smooth lie groups over local fields of positive characteristic need not be analytic

a

whichcture,twell as

eory;

nite-

further

ional

Journal of Algebra 285 (2005) 356–371

www.elsevier.com/locate/jalgebr

Smooth Lie groups over local fields of positivecharacteristic need not be analytic

Helge Glöckner

TU Darmstadt, Fachbereich Mathematik AG 5, Schlossgartenstr. 7, 64289 Darmstadt, Germany

Received 19 August 2004

Available online 19 January 2005

Communicated by Alexander Lubotzky

Abstract

We describe finite-dimensional smooth Lie groups over local fields of positive characteristicdo not admit an analytic Lie group structure compatible with the given topological group struandCn-Lie groups without a compatibleCn+1-Lie group structure for eachn � 1. We also presenexamples of non-analytic, smooth automorphisms of analytic Lie groups over such fields, asCn-automorphisms which fail to beCn+1. 2005 Elsevier Inc. All rights reserved.

Keywords:Smooth Lie group; Analytic Lie group; Totally disconnected group; Finite order differentiability;Local field; Positive characteristic; Contractive automorphism; Scale function; Tidy subgroup; Willis thNon-archimedian analysis

Introduction

It is a classical fact (the core of which goes back to Schur [14]) that every fidimensional real Lie group of classCk (wherek ∈ N ∪ {∞}) admits aCk-compatibleanalytic Lie group structure; see [12, Section 4.4] and the references given there forinformation. Based on Lazard’s characterization of analyticp-adic Lie groups within theclass of locally compact groups [10], it was shown in [6] that also every finite-dimens

E-mail address:[email protected].

0021-8693/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2004.11.018

H. Glöckner / Journal of Algebra 285 (2005) 356–371 357

heremap-

las-

in the

i-

cted,local

andositiveway, itooth,y. Forgroups

ebra)

olog-

Ck-Lie group over a local field of characteristic 0 admits aCk-compatibleK-analytic Liegroup structure. Here,Ck-maps and Lie groups are understood in the sense of [2], wa setting of differential calculus over arbitrary topological fields was developed. Aping between finite-dimensional vector spaces over a local field isC1 if and only if it isstrictly differentiable at each point [7, Remark 5.4], where strict differentiability is a csical concept, [3, 1.2.2].

In the present article, we show by example that the situation is entirely differentcase of positive characteristic. We prove (see Theorems 5.2 and 6.1):

Main Theorem. Let K be a local field of positive characteristic. Then:

(a) There exists a one-dimensional smoothK-Lie group which does not admit aK-analyticLie group structure compatible with its topological group structure.

(b) For each positive integern, there exists a one-dimensionalK-Lie group of classCn

which does not admit aK-Lie group structure of classCn+1 compatible with the giventopological group structure.

We also describe aCn-automorphism of a finite-dimensionalK-analytic Lie groupwhich is not Cn+1 (Proposition 3.4), and a non-analyticC∞-automorphism (Propostion 3.7).

As a tool, we use concepts from George Willis’ structure theory of totally disconnelocally compact groups as developed in [19,20] and analyzed for Lie groups overfields in [5,9].

While it is quite justified to restrict attention to analytic Lie groups in the realp-adic cases, our results make it clear that this habit is inappropriate in the case of pcharacteristic. Because genuinely larger classes of groups can be captured in thismay very well be of interest to study Lie groups over such fields which are merely smor merelyCn. It is not hopeless to prove substantial results in the absence of analyticitexample, it is possible to discuss contraction groups (opposite subgroups), tidy suband the scale function for automorphisms of mereC1-Lie groups [9].

1. Preliminaries and notation

We recall some definitions and basic facts from differential calculus (and linear algover local fields, and set up our notational conventions.

1.1. LetE andF be (Hausdorff) topological vector spaces over a non-discrete topical fieldK, U ⊆ E be open, andf :U → F be a map. Following [2], we callf aC1-map ifit is continuous and there exists a (necessarily unique) continuous mapf [1] :U [1] → F onU [1] := {(x, y, t) ∈ U ×E ×K: x + ty ∈ U} such thatf [1](x, y, t) = 1

t(f (x + ty)−f (x))

for all (x, y, t) ∈ U [1] with t �= 0. Thenf ′(x) :E → F , y �→ f ′(x).y := f [1](x, y,0) is acontinuous linear map for eachx ∈ U [2, Proposition 2.2]. Inductively,f is calledCn+1 ifit is C1 andf [1] is Cn; it is C∞ (or smooth) if it is Cn for all n.

358 H. Glöckner / Journal of Algebra 285 (2005) 356–371

s, the7.4ld, the.9]),

n

alhupt

s overe).

.rre-

4,15].

ete

aps

n

t

1.2. For maps between open subsets of finite-dimensional real vector spacepresent definition ofCn-maps is equivalent to the usual one (cf. [2, Propositionsand 7.7]). For maps between open subsets of an ultrametric complete valued fiepresent definition ofCn-maps is equivalent to the one in [13] (see [2, Proposition 6enabling us to use results from [13] in this case.

1.3. Compositions of composableCr -maps areCr [2, Proposition 4.5], the ChaiRule holds [2, Proposition 3.1], and beingCr is a local property for eachr ∈ N ∪ {∞}[2, Lemma 4.9]. This makes it possible to defineCr -manifolds modeled on a topologicK-vector spaceE, in the usual way. AK-Lie group of classCr is a group, equipped witaCr -manifold structure modeled on a topologicalK-vector space, which makes the grooperationsCr . If G is aCr -Lie group, we writeL(G) := T1(G) for its (geometric) tangenspace at the identity.1 Given a homomorphismα :G → H which is at leastC1, as usual weabbreviateL(α) := T1(α).

1.4. For the definition ofK-analytic maps between open subsets of Banach spacea complete valued fieldK, we refer to [3] (see also [15] for the finite-dimensional casAny K-analytic map isC∞ by [2, Proposition 7.20]. Hence everyK-analytic Lie groupmodeled on a Banach space can also be considered as aC∞-Lie group in the sense of 1.3

For further information on differential calculus over topological fields and the cosponding manifolds and Lie groups, we refer to [2,6–8]. For analytic Lie groups, see [

1.5. Recall that alocal field is a totally disconnected, locally compact, non-discrtopological field. On any local field, there exists anatural absolute value|.| :K → [0,∞[,defined via|0| := 0 and |x| := ∆K(λx) for x �= 0, whereλx :K → K, λx(y) := xy ismultiplication by x and ∆K : Aut(K,+) → ]0,∞[ the modular function. Then|K×| =〈q〉 � (]0,∞[, ·) for some prime powerq, called themodule ofK. For further informationon local fields, see [18].

The following lemma provides information concerning a special type of linear mwhich will play a crucial role in our constructions.

Lemma 1.6. Let V be a finite-dimensional vector space over a local fieldK with mod-ule q. Let |.| :K → [0,∞[ be the unique extension of the natural absolute value oK

to an absolute value on the algebraic closureK of K. Let α ∈ GL(V ) be aK-linear au-tomorphism ofV , with eigenvaluesλ1, . . . , λd in K, whered := dimK(V ). Suppose tha|λj | > 1 for eachj ∈ {1, . . . , d} and

∏dj=1 |λj | = q. ConsiderV as aK[T ]-module via

(∑n

j=0 ajTj ).v := ∑n

j=0 ajAjv. Then

(a) V is an irreducibleK[T ]-module;(b) |λj | = q1/d for eachj ∈ {1, . . . , d}; and(c) there exists an ultrametric norm‖.‖ onV such that‖α(x)‖ = q1/d‖x‖ for eachx ∈ V .

1 If r � 3, thenL(G) has a natural Lie algebra structure, but we shall not use it.


,

-

]).

ite-

to

by

Proof. (a) Let {0} �= U ⊆ V be aK[T ]-submodule ofV ; we show thatU = V . To thisend, lete := dimK(U) ∈ {1, . . . , d} and note thatλj1, . . . , λje are the eigenvalues ofα|UUin K for suitable integers 1� j1 < j2 < · · · < je � d . SetJ := {jk: k = 1, . . . , e} andr := ∏

j /∈J |λj |. Thenr � 1, with equality if and only ifJ = {1, . . . , d} (and henceU = V ).But

q = |det(α)| =d∏

j=1

|λj | = r∏j∈J

|λj |,

where 1<∏

j∈J |λj | = |det(α|UU)| ∈ qZ and thus∏

j∈J |λj | � q. Hence∏

j∈J |λj | = q

andr = 1, entailing thatU = V .(b) For eachλ ∈ σ(α) := {λj : j = 1, . . . , d}, let (V

K)λ := {v ∈ V

K: (α

K− λ)dv = 0}

be the generalized eigenspace ofαK

in VK

corresponding toλ, whereVK

is theK-vectorspace obtained fromV by extension of scalars. LetR := {|λ|: λ ∈ σ(α)}; given ρ ∈ R

defineσρ(α) := {λ ∈ σ(α): |λ| = ρ} andWρ := ⊕λ∈σρ(α)(VK

)λ. By [11, Chapter II, (1.0)p. 81], eachWρ is defined overK, i.e.,Wρ = (Vρ)

K, whereVρ := Wρ ∩ V . Note thatVρ

is α-invariant and hence aK[T ]-submodule ofV . ThusV = ⊕ρ∈R Vρ is a direct sum of

K[T ]-submodules. By (a), such a decomposition is only possible ifR = {ρ} is a singletonand thusV = Vρ . Now q = |λ1| · · · |λd | = ρd entailsρ = q1/d .

(c) We choose aK-basis v1, . . . , vd of VK

with respect to whichαK

has Jordancanonical form, and let‖.‖ be the maximum norm onV

Kwith respect to this basis. Af

ter re-labeling theλj ’s, we may assume thatvj ∈ (VK)λj

for eachj . Then αK(vj ) =

λjvj + (αK(vj ) − λjvj ) for eachj with ‖λjvj‖ = q1/d > 1 � ‖α

K(vj ) − λjvj‖. Thus

‖αK(x)‖ = q1/d‖x‖ for eachx ∈ V

K, by the ultrametric inequality (cf. [5, Eq. (1), p. 209

Now restrict‖.‖ to V . �1.7. Conventions

Throughout the article,q is a prime power,Fq a finite field of cardinalityq andK := Fq((X)) the field of formal Laurent series overFq , unless said otherwise.2 We letO := Fq [[X]] be the valuation ring ofK (the ring of formal power series overFq ). Welet |.| :K → [0,∞[ be the natural absolute value onK, given by|∑∞

k=n akXk| := q−n for

an, an+1, . . . ∈ Fq with an �= 0. We fix an algebraic closureK and use the same symbol,|.|,for the unique extension of the absolute value|.| on K to an absolute value onK. We useR andZ in their usual meanings and writeN := {1,2, . . .}, N0 := N ∪ {0}.

In the following, by a “Lie group” we always mean a Lie group modeled on a findimensional vector space. ACr -automorphismof a Cr -Lie groupG is an automorphismα :G → G such that bothα and its inverseare Cr . The same convention appliesK-analytic automorphisms,K-analytic diffeomorphisms andCr -diffeomorphisms. Givena topological groupG, its group of bicontinuous automorphisms will be denotedAut(G). If (E,‖.‖) is a normedK-vector space, we setBE

r := {x ∈ E: ‖x‖ < r} for r > 0;if E is understood, we abbreviateBr := BE

r .

2 Up to isomorphism, every local field of positive characteristic is of this form [18].


-er

at-

2. Criteria making a map Cn, or preventing it

In this section, we describe simple criteria ensuring that a mapping isCn, respectively,that it is notCn.

Lemma 2.1. Let n ∈ N and f :U → K be a map, defined on an open subsetU ⊆ K.Suppose that there exist real numbersθ > n andb > 0 such that

|f (y) − f (x)| � b|y − x|θ for all x, y ∈ U. (1)

Thenf is Cn andf ′(x) = 0 for eachx ∈ U .

Proof. This is a special case of [13, Theorem 29.12].�Lemma 2.2. LetE �= {0} andF be normedK-vector spaces andf :U → F be a mappingon an open subsetU ⊆ E. Letn ∈ N0. Suppose that there existx ∈ U , θ ∈ ]n,n + 1[ andpositive real numbersa, b such that

a‖y − x‖θ � ‖f (y) − f (x)‖ � b‖y − x‖θ for all y ∈ U. (2)

Thenf is not of classCn+1.

Proof. There exists 0�= y ∈ E such thatx + Oy ⊆ U . Theng :O → F defined byg(t) :=f (x + ty) − f (x) satisfies

A|t |θ � ‖g(t)‖ � B|t |θ for all t ∈ O, (3)

with A := a‖y‖θ , B := b‖y‖θ . It suffices to show thatg is notCn+1. To reach a contradiction, assume thatg is Cn+1. By [2, Theorem 5.1],g admits a Taylor expansion of ordn + 1 around 0:

g(t) =n+1∑j=0

aj tj + tn+1ρ(t) for all t ∈ O,

whereρ :O → F is continuous and satisfiesρ(0) = 0. But such an expansion is incompible with the asymptotic described in (3). To see this, note first thatak = 0 for k = 0, . . . , n.Indeed,g(0) = 0 entails thata0 = 0, and if we already know thata0 = · · · = ak−1 = 0, let-ting t → 0 in t−kg(t) = ak +∑n+1

j=k+1 tj−kaj + tn+1−kρ(t) we find that alsoak = 0 (since

t−kg(t) → 0, by (3)). Henceg(t) = (an+1 + ρ(t))tn+1 and thus

A � ‖g(t)‖|t |θ = ‖an+1 + ρ(t)‖ · |t |n+1−θ ,

where the right-hand side tends to 0 ast → 0. This contradictsA > 0. �


roup

f

3. Automorphisms with specific properties

In this section, we discuss a family of automorphisms of the additive topological gO := Fq [[X]], some of which have specific pathological properties.

3.1. Given a strictly ascending sequenceλ := (�j )j∈N0 of positive integers�0 < �1 <

�2 < · · · , we define a mappingβ :O → O via

β

( ∞∑j=0

ajXj

):=

∞∑j=0

ajX�j for (aj )j∈N0 ∈ F

N0q .

We defineα :O → O via α(z) := z + β(z).

Lemma 3.2. The mapβ from 3.1 is a continuous homomorphism, andα is an automor-phism of the topological group(O,+).

Proof. Clearly β and α are homomorphisms of groups. Furthermore,β is continuous,sinceO = F

N0q is equipped with the product topology and each coordinate ofβ(z) is either

constant or only depends on one coordinate ofz.To see thatα is a bijection, letz = ∑∞

j=0 ajXj andw = ∑∞

j=0 bjXj be elements ofO,

whereaj , bj ∈ Fq . Equating coefficients, we see thatα(z) = w holds if and only ifa0 = b0and

(∀i ∈ N) ai ={

bi if i �= �j for eachj ∈ N0;bi − aj if i = �j for somej ∈ N0.

(4)

Sincej < i in the second case of (4), givenw we can solve fora0, a1, . . . in turn and deducethatz ∈ O with α(z) = w exists and is uniquely determined. Thusα is an isomorphism ogroups. It is clear from the preceding formula that, for fixedi, the ith coordinateai ofz = α−1(w) only depends on finitely many coordinates,b0, b1, . . . , bi , of w. Henceα−1 iscontinuous. �Lemma 3.3. If there exist real numbersθ > 0 andK > 0 such that

jθ − K � �j � jθ + K for all j ∈ N0, (5)

then there exist positive real numbersa < b such that

a|y − x|θ � |β(y) − β(x)| � b|y − x|θ for all x, y ∈ O.

Proof. We claim that the assertion holds witha := min{q−�0, q−K} andb := qK . Sinceβ(y) − β(x) = β(y − x), we may assume without loss of generality thatx = 0. We distin-guish three cases.

1. If y = 0, then the assertion is trivial.2. If |y| = 1, then|β(y)| = q−�0 � a = a|y|θ andb|y|θ = b � 1� q−�0 = |β(y)|.


tion,3,

imple

3. If |y| = q−j with j ∈ N, then�j /j = θ + εj for someεj ∈ [−K/j,K/j ], by (5).Thus

|β(y)| = q−�j = (q−j

)�j /j = |y|�j /j = |y|θ+εj = |y|θ |y|εj ,

where|y|εj = q−jεj with jεj ∈ [−K,K] and so|y|εj ∈ [q−K,qK ]. Hence indeeda|y|θ �q−K |y|θ � |β(y)| � qK |y|θ = b|y|θ . �

The following proposition providesCn-automorphisms of the 1-dimensionalK-analyticLie group(O,+) which are notCn+1.

Proposition 3.4. Givenn ∈ N, chooseθ ∈ ]n,n + 1[ and define�j := 1 + [jθ ] for eachj ∈ N0, where[.] is the Gauß bracket(integer part). Letα :O → O be as in3.1. Thenα isa Cn-automorphism of(O,+) which is not aCn+1-map.

Proof. Sincejθ � �j � jθ + 1 for eachj ∈ N0, Lemma 3.3 provides real numbers 0<

a < b such thata|y − x|θ � |β(y) − β(x)| � b|y − x|θ for all x, y ∈ O. Henceβ is Cn

(by Lemma 2.1) but notCn+1 (by Lemma 2.2). Furthermore,β ′(x) = 0 for all x ∈ O, byLemma 2.1. Henceα = idO + β is aCn-map which is notCn+1, and such thatα′(x) = 1for eachx ∈ O. As α is an automorphism of groups by Lemma 3.2 and hence a bijecthe Inverse Function Theorem forCn-maps ([7, Theorem 7.3 and Remark 5.4] or [1Theorems 27.5 and 77.2]) implies thatα is aCn-diffeomorphism. �

To identify suitable smooth mappings as non-analytic, we shall use the following sfact (see [3, 4.2.3 and 3.2.5]):

Lemma 3.5. Let E and F be ultrametric Banach spaces overK, U ⊆ E be an open0-neighborhood andf :U → F be aK-analytic function such that

limx→0

‖f (x)‖‖x‖n

= 0 for eachn ∈ N0.

Then there exists a0-neighborhoodW ⊆ U such thatf |W = 0.

We find it convenient to formulate another preparatory lemma.

Lemma 3.6. In 3.1, chooseλ = (�j )j∈N0 such that�j /j → ∞ asj → ∞. Then, for eachn ∈ N, there exists a real numbercn > 0 such that

|β(y) − β(x)| � cn|y − x|n for all x, y ∈ O. (6)

Proof. Let n ∈ N. Since�j /j → ∞, there existsk ∈ N such that�j /j � n for all j � k.We claim that (6) holds withcn := qkn. It suffices to verify (6) forx = 0, and we mayassume thaty �= 0 (cf. proof of Lemma 3.3). If|y| = q−j with j � k, then|β(y)| = q−�j =(q−j )�j /j � (q−j )n = |y|n � cn|y|n, as required. If|y| = q−j with j ∈ {0,1, . . . , k − 1},then|β(y)| � 1� qknq−jn = cn|y|n as well. �


sional

d,

r ob-

s Ex-e

in Liedo notroup

to thecon-loped

efly

We now provide examples of smooth, non-analytic automorphisms of the 1-dimenK-analytic Lie group(O,+).

Proposition 3.7. In 3.1, chooseλ = (�j )j∈N0 such that�j /j → ∞ as j → ∞. Then thefollowing holds:

(a) α is aC∞-automorphism of(O,+).(b) α is notK-analytic.

Proof. (a) Lemmas 3.6 and 2.1 show thatβ is smooth, withβ ′(x) = 0 for eachx ∈ O. Asin the proof of Proposition 3.4, we infer thatα is aC∞-automorphism.

(b) Sinceβ = α − idO where idO is K-analytic, it suffices to show thatβ is notK-analytic. However, ifβ wasK-analytic, thenβ would vanish on some 0-neighborhooby (6) and Lemma 3.5. Asβ is injective, we have reached a contradiction.�

Note that functions of one variable whose derivatives vanish identically are familiajects in non-archimedian analysis. Examples of such functions in thep-adic case (whichare not homomorphisms) are described (or suggested) in Example 26.4 as well aercises 28.F, 29.F and 29.G in [13]. Our definition ofβ in 3.1 was stimulated by thesexamples.

4. Calibrations obtained from contractions

In this section, we provide techniques which shall enable us to check that certagroups constructed with the help of the pathological automorphisms from Section 3admit better-behaved Lie group structures compatible with their given topological gstructure. The basic idea is that suitable contractive automorphisms ofK-Lie groups de-termine a “calibration” which can be used to estimate the norms of elements closeidentity (with respect to a given chart taking 1 to 0). As a tool, we shall employcepts from the structure theory of totally disconnected, locally compact groups devein [19,20] and analyzed for Lie groups over local fields in [5,9] (cf. also [1]). We brirecall the relevant definitions.

4.1. Let G be a totally disconnected, locally compact topological group andα be abicontinuous automorphism ofG. Then there exists a compact open subgroupU ⊆ G

which istidy for α, meaning that the following holds:

(T1) U = U+U−, whereU± := ⋂k∈N0

α±k(U);(T2) the subgroupU++ := ⋃

k∈N0αk(U+) is closed inG.

Hereα(U+) ⊇ U+ andα(U−) ⊆ U−. The indexsG(α) := [α(U+) : U+] is finite and isindependent of the tidy subgroup; it is called thescale ofα. If U is tidy for α, then it isalso tidy forα−1. (See [19,20].)


of

om-

,

ts of

Recall that a bicontinuous automorphismα :G → G is contractiveif αn(x) → 1 asn → ∞ for eachx ∈ G. A sequence(Un)n∈N0 of identity neighborhoods is called afun-damental sequenceif U0 ⊇ U1 ⊇ U2 ⊇ · · · and every identity neighborhood ofG containssomeUn.

4.2. If α is contractive andU ⊆ G a compact identity neighborhood, then{αn(U):n ∈ N0} is a basis for the filter of identity neighborhoods ofG, as a consequence[17, Proposition 2.1].

Lemma 4.3. Let α be a contractive automorphism of a totally disconnected, locally cpact topological groupG. ThensG(α) = 1, and the following holds:

(a) There exists a compact, open subgroupU ⊆ G such thatα(U) ⊆ U .(b) A compact, open subgroupU ⊆ G is tidy for α if and only ifα(U) ⊆ U . In this case,

U− = U ,

U+ :=⋂

k∈N0

αk(U) = {1}, and U−− :=⋃

k∈N0

α−k(U−) = G. (7)

(c) (αn(U))n∈N0 is a fundamental sequence of identity neighborhoods forG, for eachsubgroupU ⊆ G which is tidy forα. Each of the groupsαn(U) is tidy forα. HenceGhas arbitrarily small subgroups tidy forα.

(d) If G is a C1-Lie group over a local fieldK andα is a contractiveC1-automorphismlet β := L(α) be the associatedK-linear automorphism ofL(G). Then|λ| < 1 forevery eigenvalueλ of β in K.

Proof. (a) As tidy subgroups always exist, (a) follows from (b).(b) If U ⊆ G is tidy for α, thenαn(x) → 1 asn → ∞ for x ∈ U entails thatx ∈ U− (cf.

[19, Lemma 9]). HenceU = U− and henceα(U) ⊆ U . If, conversely,U ⊆ G is a compactopen subgroup such thatα(U) ⊆ U , thenα−1(U) ⊇ U and thusU− = U . SinceG isHausdorff, 4.2 shows thatU+ = ⋂

n�0 αn(U) = {1} and henceU++ = {1}, which is closed.ThusU satisfies (T1) and (T2). SinceU+ = {1}, furthermoresG(α) = [{1} : {1}] = 1. Tocomplete the proof of (7), letx ∈ G. Then αn(x) ∈ U for somen ∈ N0 and thusx ∈α−n(U) = α−n(U−) ⊆ U−−. HenceU−− = G.

(c) Asαn+1(U) ⊆ αn(U) for eachn, the claims follow from (b) and 4.2.(d) As G has small subgroups tidy forα, results from [9] show thatGα := {x ∈ G:

αn(x) → 1 asn → ∞}, Gα−1 := {x ∈ G: α−n(x) → 1 asn → ∞} and G0 := {x ∈ G:αZ(x) is relatively compact} areC1-Lie subgroups ofG, and that the multiplication map

Gα × G0 × Gα−1 → G, (x, y, z) �→ xyz,

is a C1-diffeomorphism onto an open subset ofG. As Gα = G, this implies thatG0 =Gα−1 = {1}, whenceL(G0) = L(Gα−1) = {0}. However, it is also shown in [9] thaL(Gα−1) (respectivelyL(G0) ) is the direct sum of all the generalized eigenspace

K K


ch

s that

-

βK

to eigenvalues of absolute value> 1 (respectively, of absolute value 1). Hence sueigenvalues cannot exist.�

In view of Lemma 4.3, (b) and (c), the following definition makes sense:

Definition 4.4. Let G be a totally disconnected, locally compact group,α be a contrac-tive automorphism ofG andU ⊆ G a compact, open subgroup such thatα(U) ⊆ U . Thecalibration ofG associated withα andU is the functionκ :G → Z ∪ {∞} defined via

κ(x) :={∞ if x = 1;

n if x ∈ αn(U) \ αn+1(U).(8)

Cf. [16, 3.1–3.3] for Lemma 4.3, (a), (c), and the preceding definition.

4.5. Let G be aC1-Lie group overK = Fq((X)) andα :G → G be a contractiveC1-automorphism ofG such thatsG(α−1) = q. Let φ :P → Q ⊆ L(G) be a chart forGaround 1 such thatφ(1) = 0 andT1(φ) = idL(G). Let ‖.‖ be an ultrametric norm onL(G).For eachr > 0, letBr be the ball of radiusr in (L(G),‖.‖); let ρ > 0 such thatBρ ⊆ Q,and setWr := φ−1(Br) for 0 < r � ρ. Let U be a compact open subgroup ofG such thatα(U) ⊆ U .

Lemma 4.6. In the situation of4.5, there existN ∈ N0 and real numbersa � b such thatqb � ρ and

Wqa−n/d ⊆ αn+N(U) ⊆ Wqb−n/d for all n ∈ N0. (9)

Proof. Let λ1, . . . , λd (whered := dimK L(G)) be the eigenvalues ofL(α−1) in K, eachof which has absolute value> 1 as a consequence of Lemma 4.3(d). SinceG has smallsubgroups tidy forα (Lemma 4.3(c)), it also has small subgroups tidy forα−1. Hence

q = sG(α−1) =

∏j∈{1,...,d}s.t. |λj |�1

|λj | =d∏

j=1

|λj |, (10)

using the main result of [9] to obtain the second equality. Now Lemma 1.6 show|λj | = q1/d for eachj ∈ {1, . . . , d}, and it provides an ultrametric norm‖.‖′ onL(G) suchthat‖L(α−1).x‖ = q1/d‖x‖′ for eachx ∈ L(G). Hence

‖β(x)‖′ = q−1/d‖x‖′ for eachx ∈ L(G),

whereβ := L(α). SetB ′r := {x ∈ L(G): ‖x‖′ < r} for r > 0. There exists an open 0

neighborhoodR ⊆ Q such thatα(φ−1(R)) ⊆ P . Then

γ :R → Q, γ (x) := φ(α(φ−1(x)

)),


he

t

ed

expressesα in local coordinates. We haveγ ′(0) = T1(φ)L(α)T1(φ)−1 = L(α) = β asT1(φ) = idL(G). Thusγ ′(0) = β is invertible. Now the Inverse Function Theorem in tquantitative form [7, Proposition 7.1(b)′] shows that there existsδ > 0 with B ′

δ ⊆ R suchthatγ (B ′

r ) = β(B ′r ) = B ′

rq−1/d for eachr ∈ ]0, δ] and hence

γ n(B ′r ) = B ′

rq−n/d for all r ∈ ]0, δ] andn ∈ N0. (11)

All ultrametric norms on the finite-dimensionalK-vector spaceL(G) being equivalen[13, Theorem 13.3], there existsc � 1 such thatc−1‖.‖ � ‖.‖′ � c‖.‖ and thusBc−1r ⊆B ′

r ⊆ Bcr for eachr > 0. Setv := min{1, ρ/(cδ)}. ThenB ′vδ ⊆ B ′

δ ⊆ R. ChooseN ∈ N

such thatαN(U) ⊆ φ−1(B ′vδ). There existsu ∈ ]0, v] such thatB ′

uδ ⊆ φ(αN(U)). Then

B ′uδq−n/d ⊆ γ n

(φ(αN(U)

)) ⊆ B ′vδq−n/d for eachn ∈ N0,

by (11). ThereforeBc−1uδq−n/d ⊆ γ n(φ(αN(U))) ⊆ Bcvδq−n/d . Using thatγ n(φ(αN(U)))=φ(αn+N(U)), we obtain

Bc−1uδq−n/d ⊆ φ(αn+N(U)

) ⊆ Bcvδq−n/d for eachn ∈ N0. (12)

Note thatcvδq−n/d � cvδ � cδρ/(cδ) = ρ be definition ofv. Applying φ−1 to (12), wesee that (9) holds witha := logq(uδ/c) andb := logq(cvδ). �

In the following lemma, we retain the situation of Lemma 4.6; in particular,d :=dimK L(G). For ‖.‖ as in 4.5, defineµ :L(G) → R ∪ {∞}, µ(x) := − logq ‖x‖ (wherelogq(0) := −∞). Thus‖x‖ = q−µ(x). The calibrationκ :G → Z ∪ {∞} associated withα andU (as in (8)) can be used to approximateµ on a 0-neighborhood, up to a bounderror:

Lemma 4.7 (Calibration Lemma). There existsR ∈ [0,∞[ such that

µ(φ(x)

)d − R � κ(x) � µ

(φ(x)

)d + R for eachx ∈ αN(U). (13)

Proof. Let x ∈ αN(U) \ {1}. ThenN � κ(x) < ∞, entailing thatn := κ(x) − N ∈ N0.Using (9), we obtain:

x ∈ ακ(x)(U) = αn+N(U) ⊆ Wqb−n/d and

x /∈ ακ(x)+1(U) = αn+1+N(U) ⊇ Wqa−(n+1)/d .

Henceqa−(n+1)/d � ‖φ(x)‖ < qb−n/d and thus(n + 1)/d − a � µ(φ(x)) > n/d − b,whencen + 1− ad � µ(φ(x))d > n − bd and therefore

κ(x) − N + 1− ad � µ(φ(x)

)d > κ(x) − N − bd.

Hence the assertion holds withR := max{0,1− ad − N,N + bd}. �


od

nys.

n

pson.

The following immediate consequence of Lemma 4.7 is what we actually need.

Proposition 4.8. Let G and G∗ be C1-Lie groups overK, of dimensionsd and d∗,respectively, such thatG = G∗ as a topological group. Letφ :P → Q ⊆ L(G) andφ∗ :P ∗ → Q∗ be charts ofG andG∗ around1 taking1 to 0 and such thatT1(φ) = idL(G)

andT1(φ∗) = idL(G∗). Let‖.‖ and‖.‖∗ be ultrametric norms onL(G) andL(G∗), respec-

tively. If there exists a contractive automorphismα of G which is aC1-automorphism forboth G and G∗ and such thatsG(α−1) = q, then there exists an identity neighborhoW ⊆ P ∩ P ∗ and positive real numbersa < b such that

a‖φ(x)‖d/d∗ � ‖φ∗(x)‖∗ � b‖φ(x)‖d/d∗for eachx ∈ W .

It may very well happen thatd �= d∗ in the situation of Proposition 4.8, and in fact acombination of positive integers(d, d∗) ∈ N

2 can occur, as the following example show

Example 4.9. Let G := (K,+), equipped with its usual one-dimensionalK-analytic Liegroup structure. Then the (linear) map

τ :K → K, τ (z) := X · z,is aK-analytic automorphism ofG. The isomorphism of topological groups

φ∗ :K → K2,

∑j

ajXj �→

(∑j

a2jXj ,

∑j

a2j+1Xj

),

can be used as a global chart forK turning it into a 2-dimensionalK-analytic Lie groupG∗isomorphic to(K2,+). Note thatτ also is aK-analytic automorphism ofG∗, becauseφ∗ ◦ τ ◦ (φ∗)−1 :K2 → K2 is theK-linear map(

u

v

)�→

(0 X

1 0

)(u

v

).

By the following lemma,τ is contractive andsG(τ−1) = q. Hence, we are in the situatioof Proposition 4.8.

Variants: Given any positive integerd∗, we obtain an isomorphism of topological grouK ∼= K

d∗if we replace 2Z and 2Z+1 with d∗

Z and its cosets in the preceding constructiThis enables us to turnK into aK-analytic Lie groupG∗ of dimensiond∗, such thatτ is aK-analytic automorphism ofG∗.

We used the following simple fact.

Lemma 4.10. τ is a contractive automorphism of(K,+) with sK(τ−1) = q.

Proof. Because|τn(z)| = |Xnz| = q−n|z| for eachz ∈ K andn ∈ N, clearlyτ is contrac-tive. Sinceτ(O) = XO ⊆ O, Lemma 4.3(b) shows thatO is tidy for τ , with O+ = {0} andO− = O. ThereforesK(τ−1) = [τ−1(O) : O] = [X−1

O : O] = |Fq | = q. �


d

d

r-

n

og-

e

-

ani-

ny.p

5. Examples of non-analytic C∞-Lie groups

In this section, we present examples of smoothK-Lie groups which cannot be turneinto K-analytic Lie groups.

5.1. Let λ = (�j )j∈N0, α andβ be as in 3.1. Every elementz = ∑∞j=−n ajX

j (where

n ∈ N) of K can be written asz = z′ + z′′ with z′ := ∑−1j=−n ajX

j andz′′ := ∑∞j=0 ajX

j .We define

β :K → K, β(z) := β(z′′), and α :K → K, α(z) := z′ + α(z′′).

Thusα|O = α, β|O = β, andα = idK + β. Clearly β is a continuous homomorphism anα a bicontinuous automorphism of(K,+). We define

τ :K → K, τ (z) := X · z,

and recall from Example 4.9 and Lemma 4.10 thatτ is aK-analytic, contractive automophism of(K,+) such thatsK(τ−1) = q. We equip

G := K � 〈α, τ 〉 (14)

with the uniquely determined topology making it a topological group withK (in its usualtopology) as an open, normal subgroup; here〈α, τ 〉 denotes the subgroup of Aut(K,+)

generated byα andτ .

Theorem 5.2. For λ = (�j )j∈N0 as in Proposition3.7, the topological groupG describedin (14)has the following properties:

(a) G admits a1-dimensional smoothK-Lie group structure compatible with the givetopological group structure.

(b) G does not admit aK-analytic Lie group structure compatible with the given topolical group structure.

Proof. (a) We equip(K,+) with its usual 1-dimensionalC∞-Lie group structure. Sincthe restriction ofα to the open subsetO ⊆ K coincides withα, which is aC∞-auto-morphism ofO by Proposition 3.7, we see thatα is aC∞-automorphism of(K,+). Alsoτ is aC∞-automorphism of(K,+), and hence so is eachγ ∈ 〈α, τ 〉. Now standard arguments provide a (uniquely determined)C∞-manifold structure onG making it a smoothLie group and such thatK, with its given smooth manifold structure, is an open submfold of G (see [8, Proposition 1.18]).

(b) The proof has two parts, the first of which can be re-used later.

5.3. Let us assume, to the contrary, that there exists aK-analytic manifold structure othe groupG making it aK-analytic Lie groupG∗ and compatible with the given topologWe letL(G∗) be the Lie algebra ofG∗ and setd := dimK L(G∗) ∈ N. The open subgrou


ric

at

ner

locale.

K of G∗ inherits aK-analytic Lie group structure fromG∗. We writeH for K, equippedwith this d-dimensionalK-analytic Lie group structure. ThusL(H) = L(G∗), and bothτand α areK-analytic automorphisms ofH . We letφ :P → Q ⊆ L(H) be a chart forHaround 0 such thatφ(0) = 0 andT0(φ) = idL(H). Furthermore, we choose an ultrametnorm‖.‖ onL(H). Using Proposition 4.8, we findN ∈ N0 and positive real numbersa < b

such thatXNO ⊆ P and

a‖φ(z)‖d � |z| � b‖φ(z)‖d for eachz ∈ XNO. (15)

Set W := φ(XNO). Since β(XN

O) ⊆ XNO, we can defineγ :W → W , γ (w) :=

φ(β(φ−1(w))).

5.4. Using Lemma 3.5, we want to show thatγ is not analytic. Givenn ∈ N, thereexistsk � N such that�j � (n + 1)j for eachj � k and hence

|β(z)| � |z|n+1 for eachz ∈ XkO. (16)

For w ∈ W such that‖w‖ � q−k/db−1/d , we have|φ−1(w)| � b‖w‖d � q−k (using (15))and hence|β(φ−1(w))| � |φ−1(w)|n+1, by (16). As a consequence,

‖γ (w)‖ = ∥∥φ(β(φ−1(w)

))∥∥ � a−1/d∣∣φ−1(w)

∣∣(n+1)/d � a−1/db(n+1)/d‖w‖n+1,

entailing that‖γ (w)‖/‖w‖n → 0 asw → 0. Sinceγ is injective, Lemma 3.5 shows thγ cannot be analytic. As a consequence,β :H → H is not analytic either, nor isα =idH + β :H → H . But α :H → H has to be analytic, being the restriction of an inautomorphism of the analytic Lie groupG∗ to its open, normal subgroupH . We havereached a contradiction.�Remark 5.5. In particular, Theorem 5.2 shows that there exist smooth Lie groups overfields of positive characteristic without asmoothlycompatible analytic Lie group structur

6. Examples of Cn-Lie groups that are not Cn+1

A variant of the proof of Theorem 5.2 shows the following:

Theorem 6.1. Givenn ∈ N, let θ and λ = (�j )j∈N0 be as in Proposition3.4. Then thetopological groupG from (14)has the following properties:

(a) G admits a1-dimensionalK-Lie group structure of classCn compatible with the giventopological group structure.

(b) G does not admit aK-Lie group structure of classCn+1 compatible with the giventopological group structure.


(viz.” with. Towant,

s a

p

ugust. Thek 5.5

ted lo-

sition.

Proof. The proof of Theorem 5.2(a) and the first half of the proof of Theorem 5.2(b)paragraph 5.3) carry over to the present situation, if we replace the word “smooth“Cn” and “K-analytic” with “Cn+1” there. Thus Part (a) of the present theorem holdscomplete the proof of (b), we retain the notations from 5.3. Using Lemma 2.2, weto show thatγ is notCn+1 (whence neither isβ, nor α, which is absurd). By Lemma 3.3there exist positive real numbersA < B such that

A|z|θ � |β(z)| � B|z|θ for eachz ∈ O. (17)

For eachw ∈ W , we then have

‖γ (w)‖ = ∥∥φ(β(φ−1(w)

))∥∥ � a−1/d∣∣β(

φ−1(w))∣∣1/d � a−1/dB1/d

∣∣φ−1(w)∣∣θ/d

� a−1/dbθ/dB1/d‖w‖θ and

‖γ (w)‖ = ∥∥φ(β(φ−1(w)

))∥∥ � b−1/d∣∣β(

φ−1(w))∣∣1/d � b−1/dA1/d

∣∣φ−1(w)∣∣θ/d

� b−1/dA1/daθ/d‖w‖θ ,

using (15) and (17). SetC := b−1/dA1/daθ/d andD := a−1/dbθ/dB1/d . Then

C‖w‖θ � ‖γ (w)‖ � D‖w‖θ for eachw ∈ W ,

whence Lemma 2.2 shows thatγ is notCn+1, a contradiction. �The following problems remain open.

Problem 1. Does every smoothK-Lie group have an open subgroup which admitsmoothly compatible analytic Lie group structure?

Problem 2. Given n ∈ N, does everyK-Lie group of classCn have an open subgrouwhich admits aCn-compatibleK-Lie group structure of classCn+1?

Acknowledgments

The present research was carried out at the University of Newcastle (N.S.W.) in A2004, supported by DFG grant 447 AUS-113/22/0-1 and ARC grant LX 0349209author profited from discussions with George Willis and Udo Baumgartner. Remaranswers a question posed to the author by Linus Kramer (Darmstadt) in 2003.

References

[1] U. Baumgartner, G.A. Willis, Contraction groups and scales of automorphisms of totally disconneccally compact groups, Israel J. Math. 142 (2004) 221–248.

[2] W. Bertram, H. Glöckner, K.-H. Neeb, Differential calculus over general base fields and rings, ExpoMath. 22 (2004) 213–282.


rint

eprint

4.

f. JFM

–363.(2001)

[3] N. Bourbaki, Variétés différentielles et analytiques. Fascicule de résultats, Hermann, Paris, 1967.[4] N. Bourbaki, Lie Groups and Lie Algebras, Chapters 1–3, Springer-Verlag, 1989.[5] H. Glöckner, Scale functions onp-adic Lie groups, Manuscripta Math. 97 (1998) 205–215.[6] H. Glöckner, Every smoothp-adic Lie group admits a compatible analytic structure, prep

math.GR/0312113; Forum Math., in press.[7] H. Glöckner, Implicit functions from topological vector spaces to Banach spaces, pr

math.GM/0303320, March 2003.[8] H. Glöckner, Lie groups over non-discrete topological fields, preprint math.GR/0408008, August 200[9] H. Glöckner, Scale functions on Lie groups over local fields of positive characteristic, in preparation.

[10] M. Lazard, Groupes analytiquesp-adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965) 389–603.[11] G.A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, 1991.[12] D. Montgomery, L. Zippin, Topological Transformation Groups, Interscience, New York, 1955.[13] W.H. Schikhof, Ultrametric Calculus, Cambridge Univ. Press, Cambridge, 1984.[14] F. Schur, Zur Theorie der endlichen Transformationsgruppen, Math. Ann. 38 (1891) 263–286, c

23.0381.01.[15] J.-P. Serre, Lie Algebras and Lie Groups, Springer-Verlag, 1992.[16] E. Siebert, Contractive automorphisms of locally compact groups, Math. Z. 191 (1986) 73–90.[17] J.S.P. Wang, The Mautner phenomenon forp-adic Lie groups, Math. Z. 185 (1984) 403–412.[18] A. Weil, Basic Number Theory, Springer-Verlag, 1967.[19] G.A. Willis, The structure of totally disconnected, locally compact groups, Math. Ann. 300 (1994) 341[20] G.A. Willis, Further properties of the scale function on a totally disconnected group, J. Algebra 237

142–164.

smooth lie groups over local fields of positive characteristic need not be analytic

Documents