natural filtration. Then D(X,E) is a filteredK-algebra.

Other more explicit descriptions

The Mahler basis. Let C(Zp ,oE)∗ be the continuous dual of the space C(Zp ,oE) ofcontinuous functions f : Zp → oE. Then Clc(Zp ,oE) := {f : Zp → oE loc. cst. } ⊆C(Zp ,oE) is dense and corresponding to Clc(Zp ,oE) = ∪n≥0oE[Z/pnZ], we haveC(Zp ,oE)∗ = oE[[Zp]]. The Iwasawa isomorphism

oE[[Zp]]→ oE[[X]]111 7→ 1 + X

gives an isomorphism of topological rings where the left-hand side is endowed withthe topology of pointwise convergence. Define an oE-Banach space to be a completenormed torsionfree oE-module. Given an oE-Banach space V let

Vd := HomoE−cts.(V,oE)

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be the compact torsionfree oE-module of oE-linear continuous functionals with itstopology of pointwise convergence. Given a torsionfree compact oE-module M let

Md := HomoE(M,oE)be the oE-Banach space whose norm is given by ‖·‖sup. Then these functors are(quasi-)inverses between the categories

{ torsionfree compact oE-modules }↔ { oE-Banach spaces V with |oE | ⊇ ‖V‖}.Let C0(N,oE) be the zero sequences with entries in oE. We conclude

C(Zp ,oE) C0(N,oE)↓ ·d ↓ ·d

C(Zp ,oE)d ∼→ oE[[Zp]] ∼

→ oE[[X]]↓ ·d ↓ ·d

C(Zp ,oE) = C(Zp ,oE)dd ← C0(N,oE) = C0(N,oE)dd .The bottom isomorphism is then given by en := (. . . ,0,1,0 . . . ) 7→ �

∗

n

�with en being

the sequence whose sole nonzero entry is 1 at the n-th position. We can therefore writef =

∑n an

�∗

n

�∈ C(Zp ,E) and call {an} its Mahler coefficients.

The following proposition identifies the condition on the Mahler coefficients of thesubset Cr (Zd

p ,K) ⊆ C0(Zdp ,K) under this isomorphism given by this orthogonal basis.

Proposition 5.11. We find f ∈ Cr (Zdp ,K) if and only if |an | |n |r → 0 (with |n | =

n1 + · · · + nd ).

Proof: Use that their K-linear span given by all polynomial functions is dense by theprevious general result. Their orthogonality is quickly checked and their norm ‖

�∗

n

�‖Cr

directly computed. �

Remark. In fact, in [BB10], the authors defined f ∈ Cr (Zp ,E) for r ∈ R≥0 bydemanding |an |nr → 0.

The Taylor polynomial.

Proposition 5.12. Let X ⊆ K be open and f : X → K. Then f ∈ Cr (X,K) if and only ifthere are functions D0 f , . . . ,Dν f : X→ K such that putting

f (x + y) = D0 f (x) + D1 f (x)y + · · · + Dν f (x)yν + Rν f (x + y,x)there is for all a ∈ X and ε > 0 some neighborhood Uε 3 a such that

|Rν f (x + y,x)| ≤ ε|y |r for all x + y,x ∈ Uε.

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Proof: Express f [ν] through Taylor polynomials. �

Remark. In fact, one can show this definition to be equivalent to the one given in [Col10]over the domain X = Zp .

Definition 5.13. Let X ⊆ Kd be an open subset and k ∈ {1, . . . ,d}. We will speak of aCr ·ekT -function f : X→ K if there are continuous functions D0 f ,D1·ek f , . . . ,Dν·ek f : X→

K such that if one defines Rν·ek f : X[ek ] → K on X[ek ] := {(x ; t) ∈ X×K with x+t ·ek ∈

X} byRν·ek f (x ; t) := f (x + t · ek ) −

∑i=0, ...,ν

Di ·ek f (x)t i ,

then for every point a ∈ X and any ε > 0, there will exist a neighborhood U 3 a suchthat

|Rν·ek f (x ; t)| ≤ ε|t |r for all x + t · ek ,x ∈ U.

We will denote the set of all Cr ·ekT -functions f : X→ K by C

r ·ekT (X,K).

Corollary 5.14. Let X ⊆ Qdp be open. Then f ∈ Cr (X,K) if and only if f ∈ C

r ·e1T (X,K)∩

. . . ∩ Cr ·edT (X,K).

Proof: (i) We have f ∈ Cr (Zdp ,K) if and only if f =

∑n an

�∗

n

�with |an | |n |

r → 0.

(ii) We find |an | |n |r → 0 if and only if |an |nr1 → 0, . . . , |an |nrd → 0.

(iii) We have CrT(X,K) = Cr (X,K) and Cr (Zp ,K) = {f = ∑

n an�∗

n

�: |an |nr → 0}.

(iv) We have

C(r ,0, ...,0)T = Cr

T ⊗[C0 ⊗ · · · ⊗ C0] = {f =∑n

an

(∗

n

): |an |n

r1 → 0}

References

[BB10] L. Berger and C. Breuil, Sur quelques représentations potentiellement cristallinesde GL2(Qp), Astérisque 330 (2010), 155–211.

[Bor91] A. Borel, Linear algebraic groups, second ed., Graduate Texts in Mathematics,vol. 126, Springer-Verlag, New York, 1991. MR 1102012. DOI 10.1007/978-1-4612-0941-6.

[BS07] C. Breuil and P. Schneider, First steps towards p-adic Langlands func-toriality, J. Reine Angew. Math. 610 (2007), 149–180. MR 2359853.DOI 10.1515/CRELLE.2007.070.

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[Col10] P. Colmez, Fonctions d’une variable p-adique, Astérisque (2010), no. 330, 13–59.MR 2642404.

[Nag11] E. Nagel, Fractional non-Archimedean differentiability, Univ. Mün-ster, Mathematisch-Naturwissenschaftliche Fakultät (Diss.), 2011. zb-MATH 1223.26011. Confer http://nbn-resolving.de/urn:nbn:de:hbz:

6-75409405856.

[Sch84] W. H. Schikhof, Ultrametric calculus, Cambridge Studies in Advanced Mathe-matics, vol. 4, Cambridge University Press, Cambridge, 1984, An introductionto p-adic analysis. MR 791759.

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