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ath.
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Nov
200
2
Introduction top-adicq-difference equations(weak Frobenius structure and transfer theorems)
Lucia Di Vizio
Laboratoire Emile Picard, Universite Paul Sabatier, U.F.R. M.I.G.,118, route de Narbonne, 31062 Toulouse CEDEX4, France
Abstract
Inspired by thep-adic theory of differential equations, this paper introduces an analogue theory forq-difference equations over a local field, when|q|= 1. We define some basic concepts, for instance thegenericradius of convergence, introduce technical tools, such as atwisted Taylor formulafor analytic functions, andprove some fundamental statements, such as aneffective bound theorem, theexistence of a weak Frobeniusstructureand atransfer theoremin regular singular disks.2000 Mathematics Subject Classification: 12H10, 12H25, 39A13, 65Q05.
Table of contents
I Basic definitions and properties ofp-adic q-difference systems 41 Theq-difference algebra of analytic functions over an open disk. . . . . . . . . . . . . . . . 42 Gauss norms andq-difference operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Analytic solutions ofq-difference systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Generic points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 13
II Effective bounds for q-difference systems 155 Effective bound theorem forq-difference systems . . . . . . . . . . . . . . . . . . . . . . . . 156 Some consequences: a transfer theorem in ordinary points and a corollary aboutq-deformations 19
III Weak Frobenius structure over a disk 227 Frobenius action onq-difference systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 Normal form for a system inH ℓ
q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Proof of (7.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 26
IV Transfer theorems in regular singular disks 2810 An analogue of Christol’s theorem . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 2811 A first rough estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 2912 A sharper estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 3213 More general statements . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 33
A Twisted Taylor expansion of p-adic analytic functions 3314 Analytic functions over aq-invariant open disk . . . . . . . . . . . . . . . . . . . . . . . . . 3315 Analytic functions over nonconnected analytic domain . .. . . . . . . . . . . . . . . . . . . 3516 Proof of proposition 15.3 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 36
The author acknowledges the Institute for the Advanced Study for hospitality and the National Science Foundation for support duringthe latter stages of preparation of this paper.
1
B Basic facts about regular singularity ofq-difference systems 3917 Regular singularq-difference systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3918 Fromq-difference systems toq-difference equations . . . . . . . . . . . . . . . . . . . . . . . 40
C The q-type of a number 4119 Basic properties of theq-type of a number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4120 Radius of convergence of1Φ1(q;αq;q;(1−q)x) . . . . . . . . . . . . . . . . . . . . . . . . . 42
Introduction
Since the late 1940’s,q-difference equations have been almost forgotten. In the last ten years the field hasrecovered its original vitality and the theory has witnessed substantial advances. Authors have also consideredthese functional equations both from an arithmetical and ap-adic point of view (cf. for instance [BB92],[And00b] or [DV]). This paper aims to ameliorate a gap in the literature, offering a systematic introduction top-adicq-difference equationswhen|q|= 1.
Our motivation to start this work was Sauloy’s result onq-deformation of the local monodromy of fuchsiandifferential system overP1
C(cf. [Sau00a] and [And02] for a survey on the topic). Sauloy considers a fuchsian
differential system
(S) xdYdx
= G(x)Y(x)
such thatG(x) ∈ Mµ(C(x)). More precisely he supposes that the matrixG(x) has no poles at 0 and∞, that thedifference of any two eigenvalues ofG(0) (resp.G(∞)) is not a non zero integer and that all the polesx1, . . . ,xs
of G(x) are simple. Under these assumptions one can chooseq∈C, with |q| 6= 1, such thatxiqR∩x jqR = /0 foreveryi 6= j and construct for any 0< ε << 1 aqε-difference system of the form
(S)qε Y(qεx) =[Iµ+(qε −1)xGε(x)
]Y(x)
such that the matrixGε(x) ∈ Mµ(C(x)) converges uniformly toG(x) overΩ = C∗r∪ixiqR. One can suppose,for instance by takingGε(x)=G(x) for all ε, that there exists a matrixQε ∈Glµ(C) such thatQ= limε→∞ Qε andbothQεGε(0)Q−1
ε andQG(0)Q−1 are upper triangular matrices, and that an analogous hypothesis is verified at∞. Then, forε sufficiently closed to 0, Sauloy constructs two canonical solutionsYε,0(x) andYε,∞(x) of (S)q,respectively in a neighborhood of 0 and∞: they both turn out to be meromorphic onC∗. Therefore the BirkhoffmatrixPε(x) =Yε,∞(x)−1Yε,0(x) is also meromorphic overC∗ and moreover it is elliptic:Pε(x) = Pε(qεx).
Theorem 0.1 [Sau00a, §4]The matrixPε(x) tends to a locally constant matrixP(x) overΩ, whenε → 0. LetP′ andP′′ be the values ofP(x) over the two connected components ofΩ whose closure contains the polexi ofG(x). Then the local monodromy of(S) aroundxi is generated byP′P′′−1.
Christol and Mebkhout have developed a theory ofp-adic monodromy, answering to questions first raisedby Robba: due to the lack of analytic continuation it is much more complicated than the complex theory,therefore we think it would be interesting to study the properties ofq-deformations in thep-adic setting.
There is a fundamental difference between complex andp-adic q-deformations: while inC one can letq → 1 avoiding the unitary circle, this is not possible in thep-adic world. In other words, to studyp-adicq-deformation of differential equations one has to deal withthe case|q|= 1.
We are emphasizing the “|q| = 1”, since the literature is entirely devoted to the case|q| 6= 1, to avoid thesmall divisor problem. In fact, consider aq-difference equations
(E)q y(qµx)+aµ−1(x)y(qµ−1x)+ · · ·+a0(x)y(x) = 0
whereq is an element of a normed field, archimedean or not, such that|q| 6= 1, and theai(x)’s are rationalfunctions inK(x). Then any convergent solution∑n≥0anxn of (E)q is the expansion at 0 of a meromorphicfunction overA1
K , since the equation itself allows for a meromorphic continuation of the solution. Moreover,the Adams lemma state that even the most irregularq-difference equations always have at least one solutionanalytic at zero (cf. [Ada29] and [Sau02b, 1.2.6]). On the other hand, when|q| = 1, one has to deal with
2
the problem of estimating terms of the form 1−qn, the so called small divisors problem, which can make theprediction of the existence of a convergent solution of(E)q very difficult.
We point out that we are not distinguishing between the archimedean and the ultrametric case: the di-chotomy in theq-difference world is given by the two cases|q| = 1 and|q| 6= 1, very differently from thedifferential equation theory, where there are very few analogies between the complex and thep-adic frame-work.
Nowadays, the complex theory ofq-difference equations for|q| 6= 1 has reached a “degree of completeness”comparable to the differential equation theory one, as Birkhoff and Guenter hoped [BG41]. Moreover it seemsthat those results should be true also in thep-adic case with very similar proofs (cf. for instance [Bez92] versus[BB92] and [Sau00b, I, 2.2.4]). In the meantime very few pages are voted to the study of the case|q| = 1,which, apart from the small divisor problem, is characterized by essentially two difficulties:
1) the meromorphic continuation of solutions not working any longer, one needs a good notion of solution at apointξ 6= 0. This is a key-point of thep-adic approach, since we cannot image ap-adic theory ofq-differenceequations without an analogue of the notion of Dwork-Robba generic point and generic radius of convergence.
2) Sauloy constructs his canonical solution using the classical Theta function
Θ(x) = ∑n∈Z
q−n(n−1)
2 xn .
Of course it does not converge for|q|= 1. Moreover all the infinite products, playing such an important role inthe theory ofq-series (cf. for instance [GR90]), and that would be very useful to write meromorphic solutions,do not converge either.
The lesson of the results contained in this paper is thatp-adicq-difference equations with|q|= 1 presents thesame pathologies asp-adic differential equations, namely the uniform part of solutions at a regular singularpoints can be divergent, according to the type of the exponents. In fact, theq-difference theory is precisely aq-deformation of the differential situation. Sincep-adic differential equation theory has geometrical implications,this allows to image thatp-adicq-difference equations should have some “non commutativep-adic geometricimplications”.
∗ ∗ ∗
Concerning the content of the present work, we have chosen tointroduce the notions ofq-difference algebraas required by the paper. Anyway a systematic presentation with highly compatible notation can be found in[DV, §1]. The paper is organized as follows:
Chapter I is an introduction to basic tools. First of all we study the properties of theq-difference algebra ofanalytic functions over a disk and the properties ofq-difference operators with respect to the so calledGauss norms. We also state a result, proved in Appendix A, about the existence of aq-expansion ofanalytic functions, which we use to establish a good notion of solutions of aq-difference equation at apoint ξ 6= 0,1. In particular this allows for the definition of generic radius of convergence at a Dwork-Robba generic point.
In Chapter II we prove an effective bound theorem in the wake of the Dwork-Robba theorem, from which wededuce a transfer theorem in ordinaryq-orbits and a corollary on theq-deformations ofp-adic differentialequations.
Following Christol [Chr81], in Chapter III we construct a weak Frobenius structure forq-difference systemshaving a regular singularity at 0.
In Chapter IV we prove aq-analogue of the Christol-Andre-Baldassarri-Chiarellotto transfer theorem in aregular singular disk, relying on the result of Chapter III.
The Appendix is divided in three independent parts.
Appendix A contains a proof the twisted Taylor formula stated in §1 for analytic functions over a diskand a generalization to analytic functions over non connected analytic domains. The proof is completelyelementary and the technics used in it do not play any role in the paper.
3
In Appendix B we quickly recall some basic facts about regular singularq-difference systems that weuse in Chapters III and IV.
Finally in Appendix C we have grouped some technical estimates of theq-type, which are used in ChapterIV.
I Basic definitions and properties ofp-adic q-difference systems
Until the end of the paperK will be an algebraically closed field of characteristic zero, complete with respectto a non-archimedean norm| |, inducing ap-adic norm overQ → K. We fix the normalization of| | by setting|p|= p−1.
Moreover, we fix an elementq∈ K, such that
1. |q|= 1;
2. q is not a root of unity;
3. the image ofq in the residue field ofK generates a finite cyclic group (i.e. it is algebraic overFp).
Let us consider the ring
AD =
∑n≥0
an(x− ξ)n : an ∈ K, lim infn→∞
|an|−1/n ≥ ρ
,
of analytic functions (with coefficients inK) over the open disk
D = D(ξ,ρ−) = x∈ K : |ξ− x|< ρ
of centerξ ∈ A1K and radiusρ ∈ R, ρ > 0, and the field of meromorphic functionsMD = Frac(AD) over
D. Sometimes we will writeAD,K (resp. MD,K ) to stress the fact that we are considering analytic (resp.meromorphic) functions with coefficients inK.
If D is q-invariant (i.e. if D is invariant for the isometryx 7−→ qx, or, equivalently, if|(q− 1)ξ| < ρ), itmakes sense to consider aq-difference systems (of order µ, with meromorphic coefficients over D, defined overK):
(S)q Y(qx) = A(x)Y(x) , with A(x) ∈ Glµ(MD,K).
The main purpose of this chapter is to introduce the basic properties ofq-difference systems. First we study theproperties of theq-difference algebra of analytic functions and ofq-difference operators with respect to Gaussnorms. Then we construct analytic solutions ofq-difference systems, when they exists. Finally, we define thenotions of generic point and generic radius of convergence.
1 The q-difference algebra of analytic functions over an open disk
Let the diskD = D(ξ,ρ−) beq-invariant. Then theq-difference operator
σq : f (x) 7−→ f (qx)
is aK-algebra isomorphism ofAD: we say thatAD is aq-difference algebra.
One can also define aq-derivation
dq( f )(x) =f (qx)− f (x)(q−1)x
,
satisfying the twisted Leibniz rule
dq( f g) = σq( f )dq(g)+dq( f )g .
Lemma 1.1 The operatordq acts overAD.
4
Proof. Let 0 6∈ D. Then by definitiondq( f ) ∈ AD for all f ∈ AD.If 0 ∈ D, we can supposeξ = 0. Remark that
(1.1.1) dqxn =(1+q+ · · ·+qn−1)xn−1 ∀n≥ 1.
Then for any∑n≥0anxn ∈ AD we have:
dq
(
∑n≥0
anxn
)= ∑
n≥1
(1+q+ · · ·+qn−1)anxn−1 ∈ AD .
⊓⊔
Motivated by (1.1.1), we recall the classical definition ofq-factorialsandq-binomial coefficients, namelyfor any couple of integersn≥ i ≥ 0 we set:
[0]q = 0 , [n]q = 1+q+ · · ·+qn−1 = 1−qn
1−q ,
[0]q! = 1 , [n]q! = 1q · · · [n]q ,
(n0
)
q=
(nn
)
q= 1 ,
(ni
)
q=
[n]q![n− i]q![i]q!
=[n]q[n−1]q · · · [n− i +1]q
[i]q!.
They satisfy the relation(
ni
)
q=
(n−1i −1
)
q+
(n−1
i
)
qqi =
(n−1i −1
)
qqn−i +
(n−1
i
)
q, for n≥ i ≥ 1,
and
(1.1.2) (1− x)(1−qx) · · ·(1−qn−1x) =n
∑i=0
(−1)i(
ni
)
i(i−1)2 xi .
One verifies directly the following basic properties ofσq anddq:
Lemma 1.2 (cf. [DV, 1.1.8÷10]) For any couple integersn, i ≥ 1 and anyf ,g∈ AD we have:
1.dn
q
[n]q!xi =
(ni
)qxn−i , if n≥ i,
0, otherwise;
2. dnq( f g)(x) =
n
∑j=0
(nj
)
qdn− j
q ( f )(q j x)d jq(g)(x);
3. σnq =
n
∑j=0
(nj
)
q(q−1) jq j( j−1)/2x jdi
q;
4. dnq =
(−1)n
(q−1)nxn
n
∑j=0
(−1) j(
nj
)
q−1q−
j( j−1)2 σ j
q.
The topological basis((x− ξ)n)n≥0 of AD is not adapted to study the action of theq-derivation overAD asthe relation
dq(x− ξ)n =(qx− ξ)n− (x− ξ)n
(q−1)x
clearly shows. So, rather than(x− ξ)n, one classically consider the polynomials
(x− ξ)q,0 = 1(x− ξ)q,n = (x− ξ)(x−qξ) · · ·(x−qn−1ξ), for any integern≥ 1.
5
One immediately verifies thatdq(x− ξ)q,n = [n]q(x− ξ)q,n−1 .
Remark that this generalizes (1.1.1).
The following proposition states that the natural map
(1.2.1) Tq,ξ : f (x) 7−→ ∑n≥0
dnq f
[n]q!(ξ)(x− ξ)q,n
defines an isomorphism ofq-difference algebras (i.e. an isomorphism ofK-algebras commuting to the actionof dq) betweenAD and
Kx− ξq,ρ =
∑n≥0
an(x− ξ)q,n : an ∈ K, lim infn→∞
|an|−1/n ≥ ρ
.
We will call the mapTq,ξ q-expansionor twisted Taylor formula.
Proposition 1.3 Let D = D(ξ,ρ−) be aq-invariant open disk. The map
Tq,ξ : AD −→ Kx− ξq,ρ
f (x) 7−→ ∑n≥0dn
q f
[n]q!(ξ)(x− ξ)q,n
is a q-difference algebras isomorphism. Moreover for allf ∈ AD and for allx0 ∈ D, the seriesTq,ξ( f )(x0)converges andTq,ξ( f )(x0) = f (x0).
The proof of (1.3) can be found in the Appendix (cf. §14). As a corollary we obtain the more usefulstatement:
Corollary 1.4 Let f (x) = ∑n≥0an(x− ξ)q,n be a series such thatan ∈ K and lim infn→∞
|an|−1/n = ρ. Then the
f (x0) converges for anyx0 ∈ D(ξ,ρ−) and its sum coincide with the sum of an analytic function inAD(ξ,ρ−) ifand only ifρ > |(q−1)ξ|.
Proof. If ρ > |(q−1)ξ|, the seriesf (x) converges by (1.3).Suppose thatρ < |(q− 1)ξ|. Let n0 be the smallest positive integer such that|(qkn0 − 1)ξ| ≤ ρ for any
integerk and|(qn−1)ξ|> ρ for any integern such thatn0 6 |n. For any|x0− ξ|< ρ we have
|(x0− ξ)n,q| =n−1
∏i=0
|(x0− ξ)+ ξ(1−qi)|
> ρn−[
nn0
]−1 ∏
0<i≤[
n−1n0
]|x0− ξ+ ξ(1−qin0)| .
We conclude thatsup
|x0−ξ|<ρ|(x0− ξ)n,q|> ρn
and hence that∑n≥0an(x0− ξ)q,n does not converge overD(ξ,ρ−).Let ρ = |(q−1)ξ|. Since∑n≥0an(x− ξ)q,n is convergent, the series
f (qx) = a0+(qx− ξ)q ∑n≥1
an(x− ξ)q,n−1
must also converge. By induction we conclude that that for any x0 ∈ D(ξ,ρ−) and any integern≥ 0, the sumf (qnx0) is convergent. Hencef (x) converges over a disk of radius strictly greater thanρ, which, by (1.3),means that liminf
n→∞|an|
−1/n > ρ. ⊓⊔
6
2 Gauss norms andq-difference operators
Let D = D(ξ,ρ−). We recall thatAD comes equipped with a family of non archimedean norms| |ξ(R), theso-called Gauss norms, (cf. for instance [Rob00, §6, 1.4])
∥∥∥∥∥∑n≥0
fn(x− ξ)n
∥∥∥∥∥ξ
(R) = supn≥0
| fn|Rn ,
defined for anyR∈ (0,ρ) and any∑n≥0 fn(x−ξ)n ∈ AD. It follows by Gauss lemma that they are multiplicativenorms. IfR∈ |K|, then (cf. [DGS94, IV, 1.1])
∥∥∥∥∥∑n≥0
fn(x− ξ)n
∥∥∥∥∥ξ
(R) = supx0∈K, |x0−ξ|<R
∣∣∣∣∣∑n≥0bn(x0− ξ)n
∣∣∣∣∣ .
The norm‖ ‖ξ(R) plays a central role in the study ofp-adic differential equation as well as ofp-adic q-
difference equations, therefore it is crucial to calculatethe norm of theK-linear operatordk
q[k]q! with respect to
| |ξ(R) as well as to be able to calculate the norm off ∈ AD looking at itsq-expansion:
Proposition 2.1 Let f (x) ∈ AD(ξ,ρ−). If |(1−q)ξ| ≤ R< ρ, the norm‖ ‖ξ(R) verifies
(2.1.1)
∥∥∥∥∥dk
q
[k]q!f (x)
∥∥∥∥∥ξ
(R)≤1Rk ‖ f (x)‖ξ (R) for all k∈ Z≥0;
and
(2.1.2) ‖ f (x)‖ξ(R) = supn≥0
∣∣∣∣dn
q( f )
[n]q!(ξ)∣∣∣∣R
n .
Remark. Inequality (2.1.1) generalizes [DV, 4.2.1], where we considered the caseξ = 0 andR= 1.
Proof. Let us prove (2.1.1). Sincedk
q
[k]q!f (x) = ∑
n≥k
fndk
q
[k]q!(x− ξ)n ,
it is enough to prove that∥∥∥∥∥
dkq
[k]q!(x− ξ)n
∥∥∥∥∥ξ
(R)≤1Rk ‖(x− ξ)n‖ξ (R) = Rn−k , for anyn≥ k.
We proceed by double induction overk,n.Let k= 1. If n is an odd positive integer the inequality immediately follows by
dq(x− ξ)n =(qx− ξ)n− (x− ξ)n
(q−1)x=
n−1
∑i=0
(qx− ξ)n−1−i(x− ξ)i .
If n is an even positive integer we reduce to the precious case observing that
dq(x− ξ)n =(qx− ξ)n− (x− ξ)n
(q−1)x
=(qx− ξ)n/2− (x− ξ)n/2
(q−1)x
((qx− ξ)n/2+(x− ξ)n/2
).
7
Let nowk> 1. The inequality is clear forn= k. It follows from the twisted Leibniz Formula (1.2) that for anyn≥ k we have ∥∥∥∥
dkq
[k]q! (x− ξ)n+1
∥∥∥∥ξ(R)
=
∥∥∥∥(qkx− ξ) dkq
[k]q! (x− ξ)n+dk−1
q[k−1]q! (x− ξ)n
∥∥∥∥ξ(R)
≤ sup
(∥∥∥∥(qkx− ξ) dkq
[k]q! (x− ξ)n
∥∥∥∥ξ(R),
∥∥∥∥dk−1
q[k−1]q! (x− ξ)n
∥∥∥∥ξ(R)
)
≤ Rn+1−k ,
which achieves the proof of (2.1.1).Clearly (2.1.1) implies that
‖ f (x)‖ξ(R)≥ supn≥0
∣∣∣∣dn
q( f )
[n]q!(ξ)∣∣∣∣R
n .
So it is enough to prove the opposite inequality to obtain (2.1.2). By (1.3) we havef (x0) = Tq,ξ( f )(x0), for anyx0 ∈ D. Then it is enough to remark that
∥∥(x− ξ)q,n∥∥
ξ (R) =
∥∥∥∥∥n−1
∏i=0
[(x− ξ)+ ξ(1−qi)
]∥∥∥∥∥
ξ
(R)≤ Rn for any integern≥ 0,
to conclude that
‖ f (x)‖ξ(R)≤ supn≥0
∣∣∣∣dn
q( f )
[n]q!(ξ)∣∣∣∣∥∥(x− ξ)q,n
∥∥ξ (R)≤ sup
n≥0
∣∣∣∣dn
q( f )
[n]q!(ξ)∣∣∣∣R
n .
This achieves the proof. ⊓⊔
The following lemma will be useful in (6.3), where we will consider some properties of families ofq-difference systems deforming a differential system.
Lemma 2.2 Let f (x) = ∑n≥0an(x− ξ)n andg(x) = ∑n≥0bn(x− ξ)q,n be two analytic bounded functions overD(ξ,1−), with |ξ| ≤ 1, and let0< |1−q| ≤ ε.
Suppose‖ f −g‖ξ (1)≤ ε. Then∥∥∥d f
dx −dq(g)∥∥∥
ξ(1)≤ ε
Proof. Remark that for any positive integern we have
|[n]q−n|= |(q−1)+ · · ·+(qn−1−1)| ≤ supi=1,...,n−1
|qi −1| ≤ |q−1|< ε
and‖(x− ξ)n‖ξ (1) =
∥∥(x− ξ)q,n∥∥
ξ (1) = 1 .
We can assume that both‖ f‖ξ(1)≤ 1 and‖g‖ξ(1)≤ 1. Therefore one conclude that∥∥∥d f
dx −dq(g)∥∥∥
ξ(1) =
∥∥∑n≥1ann!(x− ξ)n−1−bn[n]q!(x− ξ)q,n−1∥∥
ξ (1)
≤ supn≥1
(|ann! −bnn!| ,
∣∣bnn! −bn[n]q!∣∣ ,
∣∣bn[n]q!∣∣∥∥(x− ξ)n−1− (x− ξ)q,n−1
∥∥ξ (1)
)
≤ sup(|n!|ε, |(1−q)bn|,
∣∣bn[n]q!(1−q)∣∣)
≤ sup(ε, |1−q|)< ε .
⊓⊔
8
3 Analytic solutions ofq-difference systems
Let us consider aq-difference system of orderµ
(S)q Y(qx) = A(x)Y(x) ,
whose coefficients are meromorphic functions over aq-invariant open diskD(ξ,ρ−). The system(S)q can berewritten in the form
(S)′q dq(Y)(x) = G(x)Y(x), with G(x) =A(x)− Iµ
(q−1)x,
whereIµ is the identity matrix of orderµ. One can iterate(S)′q obtaining
dnq(Y)(x) = Gn(x)Y(x) ,
with
(3.0.1)G0(x) = Iµ, G1(x) = G(x) andGn+1(x) = Gn(qx)G(x)+dq(Gn)(x), for any integern≥ 1.
If G(x) does not have any pole inqNξ it can be identified to an element of the ring
K [[x− ξ]]q =
∑n≥0
an(x− ξ)q,n : an ∈ K
by considering itsq-expansion. Then a formal solution matrix of(S)q, or equivalently of(S)′q, atξ is given by
(3.0.2) Y(ξ,x) = ∑n≥0
Gn(ξ)[n]q!
(x− ξ)q,n .
The fact thatY(ξ,ξ) = Iµ does not allow to conclude thatY(ξ,x) is an element ofGlµ(K [[x− ξ]]q), sinceK [[x− ξ]]q is not a local ring (cf. Appendix A, §15). Actually we need a stronger assumption:
Lemma 3.1 The system(S)q has a formal solution matrix inGlµ(K [[x− ξ]]q) if and only if the matrixA(x)
does not have any pole inqNξ anddetA(x) does not have any zero inqNξ.
Remark 3.2
1) If the conditions of the lemma above are verified, then (3.0.2) is the only solution of(S)q in Glµ(K [[x− ξ]]q)such thatY(ξ,ξ) = Iµ and all other solution matrices of(S)q in Glµ(K [[x− ξ]]q) are obtained by multiplingY(ξ,x) by an element ofGlµ(K).
2) Remark that, if(S)q has a solution matrixY(x) ∈ Glµ(AD) over aq-invariant diskD, the matrixA(x) =Y(qx)Y(x)−1 is an element ofGlµ(AD). Hence neitherA(x) has a pole inqZξ nor detA(x) has a zero inqZξ.It follows by the statement above that(S)q can have a formal solution inGlµ(K [[x− ξ]]q) which is not theq-expansion of an analytic solution.
3) Suppose thatY(ξ,x) is theq-expansion of an analytic solution of(S)q converging overD(ξ,ρ−) and let|ζ− ξ|< ρ. Then necessary we have
(3.2.1) Y(ζ,x) =Y(ξ,x)Y(ξ,ζ)−1 ,
since both matrices are analytic solution matrix of(S)q atζ, of maximal rank, having valueIµ atζ.
Proof. By the remark above, the system(S)q has a formal solution matrix inGlµ(K [[x− ξ]]q) if and only ifY(ξ,x) is in Glµ(K [[x− ξ]]q).
For any non negative integerk it makes sense to evaluateY(ξ,x) atqkξ:
Y(ξ,qkξ) = ∑n≥0
Gn(ξ)[n]q!
ξn(qk−1)q,n ,
9
since the sum on the right hand side is actually finite. Of course there are precise relations between the se-
quences(
Gn(ξ)[n]q!
)k≥0
and (Y(ξ,qkξ))k≥0 that one can easily deduce by (1.2, 3 and 4). It turns out that an
elementY(x) ∈ Mµ×µ(K [[x− ξ]]q) is uniquely determined by(Y(qkξ))k≥0. ThereforeY(ξ,x) ∈ Glµ(K [[x− ξ]]q)if and only if Y(ξ,qkξ) ∈ Glµ(K) for all k ≥ 0, the inverse ofY(ξ,x) being the element ofGlµ(K [[x− ξ]]q)associated to the data(Y(ξ,qkξ)−1)k≥0.
To conclude it is enough to remark that
Y(ξ,qkξ) = A(qk−1ξ)A(qk−2ξ) · · ·A(ξ)Y(ξ,ξ)= A(qk−1ξ)A(qk−2ξ) · · ·A(ξ) .
⊓⊔
In the next corollaries we give some sufficient conditions for having a fundamental analytic solution matrix,i.e. an invertible solution matrixY(x) such thatY(x) andY(x)−1 have analytic coefficients over a convenientq-invariant open disk. It is just a partial result and we will reconsider the problem of the existence of analyticsolutions for(S)q in the next sections.
By (1.3) one immediately obtain the corollary:
Corollary 3.3 The system(S)q has a fundamental analytic solution matrix atξ if and only if
- the matrixA(x) does not have any poles inqNξ,
- detA(x) does not have any zeros inqNξ,
- limsupn→∞
∣∣∣∣Gn(ξ)[n]q!
∣∣∣∣1/n
< |(q−1)ξ|−1.
Before stating the following result we need to introduce thenumberπq, playing a role analogous to theπof Dwork for p-adic differential equation system. We recall thatπ is an element ofK such thatπp−1 =−p.
Definition 3.4 Let κ the smallest positive integer such that|1−qκ|< |π|. Thenπq is an element ofK definedby
πp−1q =−[p]q, if κ = 1, andπκ
q = [κ]qπqκ , if κ > 1.
It follows by [DV, 4.1.1] that|πqκ |= |π| and that
limn→∞
|[n]q!|1/n = |πq| .
Corollary 3.5 Let ρ ≤ 1, ρ|π| > |(1−q)ξ| andD = D(ξ,ρ−). Suppose we are given a square matrixG(x)analytic overD such that
supx∈D
|G(x)| ≤1ρ
and that the determinant ofA(x) = (q−1)xG(x)+ Iµ does not have any zero inqNξ. ThenY(qx) = A(x)Y(x)has an analytic fundamental solution atξ.
Proof. It follows by (2.1.1) and (3.0.1) that
|Gn(ξ)| ≤1ρ
sup(1,‖Gn−1(ξ)‖)≤1ρn , for anyn≥ 1,
which implies that
limsupn→∞
∣∣∣∣Gn(ξ)[n]q!
∣∣∣∣1/n
=1
|πq|limsup
n→∞|Gn(ξ)|1/n ≤
1ρ|πq|
.
Sinceρ|πq|> |(1−q)ξ|, the matrixY(x,ξ) in (3.0.2) is theq-expansion of an analytic fundamental solution.⊓⊔
10
3.6 Iteration of (S)q and existence of analytic solutions.
In the rest of the paper we will often assume that(S)q has an analytic fundamental solution at some pointζ orthe|1−q| is smaller than some constant: this is not always true.
In some cases one can easily reduce to the assumption of having a fundamental analytic solution by iteratingtheq-difference system. In the same way one can reduce to the caseof a q∈ K such that|1−q|<< 1.
Let us analyze the situation in detail. Suppose that(S)q does not have an analytic fundamental solution atζ. Then it may happen that there existsn0 > 1 such that the system
(S)qn0 Y(qn0x) = An0(x)Y(x) , with An0(x) = A(qn0−1x)A(qn0−2x) · · ·A(x),
obtained from(S)q by iteration, has a fundamental analytic solutionY(x) over a qn0-invariant open diskD(ζ,η−). If n0 is the smallest positive integer having this property, thenone can construct a fundamentalsolutionF(x) of (S)q, analytic over the non connectedq-invariant analytic domain
(3.6.1) D(ζ,η−)∪D(qζ,η−)∪·· ·∪D(qn0−1ζ,η−)
by settingF(qix) = Ai(x)Y(x) = A(qi−1x)A(qi−2x)A(qi−1x) · · ·A(x)Y(x) ,for anyx∈ D(ζ,η−) and anyi = 0, . . . ,n0−1.
Remark that the restriction ofF(x) to D(qiζ,η−), for anyi ∈ Z, is an analytic fundamental solution of(S)qn0 .So, if theq-difference system has an analytic solution over a non connected analytic domain as above, it is
enough to consider a system obtained by iteration to reduce to the case of a system having an analytic solutionover aq-invariant open disk. In the appendix we will consider theq-expansion of analytic functions over nonconnected domain of the form (3.6.1).
The same trick allows to reduce to the case of a small|1−q|, knowing that liminfn→∞ |1−qn|= 0.
3.7 Removing apparent and trivial singularities.
In this subsection we will considerq-difference system having meromorphic solutions or analytic solutionswith meromorphic inverse. Our purpose is to explain how to reduce by gauge transformation to the assumptionof having an analytic fundamental solution.
Once again we consider aq-difference system
(S)q Y(qx) = A(x)−1Y(x)
with meromorphic coefficients over aq-invariant diskD = D(ξ,ρ−), defined overK. For any matrixF(x) ∈Glµ(MD), the matrixZ(x) = F(x)Y(x) is solution of
(3.7.1) Z(qx) = A[F](x)Z(x) , with A[F](x) = F(qx)A(x)F(x)−1.
The matrixF(x) is usually called ameromorphic gauge transformation matrix. Remark that
(3.7.2)
G[F ](x) =A[F](x)− Iµ
(q−1)x
= F(qx)A(x)− Iµ
(q−1)xF(x)−1+dq(F)(x)F(x)−1
= F(qx)G(x)F(x)−1+dq(F)(x)F(x)−1.
Following the classical terminology ofp-adic differential equations (cf. for instance [DGS94, page 172])we give the definition:
Definition 3.8 We say thatqZξ ⊂ D is an ordinaryq-orbit (resp. trivial singularity, apparent singularity)for(S)q, if (S)q has a solution inGlµ(AD′) (resp.Glµ(MD′)∩Mµ×µ(AD′), Glµ(MD′)), whereD′ ⊂D is aq-invariantanalytic domain of the form (3.6.1) containingξ.
11
The following statement is aq-analogue of the Frobenius-Christol device (cf. [Chr81, II, §8]) to removeapparent and trivial singularities over
D× =
D if 0 6∈ D,
D× = Dr 0 otherwise.
Proposition 3.9 We assume that
(3.9.1) the system(S)q has at worst a finite number of apparent singularities inD×.
Then there existsH(x) ∈ Glµ(K(x)) such that theq-difference systemY(qx) = A[H](x)Y(x) has only ordinaryorbits inD×.
The proposition immediately follows by the more precise statement:
Proposition 3.10 Suppose that (3.9.1) is verified. The the following propositions hold:1) There exists a polynomialP(x)∈K[x], with P(0) 6= 0, such that theq-difference systemY(qx)=A[PIµ](x)Y(x)has only trivial singularities inD×.2) Suppose that(S)q has only trivial singularities inD×. Then there existsH(x) ∈ Glµ(K(x)) such that- H(x) does not have a pole at0,- H(0) ∈ Glµ(K),- ‖H(x)‖0,ρ = ρ−1 and
∥∥H(x)−1∥∥
0,ρ = ρ,
- theq-difference systemY(qx) = A[H](x)Y(x) has only ordinary orbits inD×.
Proof. Let P(x) ∈ K[x] be a polynomial such that for anyζ ∈ D× and any solution matrixUζ(x) meromorphicon a convenientq-invariant analytic domain of centerζ, the matrixP(x)Uζ(x) is analytic atζ. ThenP(x)Uζ(x)is a solution matrix of theq-difference system associated toA[PIµ](x), which has only trivial singularities inD×.This achieves the poof of the first part of the statement.
Now we prove2). Let ζ ∈ D× and letY(x) ∈ Mµ×µ(AD′)∩Glµ(MD′) be a solution atζ of (S)q. If (S)q
has a trivial singularity atqZζ, then necessarily detY(x) has a zero inqZζ. Let qnζ be a zero of orderk> 0 ofdetY(x) and let~B= t(B1, . . . ,Bµ) is a non zero vector inKµ such that~BY(qnζ) = 0. We fix 1≥ ι ≤ µ such that|Bι| = maxj=1,...,µ |B j | > 0. Of course one can suppose thatBι = 1. Let us consider the gauge transformationmatrix
H(x) =
Ii−1
0...0
0
B1
x−qnζ· · ·
Bι−1
x−qnζ1
x−qnζBι+1
x−qnζ· · ·
Bµ
x−qnζ
0
0...0
Iµ−i
.
Since~BY(qnζ) = 0, the matrixH(x)Y(x) is still analytic atqnξ and
H(x)−1 =
Ii−1
0...0
0
−B1 · · · −Bι−1 x−qnζ −Bι+1 · · · −Bµ
0
0...0
Iµ−i
.
12
Moreover det(H(x)Y(x)) has a zero atqnζ of orderk− 1. By iteration, one can construct a basis changesatisfying all the conditions in2). ⊓⊔
4 Generic points
We consider an extensionΩ/K of ultrametric fields such thatΩ is complete, algebraically closed, the setof values ofΩ is R≥0 and that the residue field ofΩ is transcendental over the residue field ofK (for theconstruction of such a field see for instance [Rob00, §3, 2]).For anyR∈ R≥0 the fieldΩ contains an elementtR, transcendent overK, such that|tR|= R and that the norm induced byΩ overK(tR) is necessary defined by
∣∣∣∣∣∑ait i
R
∑b j tjR
∣∣∣∣∣=supi |ai |Ri
supj |b j |Rj .
Therefore, if f (x) is an analytic function over a disc of center 0 and radiusρ > Rwe have
| f (tR)|= ‖ f (x)‖0 (R) = supx∈Ω,|x|<R
| f (x)| .
Definition 4.1 We call tR a generic point with respect toK (at distanceR from 0). A generic point tξ,R withrespect to K (at distance R fromξ ∈ K) is defined by shifting.
Dwork-Robba’s generic points play a fundamental role inp-adic differential equation theory: historically,their introduction has been the first attempt to fill in the gapleft by the absence of ap-adic analytic continuation.In fact, the radius of solutions attξ,R is a sort of global invariant for the differential equationsthat is equal tothe radius of convergence of solutions at almost any point ofthe diskD(ξ,R−) and allows for an estimate inthe other points. From a more recent point of view, one shouldthink of generic points as points of a Berkovichanalytic space.
Consider aq-difference system of orderµ with meromorphic coefficients over aq-invariant diskD(ξ,ρ−):
(S)q Y(qx) = A(x)Y(x) , with A(x) ∈ Glµ(MD,K).
As in the previous section (cf. (3.0.2)), for anyR< ρ one can consider the formal solution of(S)q at tξ,R:
(4.1.1) Y(tξ,R,x) = ∑n≥0
Gn(tξ,R)
[n]q!(x− tξ,R)q,n ∈ Glµ(Ω
[[x− tξ,R
]]q) .
Remark that, the system(S)q being defined overK, the matrixA(x) cannot have any poles inqNtξ,R and detA(x)cannot have any zeros inqNξ: it follows thatY(tξ,R,x) is necessary invertible inGlµ(Ω
[[x− tξ,R
]]q).
Definition 4.2 We call generic radius of convergence of(S)q at tξ,R the number
χξ,R(A,q) = inf
(R, lim inf
n→∞
∣∣∣∣Gn(tξ,R)
[n]q!
∣∣∣∣−1/n
).
One easily proof that:
Lemma 4.3 Let 1≥ R≥ |(1−q)ξ|. The generic radius of convergenceχξ,R(A,q) is invariant under meromor-phic gauge transformation and
χξ,R(A,q)≥R|πq|
sup(1, |G1(tξ,R)|
) .
Proof. The first assertion is proved in [DV, 4.2.3] in the caseξ = 0 andR= 1, but the same proof apply to thiscase. The second assertion is immediately deduced by the recursive relation verified by theGn(x)’s and (2.1.1),since limn→∞ |[n]q!|1/n = |πq|. ⊓⊔
13
Lemma 4.4 If A(x) is analytic over the diskD(ξ,ρ−) andρ ≥ χξ,R(A,q) > |(1−q)ξ|, then(S)q has a funda-mental analytic solution overD(ξ,χξ,R(A,q)
−).
Proof. It is enough to remark that ∣∣∣∣Gn(ξ)[n]q!
∣∣∣∣≤∣∣∣∣Gn(tξ,R)
[n]q!
∣∣∣∣ .
⊓⊔
The theorems estimating the radius of convergence ofY(ξ,x) with respect toχξ,R(A,q) are usually calledtransfer theorems: in the next chapter we will prove a transfer theorem from a disk where an analytic solutionexists to a contiguous disk, where the system has only ordinary orbits. This result is a consequence of theeffective bound theorem. The remaining part of the paper is voted to the proof of a transfer theorem for regularsingular disks.
4.5 The cyclic vector lemma and theq-analogue of the Dwork-Frobenius theorem.
It may seem that calculate a generic radius of convergence isas difficult as calculate a radius of convergence inpoints which are rational overK. This is not completely true, in fact the generic radius of convergence is veryeasy to calculate when it is small, using aq-difference equation associated to(S)q: it is theq-analogue of theDwork-Frobenius theorem [DGS94, VI, 2.1].
As in the differential world, aq-difference equation associated toY(qx)=A(x)Y(x), with A(x)∈Glµ(MD,K),is constructed using a cyclic vector lemma (cf. for instance [Sau00a, Annexe B] or [DV, 1.3]), stating the exis-tence of a meromorphic matrixH(x) ∈ Glµ(MD,K) such that
(4.5.1) A[H] =
0... Iµ−1
0
a0(x) a1(x) . . .aµ−1(x)
.
Theny(x) is a solution of theq-difference equations
(4.5.2) y(qµx)−aµ−1(x)y(qµ−1x)−·· ·−a0(x)y(x) = 0
if and only if
y(qx)y(q2x)
...y(qµx)
= A[H]
y(x)y(qx)
...y(qµ−1x)
Consider the lower triangular gauge transformation matrix
(4.5.3) H = (ai, j)i, j=0,...,µ−1 , with ai, j =
1xi
(−1)i+ j
(q−1)i
(ij
)
q−1q−
j( j−1)2 if j ≤ i;
0 otherwise.
It follows by (1.2) that
G[HH] =A[HH]− Iµ
(q−1)x=
0... Iµ−1
0
b0(x) b1(x) . . .bµ−1(x)
14
and thaty(x) is a solution of theq-difference equations (4.5.2) if and only if
dqy(x)d2
qy(x)...
dµqy(x)
= G[HH]
y(x)dqy(x)
...dµ−1
q y(x)
.
Proposition 4.6 Let |1−q|< |π|. If supi=0,...,µ−1 |ai(t0,R)|> 1 then
χ0,R(A,q) =R|π|
supi=0,...,µ−1 |bi(t0,R)|1/(µ−i).
Proof. The proposition is proved in [DV, 4.3] in the caseR= 1, ξ = 0. We deduce the statement above byrescaling. ⊓⊔
II Effective bounds for q-difference systems
In this chapter we prove an effective bound theorem forq-difference systems: it is the analogue of a theoremby Dwork and Robba (cf. [DR80] for the proof in the case of differential equations. The statement concerningdifferential systems is proved for instance in [Bom81], [And89] and [DGS94]).
Let us just explain the effective bound theorem for an analytic differential equations of order one: thetheorem is actually almost trivial in this case, but we can already point out the difference with theq-differenceversion.
Let ξ ∈ A1K , ξ 6= 0, and letu(x) be a meromorphic function over an open disk of centerξ and radiusρ > 0.
For all R∈ (0,ρ), the multiplicative norm‖ ‖ξ(R) (cf. §2) induces a norm over the field of meromorphicfunctions overD(ξ,ρ−). Let
gn(x) =1n!
dnudxn (x)u(x)
−1 , ∀n≥ 0 .
The effective bound theorem for differential equation of order 1 states that for anyR∈ (0,ρ)
‖gn(x)‖ξ(R)≤ R−n .
Of course this inequality is easy to prove, in fact
‖gn(x)‖ξ(R) ≤
∥∥∥∥1n!
dnudxn (x)
∥∥∥∥ξ(R)∥∥u−1(x)
∥∥ξ (R)
≤ R−n‖u(x)‖ξ(R)‖u(x)‖ξ(R)−1
≤ R−n .
The multiplicativity of‖ ‖ξ(R) is the key point of the inequalities above.Let us consider a meromorphicq-difference system: its solutions may be meromorphic over anon con-
nected analytic domain (as the one considered in (3.6.1)). Therefore a natural analogue of‖ ‖ξ(R) wouldbe a sup-norm over a non connected domain, which is necessarynon multiplicative. In particular the norm‖u(x)−1‖ξ(R) could be greater than‖u(x)‖ξ(R)
−1. In other words, the assumption of having a solution matrix(analytic or meromorphic) over anq-invariant disk cannot be avoid.
5 Effective bound theorem forq-difference systems
Let D = D(ξ,ρ−) be an open disk of centerξ ∈A1K andMD be the field of meromorphic functions overD. For
anyR∈ (0,ρ) the norm‖ ‖ξ(R) extends fromAD to MD by multiplicativity. For anyf (x) ∈ MD one usuallysets:
‖ f (x)‖ξ,ρ = limR→ρ
‖ f (x)‖ξ(R) .
15
This limit can be not bounded, but respects multiplication of functions, as well as‖ ‖ξ(R).We suppose thatD is q-invariant and we considerY(x)∈Glµ(MD). ThenY(x) is solution of theq-difference
system defined byGn(x) = dn
qY(x)Y(x)−1 , ∀n≥ 0.
Obviously the entries ofGn(x) are meromorphic functions overD.
Proposition 5.1 ∥∥∥∥Gn(x)[n]q!
∥∥∥∥ξ,ρ
≤ n,µ−1qp
(sup
i=0,...,µ−1‖Gi(x)‖ξ,ρ ρi
)1ρn ,
where
n,µ−1qp =
1, if n=0,
sup1≤λ1<···<λµ−1≤n,λi∈Z
1∣∣[λ1]q · · · [λµ−1]q∣∣ , otherwise.
One can also consider a matrixY(x) analytic over aq-invariant disk aroundtξ,R, with coefficients inΩ, andassume that it is solution of aq-difference system(S)q defined overK. Then:
Corollary 5.2
∣∣∣∣Gn(tξ,R)
[n]q!
∣∣∣∣≤ n,µ−1qp
(sup
i=0,...,µ−1
∣∣Gi(tξ,R)∣∣χξ,R(A,q)
i
)1
χξ,R(A,q)n .
The proof of theorem 5.1 (cf. (5.5) below) follows the proof of Dwork-Robba theorem concerning theeffective bounds forp-adic differential systems. As in the differential case, itrelies on the analogous result forq-difference equations.
5.3 Effective bounds forq-difference equations.
We consider~u= (u1, . . . ,uµ) ∈ M µD , with µ∈ Z, µ≥ 1, such thatu1, . . . ,uµ they are linearly independent over
the field of constantK of MD. By theq-analogue of the Wronskian lemma (cf. [Sau00a, Appendice] or [DV,§1.2]) this is equivalent to suppose that theq-Wronskian matrix
Wq(~u) =
u1(x) . . . uµ(x)dqu1(x) . . . dquµ(x)
.... . .
...dµ−1
q u1(x) . . . dµ−1q uµ(x)
is in Glµ(MD).We consider theq-difference equation defined by
dnq~u=~gn(x)Wq(~u) , where~gn ∈ M µ
D andn≥ 0.
Lemma 5.4
(5.4.1)
∥∥∥∥~gn
[n]q!
∥∥∥∥ξ,ρ
≤ n,µ−1qp
1ρn .
Proof. By extendingK, we can findα ∈ K such that|α| = ρ. If we sety= αx andζ = αξ, we are reduced toprove the lemma forρ = 1.
So we supposeρ = 1 and we prove the lemma by induction onµ. Forµ= 1 and anyR∈ ((q−1)ξ,1) wehave (cf. (2.1.1)) ∥∥∥∥
~gn
[n]q!
∥∥∥∥ξ(R)≤
∥∥∥∥dn
qu1
[n]q!(x)
∥∥∥∥ξ(R)‖u1(x)
−1‖ξ(R)≤1Rn .
16
LettingR→ 1, we obtain (5.4.1).Let µ> 1. Forn= 0 the inequality is trivial, so letn> 0. We set:
~u= (u1, . . . ,uµ+1) = u(1,~τ) ∈ M µ+1D , whereu= u1 and~τ = (τ1, . . . ,τµ) ∈ M µ
D .
The idea of the proof is to apply the inductive hypothesis todq~τ. We know that the vector~hn ∈ M µD , defined for
anyn≥ 0 bydn
q (dq~τ) =~hn(x)Wq (dq~τ) ,
satisfies the inequality ∥∥∥∥∥~hn
[n]q!
∥∥∥∥∥ξ,1
≤ n,µ−1qp .
Remark that~hn and~gn verify the relation
(5.4.2)
~gn
[n]q!Wq(~u) =
dnq
[n]q!~u=
dnq
[n]q!(u(1,~τ))
=dn
q
[n]q!(u)(1,~τ)+
(0,
n
∑j=1
dn− jq
[n− j]q!(u)(q jx)
1[ j]q
~h j−1
[ j −1]q!
(0,Wq(dq~τ)
)),
while Wq(~u) andWq(dq~τ) verify
Wq(~u) =Wq (u(1,~τ)) = uP
(1 ~τ0 Wq(dq~τ)
),
where
P=1
u(x)
u(x) 0 0 · · · 0
dqu(x) u(qx) 0 · · ·...
(22
)qd2
qu(x)(2
1
)qdqu(qx) u(q2x) · · ·
......
...... · · ·
......
...... · · · 0(µ
µ
)qdµ
qu(x)( µ
µ−1
)qdµ−1
q u(qx)( µ
µ−2
)qdµ−2
q u(q2x) · · · u(qµx)
.
Since∥∥∥( µ
µ−i
)qdµ−i
q u(qix)u(x)−1∥∥∥
ξ,1≤ 1 and
∥∥u(qix)u(x)−1∥∥
ξ,1 = 1, we have‖P‖ξ,1 = ‖detP‖ξ,1 = 1 and hence
‖P−1‖ξ,1 ≤ 1. This implies that
∥∥∥∥(
1 ~τ0 Wq(dq~τ)
)Wq(~u)
−1
∥∥∥∥ξ,1
= ‖u−1P−1‖ξ,1 ≤ ‖u−1‖ξ,1 .
We deduce the desired inequality by (5.4.2):
∥∥∥∥~gn
[n]q!
∥∥∥∥ξ,1
≤ supj=1,...,n
1,
∥∥∥∥∥~h j−1
[ j]q!
∥∥∥∥∥ξ,1
≤ supj=1,...,n
(1,
∥∥∥∥1[ j]q
∥∥∥∥ξ,1
j −1,µ−1qp
)
≤ j,µ−1qp .
⊓⊔
17
5.5 Proof of theorem 5.1.
We recall that we are given a matrixY(x) ∈ Glµ(MD) meromorphic over aq-invariant open diskD and that wehave set
Gn(x) = dnqY(x)Y(x)−1 ∈ Mµ×µ(MD) .
We want to prove that∥∥∥∥
Gn(x)[n]q!
∥∥∥∥ξ,ρ
≤ n,µ−1qp
(sup
i=0,...,µ−1‖Gi(x)‖ξ,ρ ρi
)1ρn .
First of all we remark that:1) as in the case ofq-difference equations it is enough to prove the inequality for ρ = 1;2) it is enough to prove the inequality above for the first row of Gn(x).So we supposeρ = 1 and we call~u= (u1, . . . ,uµ) ∈ M µ
D the first row ofY(x). Let k≤ µ be the rank ofWq(~u)andE ∈ Glµ(K) be such that
~uE= (~z,0) , with~z∈ M kD and 0= (0, . . . ,0︸ ︷︷ ︸
×(n−k)
) .
Remark that
~udq~u
...dk−1
q ~u
E = (Wq(~z),0) ∈ Mk×µ(MD) .
We set:dn
q(~z) =~hnWq(~z) .
In (5.4) we have proved that ∣∣∣∣∣~hn
[n]q!
∣∣∣∣∣ξ,1
≤ n,µ−1qp .
We need to expressdn
q[n]q! (~u) in terms of
~hn[n]q! :
dnq
[n]q!(~u)E =
dnq
[n]q!(~uE) =
dnq
[n]q!(~z,0) =
~hn
[n]q!(Wq(~z),0) =
~hn
[n]q!
~udq~u
...dk−1
q ~u
E
and therefore
dnq
[n]q!(~u) =
~hn
[n]q!
~udq~u
...dk−1
q ~u
.
Finally we deduce that
(first row of
Gn(x)[n]q!
)=
~hn
[n]q!
~udq~u
...dk−1
q ~u
Y(x)−1
and hence that
∥∥∥∥first row ofGn(x)[n]q!
∥∥∥∥ξ,1
≤∥∥∥ ~hn[n]q!
∥∥∥ξ,1
∥∥∥∥∥∥∥∥∥
~udq~u
...dk−1
q ~u
Y(x)−1
∥∥∥∥∥∥∥∥∥ξ,1
≤ n,µ−1qp
(supi=0,...,k−1‖first row ofGi(x)‖ξ,1
).
18
This achieves the proof.
6 Some consequences: a transfer theorem in ordinary points and a corollary aboutq-deformations
6.1 Transfer theorem for ordinary disks
As in p-adic differential equation theory, the following transfer result follows from the effective bound theorem:
Corollary 6.2 Let G(x) ∈ Mµ×µ(MD(ξ,ρ−)) be a meromorphic matrix over the diskD(ξ,ρ−). We suppose thatdqY(x) = G(x)Y(x) has a meromorphic fundamental solution over theq-invariant diskD(ξ,η−), with η < ρ,and that exists a pointζ such that- |ξ− ζ|= η,- G(x) is analytic overD(ζ,η−).
ThendqY(x) = G(x)Y(x) has an analytic solution inD(ζ,η−).
Proof. Let dnqY(x) = Gn(x)Y(x) for any non negative integern. By the previous theorem we have
∣∣∣∣Gn(tξ,η)
[n]q!
∣∣∣∣=∥∥∥∥
Gn(x)[n]q!
∥∥∥∥ξ,η
≤Cn,µ−1qp
1ηn ,
whereC is a constant depending only onG(x) andη. The pointtξ,η is also a generic point at distanceη fromζ, hence ∣∣∣∣
Gn(ζ)[n]q!
∣∣∣∣≤∣∣∣∣Gn(tξ,η)
[n]q!
∣∣∣∣ .
Let κ be the smallest positive integer such that|1− qκ| < |π| and let pln be the greatest integer power ofpsmaller or equal ton. Then we have
n,µ−1qp ≤
∣∣∣∣[plnκ
]q
∣∣∣∣1−µ
≤(|pln[κ]q|
)1−µ=(
p−ln|[κ]q|)1−µ
≤ nµ−1|[κ]q|1−µ .
We conclude by (1.3), since
limsupn→∞
∣∣∣∣Gn(ζ)[n]q!
∣∣∣∣1/n
≤ η−1 limsupn→∞
n(µ−1)/n|[κ]q|−(µ−1)/n = η−1 .
⊓⊔
6.3 Effective bounds andq-deformation of p-adic differential equations
Let qk ∈K, k∈N, be a sequence such thatqk → 1 whenk→ ∞ andG(k)(x) a sequence of square matrix of orderµ whose entries areanalytic bounded functionsover an open diskD of centerξ and radiusρ, with |ξ| ≤ ρ ≤ 1.By rescaling, we can assume thatD = D(ξ,1−). Suppose that the sequence of matricesG(k)(x) tends to amatrixG(x) uniformly overD. Then we say that the family of systems
(S)′qkdqkY(x) = G(k)(x)Y(x)
is aq-deformation of differential system
(S)dYdx
(x) = G(x)Y(x) .
Let G(k)n (x) andGn(x), for n≥ 0, be the matrices respectively defined by
dnqk
Y(x) = G(k)n (x)Y(x) and
dnYdxn (x) = Gn(x)Y(x) .
We assume that‖G(x)‖ξ,1 ≤ 1 .
19
Lemma 6.4 Let ε ∈R>0. There existskε >> 0 such that, if
‖G(x)−G(k)(x)‖ξ,1 < ε
for anyk≥ kε then‖Gn(x)−G(k)
n (x)‖ξ,1 < ε
for anyk≥ kε and anyn≥ 1.
Proof. We want to prove the statement by induction onn, the casen= 1 being true by assumption. Suppose
‖G(k)n (x)−Gn(x)‖ξ,1 < ε .
Then by (2.2), forkε >> 0, we have also∥∥∥∥dqkG
(k)n (x)−
ddx
Gn(x)
∥∥∥∥ξ,1
< ε .
MoreoverG(k)n (x) are analytic bounded function overD(ξ,1−), hence forkε large enough
∥∥∥G(k)n (qkx)−G(k)
n (x)∥∥∥
ξ,1≤ |1−qk|< ε .
Finally
‖G(k)n+1(x)−Gn+1(x)‖ξ,1
=∥∥∥G(k)
1 (x)G(k)n (qkx)−dqkG
(k)n (x)−G1(x)Gn(x)+ d
dxGn(x)∥∥∥
ξ,1
≤ sup(∥∥∥(G(k)
1 (x)−G1(x))G(k)n (qkx)
∥∥∥ξ,1
,∥∥∥G1(x)(G
(k)n (qkx)−G(k)
n (x))∥∥∥
ξ,1,
∥∥∥G1(x)(G(k)n (qkx)−Gn(x))
∥∥∥ξ,1
,∥∥∥dqkG
(k)n (x)− d
dxGn(x)∥∥∥
ξ,1
)
< ε .⊓⊔
Proposition 6.5 Under the assumption above we have:1) For anyk>> 0 theqk-difference systemdqkY(x) = G(k)(x)Y(x) has an analytic fundamental solutionY(k)(x)over a diskD(ξ,η−
k )⊂ D, such thatY(k)(ξ) = I.
2) Letη= lim infn→∞ ηk. ThenY(k)(x) tends pointwise overD(ξ,η−) to a fundamental solutionY(x) of dY(x)dx =
G(x)Y(x). MoreoverY(k)(x) tends uniformly toY(x) overD(ξ,(η′)−), for anyη′ < η.
Proof.1) By (3.1) the system(S)′qk
has a formal solution of the form
Y(k)(ξ,x) = ∑n≥0
G(k)n (ξ)[n]qk!
(x− ξ)qk,n .
We set
η−1k = limsup
n→∞
∣∣∣∣∣G(k)
n (ξ)[n]qk!
∣∣∣∣∣
1/n
.
20
On the other hand the differential system(S) has a solution
Y(ξ,x) = ∑n≥0
Gn(ξ)n!
(x− ξ)n
which is necessary an analytic function atξ, sinceG1(x) does not have any pole atξ. Suppose thatY(ξ,x)converges overD(ξ,η−), with η ≤ 1, and lett = tξ,η be generic point at distanceη from ξ. By [DGS94, IV,5.1], the solutionY(t,x) of (S) at the generic pointt converges overD(t,η−).
Let
h(n, t) = sups≤n
(1,
∣∣∣∣Gn(t)
n!
∣∣∣∣)
andh(n,k, t) = sups≤n
(1,
∣∣∣∣∣G(k)
n (t)[n]qk!
∣∣∣∣∣
)
By the previous lemma,h(n,k, t) converges toh(n, t) whenk → ∞, uniformly with respect ton, in fact fork>> 1 ∣∣∣∣∣
G(k)n (t)[n]qk!
−Gn(t)
n!
∣∣∣∣∣≤ |π|2∣∣∣G(k)
n (t)n! −Gn(t)[n]qk!∣∣∣
≤ |π|2sup(∣∣∣G(k)
n (t)(n! − [n]qk!)∣∣∣−∣∣∣(G(k)
n (t)−Gn(t))[n]qk!∣∣∣)
≤ sup(|1−qk|,ε)≤ ε .
Sinceh(n,k, t),h(n, t)≥ 1 we have∣∣∣h(n,k, t)1/n−h(n, t)1/n
∣∣∣≤ |h(n,k, t)−h(n, t)| .
Henceh(n,k, t)1/n tends toh(n, t)1/n uniformly with respect ton. Moreover, we know (cf. [And89, IV, §5] and[DGS94, IV, 5.1] for the differential case and [DV, 4.2.7] and (6.2) for theq-difference case) that the followinglimits exists and
η−1k = lim
n→∞h(n,k, t)1/n andη−1 = lim
n→∞h(n, t)1/n .
We conclude thatlimk→∞
limn→∞
h(n,k, t)1/n = limn→∞
limk→∞
h(n,k, t)1/n .
In other words limk→∞ ηk = η. It implies that fork>> 0 we haveηk > |(1−qk)ξ| and allows to conclude by(4.4).
2) The proof of the second assertion follows faithfully the proof of [DGS94, IV, 5.4]. Letηk be the radius ofconvergence ofY(k)(ξ,x). Lettingk→ ∞ in the effective bound estimate
∣∣∣∣∣G(k)
n (ξ)[n]qk!
∣∣∣∣∣≤ n,µ−1qkp Ck
1ηn
k, whereCk = sup
i=0,...,µ−1
∥∥∥G(k)i (x)
∥∥∥ξ,ηk
ηik ,
we obtain ∣∣∣∣Gn(ξ)
n!
∣∣∣∣ ≤ n,µ−1pC1
ηn , whereC=
(sup
i=0,...,µ−1‖Gi(x)‖ξ,η ηi
),
andn,µ−1p =
1, if n=0,
sup1≤λ1<···<λµ−1≤n
1∣∣λ1 · · ·λµ−1∣∣ , otherwise.
This proves that the solutionY(ξ,x) of (S)q at ξ converges overD(ξ,η−).As far as the uniform convergence is regard, remark that for any R∈ (0,η) andε > 0 there exist ¯n, k ∈ N
such that for anyn≥ n and anyk≥ k we have:∥∥∥∥∥
G(k)n (ξ)[n]qk!
(x− ξ)qk,n
∥∥∥∥∥ξ
(R)≤Ckn,µ−1qkp
(Rηk
)n
≤Cknµ−1(
Rηk
)n
< ε
21
and ∥∥∥∥Gn(ξ)
n!(x− ξ)n
∥∥∥∥ξ(R)≤Cn,µ−1p
(Rη
)n
≤Cnµ−1(
Rη
)n
< ε
By considering a biggerk, we can suppose that for anyk≥ k we have also∥∥∥∥∥
n
∑n=0
G(k)n (ξ)[n]qk!
(x− ξ)qk,n−n
∑n=0
Gn(ξ)n!
(x− ξ)n
∥∥∥∥∥ξ
(R)
≤ supn=0,...,n
∥∥∥∥∥G(k)
n (ξ)[n]qk!
(x− ξ)qk,n−Gn(ξ)
n!(x− ξ)n
∥∥∥∥∥ξ
(R)< ε ,
using an analogue argument to the one in the proof of (2.2). Finally we obtain the uniform convergence overanyD(ξ,R−): ∥∥Y(ξ,x)−Y(k)(ξ,x)
∥∥ξ(R)
≤ sup
∥∥∥∥∥
n
∑n=0
G(k)n (ξ)[n]qk!
(x− ξ)qk,n−n
∑n=0
Gn(ξ)n!
(x− ξ)n
∥∥∥∥∥ξ
(R),
supn≥n
∥∥∥∥∥G(k)
n (ξ)[n]qk!
(x− ξ)qk,n−Gn(ξ)
n!(x− ξ)n
∥∥∥∥∥ξ
(R)
< ε .
⊓⊔
III Weak Frobenius structure over a disk
Warning. In chapters III and IV we will assume that|1−q|< |π|= |πq|: this implies that for any integer n wehave
|1−qn|= | logqn|= |n||logq|= |n||1−q| .
Moreover during the whole chapters III and IV we will consider only q-difference systems over D=D(0,1−), therefore we will suppress everywhere the index “0,1”: so we will write t for t0,1, ‖ ‖ for ‖ ‖0,1,χ(A,q) for χ0,1(A,q) and so on.
In this chapter we study the action of the Frobenius mapx 7−→ xpℓ , ℓ ∈ Z>0, on q-difference systems.Namely, following [Chr81], under convenient hypothesis, we will construct a matrixH(x), depending on the
q-difference systemY(qx) = A(x)Y(x), such thatA[H](x) is a function ofxpℓ . This means that
Y(qx) = A[H](x)Y(x)
is actually aqpℓ-difference system in the variablexpℓ . The interest of such a construction is that it changes thegeneric radius of convergence, in fact:
χ(A[H](xpℓ),qpℓ)≥ χ(A(x),q)pℓ .
The original Christol theorem in [Chr81] forp-adic differential equation, as well as its generalizationtodifferential modules over an annulus, due to Christol-Dwork [CD94], is an irreplaceable tool: we think that thetheorem above is destinated to play an analogous role forq-difference equations.
7 Frobenius action onq-difference systems
Until now we have worked with analytic and meromorphic functions: in the next sections we will restrict ourattention to analytic elements. An introduction to the theory of analytic elements can be found in [Rob00, §6,4]. We briefly recall the definition:
22
Definition 7.1 The ringE0 of the analytic elements on the diskD(0,1−) (defined overK) is the completion ofthe subring ofK(x) of all rational function not having poles inD(0,1−) with respect to the norm‖ ‖.
We denote byE′0 its quotient field.
Remark 7.21) Of course one can define analytic elements over any disk, but in this chapter we will deal with disks centeredat 0. Therefore, by rescaling, we will always consider analytic elements overD(0,1−).
2) In the sequel we will essentially use three properties of analytics elements, namely:
• By taking Taylor expansion at zero, one can identify analytic elements to bounded analytic functionsoverD(0,1−): this ring isomorphism is an isometry with respect to‖ ‖.
• Any element ofE0 has a finite numbers of zeros (cf. [DGS94, IV, 5.2]), hence any element ofE′0 has a
finite number of zeros and poles.• The two properties above pass to the limit of a sequence of analytic elements,E0 being complete.
The following theorem concerns a classH ℓq , ℓ ∈ Z>0, of q-difference systemsY(qx) = A(x)Y(x) verifying
the properties:
1. A(x) ∈ Glµ(E′0);
2. A(x) is analytic at 0 andA(0) ∈ Glµ(K);3. Y(qx) = A(x)Y(x) has only apparent singularities inD(0,1−)r 0;
4. χ(A,q)> |πq|1
pℓ−1 .
We will say that a system inH ℓq is in normal formif moreover it satisfies the conditions 1÷3 of the following
proposition (cf. §8 for the proof):
Proposition 7.3 Let Y(qx) = A(x)Y(x) be in H ℓq . Then there exists a matrixU ∈ GLµ(K(x)) such that the
q-difference system associated toA[U] is in H ℓq and moreover:
1. A[U](x) ∈ Glµ(E0);
2. any two eigenvaluesλ1, λ2 of A[U](0) satisfy eitherλ1 = λ2 or λ1 6∈ λ2qZ;
3. any eigenvalueλ of A[U](0) satisfies|λ|< p1−ℓ.
Finally we are able to state the main theorem:
Theorem 7.4 Let us consider aq-difference system inH ℓq , in its normal form
Y(qx) = A(x)Y(x)
and letχ(A,q)> χ ≥ |πq|1
pℓ−1 .Then there exists a matrixH(x) ∈ Glµ(E′
0)∩Mµ×µ(E0) such that
H(0) = Iµ;(7.4.1)
A[H](x) = F(xpℓ);(7.4.2)
A(0) = F(0).(7.4.3)
Moreover theqpℓ-difference systemV(qpℓX) = F(X)V(X), with X = xpℓ , has the following properties:
χ(F,qpℓ)≥ χpℓ ;(7.4.4)
F(X) ∈ Glµ(E0).(7.4.5)
Remark 7.5 It follows by (4.4) that a systemY(qx) = A(x)Y(x) with coefficients inE0 has only ordinaryq-orbits inD(0,1−)r0 if and only if there existsn≥ 0 such thatxnA(x)∈Glµ(E0). Therefore ifA(x)∈Glµ(E0)thenY(qx) = A(x)Y(x) has only ordinaryq-orbits inD(0,1−)r 0 and 0 is a regular singularity.Vice versa,if A(x) is analytic at 0 withA(0) ∈ Glµ(K) andY(qx) = A(x)Y(x) has only ordinaryq-orbits inD(0,1−)r 0thenA(x) ∈ Glµ(E0).
The proof of the theorem above, which follows the proof of Christol [Chr81], is the object of §9.
23
8 Normal form for a system in H ℓq
Proposition 7.3 is a consequence of the following lemma:
Lemma 8.1 If χ ≥ |πq|1
pℓ−1 , then for any eigenvalueλ of A(0) we have:
dist(λ,qZp) = infα∈Zp
|λ−qα| ≤ p1−ℓ|q−1| ,
whereqα, for α ∈ Zp, is defined as the sum of the binomial series
qα = ∑n≥0
α(α−1) · · ·(α−n+1)n!
(q−1)n .
In fact:
Proof of (7.3). There exists a gauge transformationU1(x) with coefficients inK[x, 1
x
], calledshearing trans-
formation(cf. Appendix B, in particular (17.3)), such that the eigenvalues of A[U1](0) are the eigenvalues ofA(0), multiplied by chosen powers ofq. Hence by the previous lemma we can assume that any two eigenvaluesλ1, λ2 of A[U1](0) satisfy the conditions:
1) eitherλ1 = λ2 or λ1 6∈ λ2qZ;
2)
∣∣∣∣λ1−1q−1
∣∣∣∣< p1−ℓ.
Then by (3.9) there exists a matrixU2(x), with coefficients inK(x) and analytic at 0, such thatA[U2U1](x) hasonly ordinary orbits inDr 0.
SinceA[U2U1](x) is analytic at 0 andY(qx)=A[U2U1](x)Y(x) has only ordinary orbits inDr0, we concludethatA[U2U1](x) ∈ Glµ(E0). Moreover the eigenvalues ofA[U1](0) andA[U2U1](0) coincide, sinceU2(0) ∈ Glµ(K).Therefore it is enough to setU(x) =U2(x)U1(x) to conclude.
Let us prove lemma 8.1:
Proof of (8.1).Write the system
(S)q Y(qx) = A(x)Y(x)
in the formdn
q
[n]q!= Gn(x)Y(x) , for all n≥ 0.
Then
G1(x) =1x
A(0)− Iµ
q−1+higher order terms.
By (1.2) and (1.1.2), this formula generalizes to anyn≥ 1 in the following way:
(8.1.1)
Gn(x) =1xn
(−1)n
(q−1)n[n]q!
n
∑i=0
(−1)i(
ni
)
q−1q−
i(i−1)2 A(0)i +h.o.t.
=1xn
(−1)n(1−A(0))q−1,n
(q−1)n[n]q!+h.o.t.
=q−
n(n−1)2
xn
(A(0)−1)(A(0)−q) · · ·(A(0)−qn−1
)
(q−1)(q2−1) · · ·(qn−1)+h.o.t.
We set
Gn =(A(0)−1)(A(0)−q) · · ·
(A(0)−qn−1
)
(q−1)(q2−1) · · ·(qn−1).
24
Since ∣∣∣Gn
∣∣∣≤ ‖Gn(x)‖
the seriesg(x) = ∑n≥1 Gnxn converges for|x| ≤ χ.LetC∈ Glµ(K) be a constant matrix such thatC−1A(0)C is a matrix in the Jordan normal form, then
C−1g(x)C= ∑n≥1
(C−1A(0)C−1
)(C−1A(0)C−q
)· · ·(C−1A(0)C−qn−1
)
(q−1)(q2−1) · · ·(qn−1)xn .
Hence we conclude that the eigenvalue ofA(0) satisfy the desired inequality by applying the following result:
Lemma 8.2 Let λ ∈ K and letΛ(x) = ∑n≥1λnxn, with
λn =(λ−1)(λ−q)· · ·
(λ−qn−1
)
(q−1)(q2−1) · · ·(qn−1).
Then:1) The seriesΛ(x) converges for|x|< 1 if and only if λ ∈ qZp, i.e. if and only if dist(λ,qZp) = 0.
2) If the seriesΛ(x) converges for|x| ≤ |π|1
pℓ−1 , thendist(λ,qZp)≤ p1−ℓ|q−1|.
Proof.
1) Let α ∈ Zp be such thatλ = qα. Since|qn−1| ≤ |q−1|< |π| for anyn∈ Z, by the density ofZ in Zp, wehave also|qα−n−1|< |π|. We deduce from thep-adic properties of logarithm that
(8.2.1)
|λn| =
∣∣∣∣∣(qα −1)(qα −q) · · ·
(qα −qn−1
)
(q−1)(q2−1) · · ·(qn−1)
∣∣∣∣∣
=
∣∣∣∣∣(qα −1)
(qαq−1−1
)· · ·(qαq1−n−1
)
(q−1)(q2−1) · · ·(qn−1)
∣∣∣∣∣
=| logqα|| log(qαq−1)| · · · | log(qαq1−n)|
| logq|n|n!|
=1|n!|
∣∣∣∣logqα
logq
∣∣∣∣
∣∣∣∣logqα
logq−1
∣∣∣∣ · · ·∣∣∣∣logqα
logq− (n−1)
∣∣∣∣ .
Since logqn = nlogq for anyn andZ is dense inZp, one verifies thatlogqα
logq = α. We conclude that the radius
of convergence ofΛ(x) is equal to the radius of convergence of the binomial series∑n≥0(α
n
)xn, hence is equal
to 1 [DGS94, IV, 7.5].Suppose now thatΛ(x) converges for|x|< 1. Remark that
λn =1
[n]q!λ−1q−1
(λ−1q−1
− [1]q
)· · ·
(λ−1q−1
− [n−1]q
).
Hence if∣∣∣λ−1
q−1
∣∣∣> 1 we have
|λn| ≤
∣∣∣∣λ−1q−1
∣∣∣∣n 1|n!|
≤
∣∣∣∣λ−1q−1
∣∣∣∣n
|π|−n
and the radius of convergence ofΛ(x) would be∣∣∣π q−1
λ−1
∣∣∣ < 1. We conclude thatΛ(x) convergent for|x| < 1
implies|λ−1| ≤ |q−1|< |π|. As in (8.2.1) we obtain:
|λn|=1|n!|
∣∣∣∣logλlogq
∣∣∣∣
∣∣∣∣logλlogq
−1
∣∣∣∣ · · ·∣∣∣∣logλlogq
− (n−1)
∣∣∣∣ .
25
Finally we conclude thatα = logλlogq ∈ Zp since the series∑n≥0
(logλ/ logqn
)xn converges for|x| < 1 [DGS94, IV,
7.5]. Then logλ = logqα, with |λ−1|, |qα−1|< |π|, therefore
λ = explogλ = explogqα = qα .
2) If λ ∈ qZp the assertion follows immediately from 1), hence it is enough to consider the caseλ 6∈ qZp.
Moreover, if∣∣∣ λ−1
q−1
∣∣∣ > 1 the seriesΛ(x) has radius of convergence∣∣∣π q−1
λ−1
∣∣∣ < |π|1
pℓ−1 , thereforeλ verifies
dist(λ,qZp)≤ |q−1|. Letdist(λ,qZp)
|q−1|=
∣∣∣∣λ−1q−1
∣∣∣∣= |p|k+ε ,
with k ∈ Z≥0 andε ∈ [0,1). SinceZp is compact there existsα ∈ Zp such that|λ−qα| = |q−1||p|k+ε. Let sbe a nonnegative integer such that|qs−qα|< |q−1||p|k+ε; then|λ−qs|= |q−1||p|k+ε = dist(λ,qZp). Hencewe can chooseα = s∈ Z≥0.
Let us supposes= 0. Then for anym∈ Z we have:∣∣∣∣λ−qm
q−1
∣∣∣∣=∣∣∣∣λ−1q−1
−qm−1q−1
∣∣∣∣≥∣∣∣∣λ−1q−1
∣∣∣∣
We obtain the estimate (cf. for instance [BC92b, 2.2] or [DGS94, IV, 7.3] for the analogous classical estimate):
ordp([n]q!λn) =k
∑i=1
(1+
[n−1
pi
])+ ε[
n−1pk+1
]= ordp
(λ−1q−1n
),
with (λ−1q−1n
)=
1n!
λ−1q−1
(λ−1q−1
−1
)· · ·
(λ−1q−1
−n+1
).
Since|[n]q!|= |n!|, the radius of convergence ofΛ(x) is the same as the radius of convergence of the binomial
series∑n≥0( λ−1
q−1n
)xn, namelyp
1pk
(εp−
1p−1
)
(see [BC92b, 2.2] or [DGS94, IV, 7.3] for explicit calculation). We
conclude by pointing out that|π|1
pℓ−1 ≤ p1pk
(εp−
1p−1
)
implies thatℓ−1< k+ ε.Finally if s> 0, then for anyn> swe have:
λn = λs(q−sλ−1)(q−sλ−q)· · ·
(q−sλ−qn−s−1
)
(q−1)(q2−1) · · ·(qn−s−1−1).
By takingq−sλ instead ofλ, we reduce to the cases= 0. ⊓⊔
9 Proof of (7.4)
The proof of (7.4) is divided in two steps:
Step 1: construction of the matrixH(x) such thatH(0) = Iµ.
Step 2: proof of (7.4.2÷7.4.5).
They are respectively developed in (9.1) and (9.2).
9.1 Construction of the matrix H(x) such thatH(0) = Iµ.
By assumption the solutionY(ξ,x) = ∑
n≥0
Gn(ξ)(x− ξ)q,n
26
of Y(qx) = A(x)Y(x) is defined for anyξ ∈ (D(0,1−)r0)∪D(t,1−) and converges for|x−ξ| ≤ χ. We set:
(9.1.1)
H(x) :=1pℓ ∑
ζpℓ=1
Y(x,ζx)
=1pℓ ∑
n≥0
Gn(x)xn ∑
ζpℓ=1
(ζ−1)q,n .
SinceGn(x)xn ∈ Mµ×µ(E0), with
liminfn→∞
‖Gn(x)xn‖−1/n > χ,
and
|(ζ−1)q,n| =n
∏i=0
|ζ−1+1−qi|
≤n
∏i=0
sup(|ζ−1|, |i||1−q|)
≤ |π|n
pℓ−1 ≤ χn ,
the matrixH(x) is a well defined element ofMµ×µ(E0) and it makes sense to evaluateH(x) at 0. On the otherhand the matrixY(x,ζx) is not defined forx= 0, so one should be careful in the calculation ofH(0).
Remark thatY(x,ζx)− Iµ = x(ζ−1) ∑
n≥0
Gn+1(x)xn(ζ−q)q,n .
The series of the right hand side is still a well defined analytic element inE0 since|(ζ−q)q,n| ≤ χn. We deducethat
limx0→0
∥∥Y(x0,ζx0)− Iµ∥∥ ≤ lim
x0→0|x0(ζ−1)|
∥∥∥∥∥∑n≥0
Gn+1(x)xn(ζ−q)q,n
∥∥∥∥∥
≤ C limx0→0
|x0(ζ−1)|= 0 ,
whereC is a constant depending only onG1. Finally limx→0Y(x,ζx) = Iµ and
H(0) = limx→0
H(x)
= limx→0
1pℓ ∑
ζpℓ=1
Y(x,ζx)
=1pℓ ∑
ζpℓ=1
limx→0
Y(x,ζx)
= Iµ .
9.2 The matrix H(x) verifies (7.4.2÷7.4.5).
By assumption the solutionY(t,x) converges for|x− t| ≤ χ. It follows from [DGS94, page 175] that the matrix
V(x) =1pℓ ∑
zpℓ=x
Y(t,z)
is an analytic over the closed diskD(t pℓ ,χpℓ). The formula (3.2.1) implies that
V(xpℓ) =1pℓ ∑
zpℓ=xpℓ
Y(t,z)
=1pℓ ∑
ζpℓ=1
Y(t,ζx)
=1pℓ ∑
ζpℓ=1
Y(x,ζx)Y(t,x)
= H(x)Y(t,x) .
27
Hence the matrixF(x) =V(qpℓx)V(x)−1
verifies (7.4.2), in fact:F(xpℓ) = V(qpℓxpℓ)V(xpℓ)−1
= H(qx)Y(t,qx)Y(t,x)−1H(x)−1
= H(qx)A(x)H(x)−1 .
SinceH(0) = Iµ andA(x) is analytic at zero, we immediately obtain (7.4.3):F(0) = A(0).
Let X = xpℓ . ThenV(X) is solution of
V(qpℓX) = F(X)V(X) .
SinceV(X) is a analytic aroundt pℓ andV(t pℓ) = H(t), the generic radiusχ(F,qpℓ) at t pℓ coincides with theradius of convergence ofH(t)−1V(X), hence
χ(F,qpℓ)> χpℓ .
Finally let ξ ∈ D(0,1−), ξ 6= 0. Similar calculation to the ones we have worked out forV(x) shows that ananalytic fundamental solution ofV(qpℓX) = F(X)V(X) atξpℓ is given by
1pℓ ∑
zpℓ=x
Y(ξ,z) .
SinceD(0,1−) −→ D(0,1−)
ξ 7−→ ξpℓ
is a bijective map, this proves (7.4.5).
IV Transfer theorems in regular singular disks
In §6 we have seen how the generic radius of convergence is linked to the radius of convergence of analytic so-lution at ordinaryq-orbits. By (3.9) we are able to deduce estimates also for theradius of meromorphy of solu-tions at apparent and trivial singularities. In this chapter we are going to consider aq-difference system having aregular singularity at 0 (cf. Appendix B for a summary of basic properties of regular singular q-difference equa-tions). In particular we are going to prove aq-different version of the Christol-Andre-Baldassarri-Chiarellottotheorem (cf. [Chr84], [And87], [BC92a] and [BC92b]).
10 An analogue of Christol’s theorem
A q-difference systemY(qx) = A(x)Y(x)
with coefficients inE′0 is regular singular at0 if it is regular singular regarded as aq-difference system with
coefficients in the fieldK((x)) via the canonical immersionE′0 →K((x)), i.e. if there existsU(x)∈Glµ(K((x)))
such thatA[U](x) ∈ Glµ(K [[x]]).Let r(U(x)) be the radius of converge of the matrixxNU(x), for N large enough to havexNU(x)∈Glµ(K [[x]]).
Very similarly to the differential case, the theorem below establish an estimate ofr(U(x)) with respect to thegeneric radius of convergence and a certain number attachedto the eigenvalues ofA[U](0):
Definition 10.1 We call q-type ofα ∈ K, and we writetypeq(α), the radius of convergence of
∑n≥0,α 6=qn
1−q1−qnα
xn .
28
Remark 10.2 In the p-adic theory ofq-difference equations one calls type of a numberα ∈ K the radius ofconvergence type(α) of the series
∑n≥0,n6=α
xn
n−α.
The definition above is an analogue of this notion. (see Appendix C, §19, for an estimate of theq-type in termsof the classical type and some general properties of theq-type).
The following is aq-analogue of Christol theorem [Chr84]. It is a transfer theorem for q-difference systemY(qx) = A(x)Y(x) such thatχ(A,q) = 1. Statements concerning the general situation can be foundin §13.
Theorem 10.3 Let us consider a systemY(qx) = A(x)Y(x) such that
1) A(x) ∈ Glµ(E′0);
2) the systemY(qx) = A(x)Y(x) has only apparent singularities inD(0,1−)r0;
3) there existsU(x) ∈ Glµ(K((x))) such thatA[U] ∈ Mµ×µ(K);
4) χ(A,q) = 1.
Then
(10.3.1) r(U(x))≥ ∏ typeq
(αβ−1) ,
where the product is taken over all the (ordered) couples of eigenvaluesα,β of A[U].
The proof of this theorem, as the Christol-Andre-Baldassarri-Chiarellotto theorem, which inspired it, relieson the existence of the weak Frobenius structure and it is quite long. Sections §11 and §12 are voted to theproof of some preliminary estimates ofr(U). As a corollary, in §13 we prove the transfer theorem 10.3 plusmore general statements, without any assumption overχ(A,q).
11 A first rough estimate
In this section we prove a first estimate, which is not very sharp, but crucial for the proof of (10.3):
Proposition 11.1 LetY(qx) = A(x)Y(x) be a system such that
1) A(x) ∈ Glµ(E0);
2) the systemY(qx) = A(x)Y(x) has only ordinary point inD(0,1−)r0;
3) any two eigenvaluesα,β of A(0) verify eitherα = β or αβ−1 6∈ qZ.
Then the matrixU(x) ∈ Glµ(K [[x]]) verifingA[U] = A(0) is such that
r(U(x))≥ inf (|π|,χ(A,q))µ2(detA(0))µ ∏
α,β e.v. ofA(0)
typeq
(αβ−1) .
Remark 11.2 Observe that under the assumption of the proposition above the matrixU(x) always exists (cf.Appendix B, (17.2)).
Proof. The proof is devided in two steps:
Step 1. One can assume that Y(qx) = A(x)Y(x) verifies
(11.2.1)
∥∥∥∥A(x)− Iµ
(q−1)x
∥∥∥∥≤ sup
(1,
|π|χ(A,q)
).
29
Let us setχ = χ(A,q) to simplify notation. By the cyclic vector lemma there exists a matrixH1(x) ∈ Glµ(E′0)
such that the matrixA[H1](x) is in the form
A[H1](x) =
0... Iµ−1
0
a0(x) a1(x) . . .aµ−1(x)
.
An analogous of Fuchs’ theory forq-difference equation (cf. Appendix B, §18) assure thatA[H1](x) is analytic
at 0 andA[H1](0) is an invertible constant matrix. Consider the matrixH (which is very similar to the onedefined in (4.5.3)):
H = (ai, j)i, j=0,...,µ−1 , with ai, j =
(−1)i+ j
(q−1)i
(ij
)
q−1q−
j( j−1)2 if j ≤ i;
0 otherwise.
Then we have (cf. (1.2))
G[HH1]=
A[HH1](x)− Iµ
(q−1)x
0... Iµ−1
0
b0(x) b1(x) . . .bµ−1(x)
.
Remark thatxG[HH1]is analytic at 0 andA[HH1]
(0) ∈ Glµ(K), sinceA[H1](x) has the same properties.By proposition 4.6, ifχ ≥ |π|, then ∥∥∥G[HH1]
∥∥∥≤ 1 .
On the other hand, ifχ < |π|, by (4.3) and (4.6) we immediately deduce that
χ =|π|
supi ‖bi(x)‖1/(µ−i)
.
SinceK is algebraically closed, we can findγ ∈ K such that
|γ|= supi‖bi(x)‖
1/(µ−i) ;
then an explicit calculation shows that the diagonal gauge transformation
H2 =
11/γ
. . .1/γµ−1
verifies ∥∥∥G[H2HH1](x)∥∥∥≤
|π|χ
.
We setH(x) = HH1(x) if χ ≥ |π| andH(x) = H2HH1(x) otherwise. In both casesH(x) ∈ Glµ(E′0) andA[H](x)
is analytic at 0 withA[H](0) ∈ Glµ(K), sinceA[HH1](x) has the same properties.
We proceed as in (7.3): applying successively an unimodularshearing transformationH4(x)∈Glµ(K[x, 1
x
])
constructed as in (17.3) and an unimodular gauge transformationH3(x)∈ Glµ(K(x)) as in (3.9) we obtain a ma-trix A[H4H3H](x) verifying hypothesis 1), 2) and 3) plus the inequality:
∥∥∥∥A[H4H3H](x)− Iµ
(q−1)x
∥∥∥∥≤ sup
(1,
|π|χ(A,q)
)
SetQ(x) = H4(x)H3(x)H(x) andB(x) = A[Q](x). One remarks that
30
- since bothA(x) andB(x) satisfy hypothesis 1) and 2), necessarilyQ(x) andQ(x)−1 have no poles inD(0,1−)r0;
- B[UQ−1](x) = A(0).
Once again there exists a shearing transformationP(x) ∈ Glµ(K[x, 1
x
])such thatB[PUQ−1](x) = B(0) and
PUQ−1 has its coefficients inK((x)). Finally r(U) = r(PUQ−1), soY(qx) = B(x)Y(x) satisfies condition1)÷3) plus (11.2.1) and the matrixV = PUQ−1 is such thatB[V](x) = B(0) with r(V) = r(U).
Step 2. Proof of (10.3) assuming (11.2.1).
Let us setU(x) =U0+U1x+U2x
2+ . . .
andA(x) = A0+A1x+A2x
2+ . . .
Of course one can supposeU0 = Iµ. By assumption we haveU(qx)A(x) = A0U(x). Hence a direct calculationshows that
qmUmA0−A0Um = A1Um−1+A2Um−2+ · · ·+Am .
Consider theK-linear map
(11.2.2)Φqm,A0 : Mµ×µ(K) −→ Mµ×µ(K)
M 7−→ qmMA0−A0M = qmM(A0− Iµ)−M(qmIµ−A0).
The eigenvalues ofΦqm,A0 are precisely of the formqmα−β, whereα,β are any two eigenvalues ofA0. There-fore the operatorΦqm,A0 is invertible by hypothesis 3), which means that
Um = Φ−1qm,A0
(A1Um−1+A2Um−2+ · · ·+Am) .
We need to calculate the norm ofΦ−1qm,A0
as aK-linear operator. Since
‖A(x)− Iµ‖ ≤ |q−1|sup
(∥∥∥∥A(x)− Iµ
(q−1)x
∥∥∥∥ ,1)≤ |q−1|sup
(|π|χ,1
)
we conclude that
∥∥∥Φ−1qm,A0
∥∥∥=∥∥adjΦqm,A0
∥∥∥∥detΦqm,A0
∥∥ ≤|q−1|µ
2−1sup(|π|χ ,1
)µ2−1
∏α,β e.v. ofA0|qmα−β|
.
Recursively we obtain the estimate
|Um| ≤∥∥∥Φ−1
qm,A0
∥∥∥(
supi≥1
|Ai |
)(sup
i=1,...,m−1|Ui |
)
≤m
∏i=1
∥∥∥Φ−1qm,A0
∥∥∥‖A(x)− Iµ‖m
≤sup(|π|χ ,1
)µ2m
|detA0|µm
∏
α,β e.v. ofA0
m
∏i=1
∣∣∣∣1−qmαβ−1
1−q
∣∣∣∣
−1
We deduce from (20.2) that
r(U(x)) = lim infm→∞
|Um|− 1
m ≥ inf (|π| ,χ)µ2|detA0|
µ∏α,β
typeq(αβ−1) .
⊓⊔
31
For further reference, we point out that the Step 1 above is actually a proof of the following statement:
Lemma 11.3 Let Y(qx) = A(x)Y(x) be aq-difference equation, withA(x) ∈ Glµ(E′0), having only apparent
singularities inD(0,1−)r 0 and a regular singularity at0. We suppose that there existsU(x) ∈ Glµ(K((x)))verifying A[U](x) ∈ Glµ(K). Then there exists a gauge transformation matrixH(x) ∈ Glµ(E′
0) such that:
1) A[H](x) ∈ Glµ(E0);
2) any two eigenvalueα,β of A[H](0) are either equal orαβ−1 6∈ qZ;
3) r(U(x)) = r(H(x)U(x)) .
12 A sharper estimate
In this section we are going to the deduce from (11.1) a sharper estimate forq-difference system inH ℓq (cf.
(7.3)), reposing on the existence of the weak Frobenius structure.
Proposition 12.1 Let Y(qx) = A(x)Y(x) be aq-difference system inH ℓq in its normal form and letχ(A,q) >
χ ≥ |π|1
pℓ−1 . The matrixU(x) ∈ Glµ(K [[x]]) verifingA[U] = A(0) is such that
r(U(x))≥ inf
(|π|
1pℓ ,χ
)µ2
(detA(0))µpℓ ∏
α,β e.v. ofA(0)
typeq
(αβ−1) .
Proof. By theorem 7.4 about the weak Frobenius structure, there exists a matrixH(x) ∈ Glµ(E′0)∩Mµ×µ(E0)
such that:
1) H(0) = Iµ;
2) A[H](x) = F(xpℓ);
3) A(0) = F(0) ∈ Glµ(K);
4) theqpℓ-systemV(qpℓX) = F(X)V(X), with X = xpℓ , has only ordinary orbits inD(0,1−)r 0;
5) χ(F,qpℓ)≥ χpℓ .
By (17.2) there existsV(X)∈Glµ(K [[X]]) such thatF[V](X)=F(0). On the other hand we have alsoF[UH−1](x)=
F(0). It follows by (17.4) that there exists a shearing transformationQ(x)∈Glµ(K[x, 1
x
])such thatQ(x)U(x)=
V(x)H(x), hencer(U(x)) = r(V(x)) = r(V(X))1pℓ .
By (11.1) we have
r(V(X))≥ inf(|π|,χpℓ
)µ2
(detF(0))µ ∏α,β e.v. ofF(0)
typeqpℓ
(αβ−1)
and hence
r(U(x))≥ inf
(|π|
1pℓ ,χ
)µ2
(detA(0))µpℓ ∏
α,β e.v. ofA(0)
typeqpℓ
(αβ−1) 1
pℓ .
We conclude by remarking that
typeq(α) = lim infn→∞
|1−qnα|1/n
≤ lim infn→∞
|1−qpℓnα|1/pℓn
=(
typeqpℓ (α)
) 1pℓ .
⊓⊔
32
13 More general statements
LetY(qx) = A(x)Y(x) be aq-difference system such thatA(x) ∈ Glµ(E′0), having only apparent singularities in
D(0,1−)r 0. We suppose that there existsU(x) ∈ Glµ(K((x))) verifying A[U](x) ∈ Glµ(K). Then we have:
Corollary 13.1
1) If χ(A,q)≤ |π|. Then
r(U(x))≥ χ(A,q)µ2|detA(0)|µ ∏
α,β e.v. ofA(0)
typeq
(αβ−1) .
2) Let |π|< χ(A,q)< 1 and letℓ be a positive integer such that|π|1pℓ ≥ χ(A,q)> |π|
1pℓ−1 . Then
r(U(x))≥ χ(A,q)µ2|detA(0)|
µpℓ ∏
α,β e.v. ofA(0)
typeq
(αβ−1)
3) Let χ(A,q) = 1. Thenr(U(x))≥ ∏
α,β e.v. ofA(0)
typeq
(αβ−1)
Proof. Statement 1) follows immediately by (11.3) and (11.1), while statement 2) follows by (11.3) and (12.1).
Let us prove 3). By (11.3) and (12.1), for anyℓ ∈ Z≥0, we haveχ(A,q)> |π|1
pℓ−1 , hence
r(U(x)) ≥ |π(1−q)|1pℓ |detA(0)|
µpℓ ∏
α,β e.v. ofA(0)
typeq
(αβ−1) , for anyℓ > 0.
We conclude by lettingℓ→+∞. ⊓⊔
Appendix
A Twisted Taylor expansion of p-adic analytic functions
14 Analytic functions over aq-invariant open disk
In this section we prove proposition 1.3. Let us recall its statement:
Proposition 14.1 Let D = D(ξ,ρ−) is aq-invariant open disk. Then the map
Tq,ξ : AD −→ Kx− ξq,ρ
f (x) 7−→ ∑n≥0
dnq f
[n]q!(ξ)(x− ξ)q,n
is aq-difference algebras isomorphism such that for allf ∈ AD and allx0 ∈ D the seriesTq,ξ( f )(x0) convergesand its sumTq,ξ( f )(x0) is equal to the sum off (x0).
The proof is divided in three steps:
STEP 1. The map Tq,ξ is a well defined ring homorphism.
Proof. The only non trivial point is thatTq,ξ is well defined. Letf = ∑n≥0 fn(x− ξ)n ∈ AD. By (2.1.1), for anyintegerk≥ 0 and anyη ∈ R such that|(1−q)ξ| ≤ η < ρ, we have
∣∣∣∣∣dk
q( f )
[k]q!(ξ)
∣∣∣∣∣ ≤∥∥∥∥∥
dkq
[k]q!f (x)
∥∥∥∥∥ξ
(η)≤‖ f‖ξ(η)
ηk .
33
Hence liminfn→∞
∣∣∣∣dk
q( f )[k]q! (ξ)
∣∣∣∣−1/n
≥ η, for all η such that|(1−q)ξ| ≤ η < ρ. Letting η → ρ one proves that
Tq,ξ( f ) ∈ Kx− ξq,ρ. ⊓⊔
STEP 2. Let y=∑n≥0an(x−ξ)q,n ∈Kx−ξq,ρ. Then y converges for all x0 ∈D and the sum of y is an elementof AD.
Proof. For any couple of integersi, k, such that 0≤ i ≤ k, we consider the symmetric polynomial
Sik(x1, . . . ,xk) =
1 if i = 0;
∑1≤ j1<···< j i≤k
xi1 · · ·x j i otherwise.
We notice that
(14.1.1)
(x− ξ)q,n = (x− ξ)(x−qξ) · · ·(x−qn−1ξ)
= (x− ξ) [(x− ξ)+ ξ(1−q)]· · ·[(x− ξ)+ ξ(1−qn−1)
]
=n
∑k=0
ξn−kSn−kn (0,1−q, . . . ,1−qn−1)(x− ξ)k .
Since|ξn−kSn−kn (1−q0,1−q, . . . ,1−qn−1)| ≤ |ξ(1−q)|n−k−1 ≤ Rn−k−1 < ρn−k−1, the following series con-
verges inK:
αk =∞
∑n=k
anξn−kSn−kn (1−q0,1−q, . . . ,1−qn−1) ∈ K .
We want to prove thaty(x) = ∑
k≥0
αk(x− ξ)k ∈ K [[x− ξ]]
andy(x) = ∑
n≥0
an(x− ξ)q,n ∈ Kx− ξq,ρ
have the same sum for allx∈ D.First we prove that the seriesy(x) converges overD. Since for anyx0 ∈ D we have
sup(|ξ(1−q)|, |x0− ξ|)≤ R< ρ ,
we deduce that
0 ≤ limsupk→∞
|αk(x0− ξ)k|
≤ limsupk→∞
∣∣∣∣∣∞
∑n=k
anξn−kSn−kn (1−q0,1−q, . . . ,1−qn−1)(x0− ξ)k
∣∣∣∣∣
≤ limsupk→∞
(supn≥k
|an| |ξ(1−q)|n−k|x0− ξ|k)
≤ limsupk→∞
(supn≥k
|an|Rn
)= 0
.
This proves thaty(x) converges overD, hence thaty(x) ∈ AD.We fix a positive real numberη < ρ and we setCN = supn>N |an|ηn. We observe that
limN→∞
CN = 0 ,
34
sinceη < ρ. We have∥∥∥∥∥
N
∑k=0
αk(x− ξ)k−N
∑n=0
ak(x− ξ)q,k
∥∥∥∥∥ξ
(η)
=
∥∥∥∥∥N
∑k=0
αk(x− ξ)k−N
∑n=0
an
n
∑k=0
ξn−kSn−kn (0,1−q, . . . ,1−qn−1)(x− ξ)k
∥∥∥∥∥ξ
(η)
=
∥∥∥∥∥N
∑k=0
∞
∑n=k
anξn−kSn−kn (0,1−q, . . . ,1−qn−1)(x− ξ)k
−N
∑k=0
N
∑n=k
anξn−kSn−kn (0,1−q, . . . ,1−qn−1)(x− ξ)k
∥∥∥∥∥ξ
(η)
=
∥∥∥∥∥N
∑k=0
(
∑n>N
anξn−kSn−kn (1−q0,1−q, . . . ,1−qn−1)
)(x− ξ)k
∥∥∥∥∥ξ
(η)
≤ supk=0,...,N
(ηk sup
n>N|an|ηn−k
)
≤ supk=0,...,N
(supn>N
|an|ηn)=CN .
We conclude that the two series converge to the same sum overD(ξ,η−), for all 0< η < ρ, hence overD. ⊓⊔
STEP 3. The map Tq,ξ is an isomomorphism and for all x0 ∈ D we have Tq,ξ( f )(x0) = f (x0).
Proof. By the previous step, one can define a mapS: Kx− ξq,ρ −→ AD. Sinceξ is a limit point ofqNξ, forany f (x),g(x) ∈ AD we have
dnq f
[n]q!(ξ) =
dnqg
[n]q!(ξ) , ∀n∈ Zn≥0 ⇔ f (qnξ) = g(qnξ) for all integersn≥ 0
⇔ f = g .
Then one deduce thatT−1q,ξ = Sby the fact that
dnq
[n]q!
(STq,ξ( f )
)(ξ) =
dnq f
[n]q!(ξ) and
dnq
[n]q!
(Tq,ξ S(g)
)(ξ) =
dnqg
[n]q!(ξ)
for any f ∈ AD, anyg∈ Kx− ξq,ρ and any nonnegative integern. ⊓⊔
This achieves the proof of (14.1).
15 Analytic functions over nonconnected analytic domain
Proposition 14.1 can be generalized to the case of analytic functions over convenientq-invariant nonconnectedanalytic domains:
Definition 15.1 We call q-disk of centerξ ∈ A1K and radiusη ∈ R>0 the set
qZD(ξ,η−) := ∪n∈Z
D(qnξ,η−) .
Remark 15.2 Of course, ifn0 is the smallest positive integer such that|(1− qn0)ξ| < η, the q-disk D is adisjoint union of the open disksD(qnξ,η−), with 0≤ n< n0, each one of them beingqn0-invariant.
35
The algebraAD of analytic functions over aq-disk D is the direct product of the algebras of analyticfunctions over each connected component. Since the analytic domainD is q-invariantAD has a structure ofq-difference algebra. Hence one can define aq-expansion map
Tq,ξ : AD −→ K [[x− ξ]]q =
∑n≥0an(x− ξ)q,n : an ∈ K
f (x) 7−→ ∑n≥0
dnq( f )
[n]q!(ξ)(x− ξ)n,q
,
that we still callTq,ξ.
Proposition 15.3 Let f (x) = ∑n≥0an(x− ξ)q,n be a formal series such that thean’s are elements ofK andlim inf
n→∞|an|
−1/n = ρ. Then f (x) is aq-expansion of an analytic function over aq-diskD if and only if
ρ|(q−1)ξπq|
> 1 .
Let n0 be the number of connected components ofD andρ their radius. Thenn0 is the smallest positve integersuch that
(15.3.1)ρ
|(1−q)ξπq|> |πqn0 |
1/n0
andρ, ρ andn0 are linked by the relation:
(15.3.2)ρ
|(qn0 −1)ξπqn0 |=
(ρ
|(q−1)ξπq|
)n0
.
Remark 15.4 We recall thatπq ∈ K verifies|πq|= limn→∞ |[n]q!|1/n (cf. (3.4) for the precise definition).
It follows immediately by (3.3) and (15.3) that:
Corollary 15.5 The system
Y(qx) = A(x)Y(x) , with A(x) ∈ Glµ(AD(ξ,ρ−)),
has a analytic fundamental solution over aq-disk centered atξ if and only if the matrixA(x) does not have anypoles inqNξ, detA(x) does not have any zeros inqNξ and
limsupn→∞
∣∣∣∣Gn(ξ)[n]q!
∣∣∣∣1/n
< |(q−1)ξπq|−1 .
16 Proof of proposition 15.3
The main result of this section is the proof proposition 15.3(cf. (16.5) below), that we will deduce by the moregeneral result (16.2).
Let ξ 6= 0 andf (x) = ∑n≥0an(x−ξ)q,n ∈ K [[x− ξ]]q. For any non negative integerk it makes sense to defineformally
dkq( f )
[k]q!(ξ) = ak
andf (qkξ) = ∑
n≥0αnξn(qk−1)q,n .
An elementf (x) ∈ K [[x− ξ]]q is uniquely determined by( f (qkξ))k≥0 or (ak)k≥0, knowing that these two se-quences are linked by relations that can be deduced by 1.2. ThereforeK [[x− ξ]]q is not a local ring and wehave:
36
Lemma 16.1 If ξ 6= 0 the natural morphism ofq-difference algebras, defined for anyn0 ≥ 1,
(16.1.1)
K [[x− ξ]]q −→n0−1
∏i=0
K[[
x−qiξ]]
qn0
f (x) 7−→
(
∑n≥0
dnqn0( f )(qiξ)[n]qn0 !
(x−qiξ)qn0 ,n
)
i=0,...,n0−1
is an isomorphism. In particularK [[x− ξ]]q is not a domain.
Proof. It is enough to remark that an elementf ∈ ∏n0−1i=0 K
[[x−qiξ
]]qn0 is uniquely determined by
((f (qnn0+iξ)
)n≥0
)
i=0,...,n0−1.
⊓⊔
The isomorphism (16.1.1) induces an isomorphism between “convergingq-series”:
Proposition 16.2 We setq= qn0. Let
g= ∑n≥0
gn(x− ξ)q,n ∈ K [[x− ξ]]q
and let
f =
(
∑n≥0
fi,n(x− ξ)q,n
)
0≤i<n0
∈n0−1
∏i=0
K[[
x−qiξ]]
q
be its image via (16.1.1). We set
ρ = lim infn→∞
|gn|−1 and ρ = inf
0≤i<n0
(lim inf
n→∞| fi,n|
−1).
Thenρ
|(q−1)ξπq|=
(ρ
|(q−1)ξπq|
)n0
.
In order to prove proposition 16.2, we need a technical lemma:
Lemma 16.3 1) Letg∈ K [[x− ξ]]q; then
(16.3.1)
(q−1)kξkdkq(g)(ξ) = (−1)k
n0k
∑h=0
∑h
n0≤ j≤k
(−1) j(
kj
)
q−1
·q− j( j−1)
2
(n0 jh
)
q·q
h(h−1)2 (q−1)hξhdh
q(g)(ξ) .
2) Let f = ( fl ) ∈ ∏n0−1l=0 K
[[x−qlξ
]]q; then
(16.3.2)
(q−1)kξkdkq( f )(ξ) = (−1)k
[ n0k ]
∑h=0
∑0≤l<n0
hn0+l≤k
(k
hn0+ l
)
q−1q
−(hn0+l )(hn0+l−1)2
·h
∑j=0
(hj
)
j( j−1)2 ql j (q−1) jξ jd j
q( fl )(ql ξ) .
37
Proof. The two formulas can be obtained by applying twice the formulas relatingdq andσq (cf. (1.2)), takinginto account thatdq( f (ql ξ)) = qldq( f )(ql ξ). ⊓⊔
16.4 Proof of proposition 16.2.
The proof is based on the following property of the limit of a sequence(an) of real non negative numbers (cf.[AB01, II, 1.8 and 1.9]):
(16.4.1) sup(1, limsupn→∞
a1/nn ) = limsup
n→∞
(sups≤n
as
)1/n
.
STEP 1. Proof of (16.2) when one of the (equivalent) condition- ρ < |(q−1)ξπq|,- ρ < |(q−1)ξπq|is satisfied.Suppose first thatρ < |(q−1)ξπq|. Sinceρ−1|πq|> 1, equality (16.4.1) implies that
ρ−1|πq| = |πq| limsupn→∞
∣∣∣∣dn
q(g)
[n]q!(ξ)∣∣∣∣1/n
≤ limsupn→∞
∣∣dnq(g)(ξ)
∣∣1/n
= limsupn→∞
(sups≤n
∣∣dsq(g)(ξ)
∣∣)1/n
.
Since|(q−1)ξπq|ρ−1 > 1 we deduce by (16.3.2) that
|(q−1)ξπq|ρ−1 ≤ limsupn→∞
∣∣(q−1)nξndnq(g)(ξ)
∣∣1/n
≤ limsupn→∞
sup
s≤[
nn0
],l=0,...,n0−1
∣∣∣(q−1)sξsdsq(g)(q
l ξ)∣∣∣
1/n
= sup(
1, |(q−1)ξπq|1/n0ρ−1/n0
)
= (|(q−1)ξπq|ρ)−1/n0
and hence thatρ ≥ |(q−1)ξπq|(
ρ|(q−1)ξπq|
)1/n0. Moreover the last inequality implies thatρ < |(q−1)ξπq|.
Let us suppose now thatρ < |(q−1)ξπq|. Then
ρ−1|πq| ≤ supi=0,...,n0−1
(limsup
n→∞
(sups≤n
∣∣dsq( f )(qiξ)
∣∣)1/n
).
We deduce by (16.3.1) that
|(q−1)ξπq|ρ−1 ≤ limsupn→∞
∣∣dnq( f )(ξ)
∣∣1/n
≤ limsupn→∞
(sup
s≤nn0
∣∣(q−1)sξsdsq( f )(ξ)
∣∣)1/n
≤ |(q−1)ξ|n0|p|n0/(p−1)ρ−n0
and hence thatρ ≥
(|(q−1)ξπq|
|(q−1)ξπq|1/n0
)−n0
ρn0. Remark that this inequality implies thatρ < |(q−1)ξπq|
STEP 2. Proof of (16.2) when one of the (equivalent) conditions- ρ ≥ |(q−1)ξπq|,
38
- ρ ≥ |(q−1)ξπq|is satisfied.We chooseγ ∈ K such that
|γ|> |(q−1)ξπq|−1ρ .
By settingξ = γζ andx= γt, we identifyg to an element ofK [[t − ζ]]q verifying the hypothesis of Step 1. Wededuce by the estimate in Step 1 that
|γ|−1ρ =|(q−1)ζπq|
|(q−1)ζπq|1/n0(|γ|−1ρ)1/n0
= |γ|−1 |(q−1)ξπq|
|(q−1)ξπq|1/n0ρ1/n0 .
16.5 Proof of proposition 15.3.
We briefly recall the statement of (15.3). Letf (x) = ∑n≥0an(x− ξ)q,n be a formal series such thatan ∈ Kand liminf
n→∞|an|
−1/n = ρ. We have to prove thatf (x) is a q-expansion of an analytic function over aq-disk
qZD(ξ, ρ−) if and only if
(16.5.1)ρ
|(q−1)ξπq|> 1 .
Let f (x) verify (16.5.1). Since ρ|(q−1)ξπq|
> 1 there existsn0 such that
ρ|(q−1)ξ|
=
(ρ
|(q−1)ξπq|
)n0
|πq|> 1 .
It follows from (1.3) and (15.3) thatf (x) is theq-expansion of an analytic function overq-disk qZD(ξ, ρ−)verifying (15.3.1) and (15.3.2).
Vice versa, let f (x) be theq-expansion of an analytic function over aq-disk D = qZD(ξ, ρ−). Let n0 bethe number of connected component ofD and q = qn0. Then ρ > |(q− 1)ξ|. It follows from (16.2) thatρ > |(q−1)ξπq|.
B Basic facts about regular singularity ofq-difference systems
In this chapter we briefly recall some properties of regular singularq-difference systems, that we use in III andIV. An complete exposition can be found in [vdPS97, Ch. 12], [Sau00a] and [Sau02b].
17 Regular singularq-difference systems
Let K be an algebraically closed field of characteristic 0 andK((x)) = K [[x]][
1x
]the field of Laurent series. For
anyq∈ K, it has a natural structure ofq-difference algebra, hence one can consider aq-difference system withcoefficients inK((x)):
Y(qx) = A(x)Y(x) .
Definition 17.1 The q-difference systemY(qx) = A(x)Y(x) is said to have aregular singularity at 0(or isregular singular at 0) if there existsU(x) ∈ Glµ(K((x))) such that the matrixA[U](x) = U(qx)A(x)U(x)−1 ∈Glµ(K [[x]]).
Lemma 17.2 [Sau00a, 1.1.3]SupposeA(x) ∈ Glµ(K [[x]]). If any two eigenvaluesα,β of A(0) are such thateitherα = β or αβ−1 6∈ qZ, then one can constructU(x) ∈ Glµ(K [[x]]) such thatA[U](x) = A(0).
By a convenient gauge transformation one can always assume that the assumption of the lemma above areverified:
39
Proposition 17.3 [Sau00a, 1.1.1]Suppose thatA(x) ∈ Glµ(K [[x]]) and for any eigenvalueα of A(0) choose anintegernα. Then there exists a matrixH(x) constructed by multipling alternatively constant matrices inGlµ(K)and diagonal matrices of the form
(17.3.1)
Iµ1
Iµ2x±1
Iµ3
, with µ1+µ2+µ3 = µ ,
such that, for any eigenvalueα of A(0), qnαα is an eigenvalue ofA[H](0).In particular one can construct a matrixH(x) such that any two eigenvaluesα,β of A[H](0) verify either
α = β or αβ−1 6∈ qZ.
The idea of the proof of the proposition above is that one has first to consider a constant gauge matrixQsuch thatQ−1A(x)Q has the constant term in the Jordan normal form. Then by usinga gauge matrix of theform (17.3.1) one multiply the eigenvalue of a chosen block by q±1 of A(0). By iterating the algorithm oneobtain the desired gauge transformation.
One callsshearing transformationa gauge matrix constructed as in the previous proposition.
Corollary 17.4 LetU(x),V(x) ∈ Glµ(K((x))) be two gauge transformation matrix such that bothA[U] andA[V]
are inGLµ(K). Then there exists a shearing transformationH(x) such thatHU =V.
The corollary follows immediately by proposition (17.3), knowing that the Jordan normal forms ofA[U] andA[V] coincide, modulo the fact that the eigenvalues ofA[U] are the eigenvalues ofA[V] multiplied by an integerpower ofq.
18 From q-difference systems toq-difference equations
Consider now aq-difference equation
Ly= y(qµx)−aµ−1(x)y(qµ−1x)−·· ·−a0(x)y(x) = 0 .
with ai(x) ∈ K((x)), for all i = 0, . . . ,µ−1. The origin is said to be aregular singularity ofLy= 0 is and onlyif the convex hull inR2 of (i, j) ∈ Z2 : j ≥ ordxai(x), i.e. the Newton Polygon, has only one finite slope equalto 0.
Obviously one has:
Lemma 18.1 If 0 is a regular singularity forLy = 0, thenai(x) does not have any pole at0, for any i =0, . . . ,µ−1. In particular,a0(0) 6= 0.
It follows that theq-difference system
(18.1.1) Y(qx) = A[H](x)Y(x) , with A[H](x) =
0... Iµ−1
0
a0(x) a1(x) . . .aµ−1(x)
has a a regular singularity at 0.
The contrary is also true:
Proposition 18.2 ([Sau00a, Annexe B] and [Sau02b, §2, in particular 2.2.6, (ii)])A q-difference system overK((x)) has a regular singularity at0 if and only if the associatedq-differenceequationvia the cyclic vector lemma has a regular singularity at0.
40
18.3 Second orderq-difference equations.
Consider a second order regular singularq-difference equation
(18.3.1) y(q2x)−P(x)y(qx)−Q(x)y(x) = 0 .
Then the associatedq-difference system is given by
Y(qx) = A(x)Y(x) , with A(x) =
(0 1
Q(x) P(x)
)
and the eigenvaluesα,β of A(0) are solutions of the second order equation
T2−P(0)T −Q(0) = 0 .
By (17.2), if α 6= β andαβ−1 6∈ qZ, there existsU(x) ∈ Glµ(K [[x]]) such that
U(qx)A(x)U(x)−1 =
(α 00 β
).
If we call eα(x) (resp.eβ(x)) a solution ofy(qx) = αy(x) (resp.y(qx) = βy(x)), then we have
[U(qx)−1
(eα(qx) 0
0 eβ(qx)
)]= A(x)
[U(x)−1
(eα(x) 0
0 eβ(x)
)].
If uα(x),uβ(x) is the first row ofU(x)−1, theneα(x)uα(x),eβ(x)uβ(x) is a basis of solution of (18.3.1), meaningthat eα(x)uα(x) andeβ(x)uβ(x) are linearly independent over the field of constants and theyspan the vectorspace of solutions ofLy= 0. Remark also thateα(x),eβ(x) are linearely independent overK((x)) (that can beproved as in [Sau00a, Annexe A, 4)]).
C The q-type of a number
The purpose of this appendix is to calculate the radius of convergence of the series
∑n≥0
(1−q)nxn
(1−qα) · · ·(1−qnα)
for α 6∈ q−N, which is a keypoint in the proof theorem 10.3. Actually the radius of such a series turns out to beequal to|π|typeq(α).
As in chapters III and IV, we assume that|1−q|< |π|, so that|1−q|= | logq|.
19 Basic properties of theq-type of a number
We racall that for anyα∈K we have defined theq-type ofα to be the radius of convergenceof∑n≥0,α 6=qn(1−q)nxn
1−qnα ,
while the type ofα is the radius of convergence of∑n≥0,α 6=nxn
n−α .
Proposition 19.1 For anyα ∈ K we have
typeq(α) =
type
(logαlogq
)if
∣∣∣∣α−1q−1
∣∣∣∣≤ 1 ,
1 otherwise.
41
Proof. We haveα−qn
q−1=
α−1q−1
−qn−1q−1
=α−1q−1
− [n]q .
Hence if∣∣∣α−1
q−1
∣∣∣> 1, the radius of convergence of∑n≥0q−1
α−qn xn is equal to 1.
Suppose|α−1| ≤ |q−1|< |π|. Then|α−qn| ≤ |q−1|< |π| for any integern and∣∣∣∣α−qn
q−1
∣∣∣∣=∣∣∣∣log(αq−n)
logq
∣∣∣∣=∣∣∣∣logαlogq
−n
∣∣∣∣ .
⊓⊔
20 Radius of convergence of1Φ1(q;αq;q;(1−q)x)
The series
∑n≥0
(1−q)nxn
(1−qα) · · ·(1−qnα),
whose radius of convergence we want to estimate, is a basic hypergeometric series. In the literature it is denotedby 1Φ1(q;αq;q;(1−q)x) (cf. [GR90]).
Lemma 20.1 For anyα ∈ KrqZ we have:
∑n≥0
(1−q)nxn
(1−qα) · · ·(1−qnα)=
1−α1−q
(
∑n≥0
xn
[n]q!
)(
∑n≥0
qn(n+1)
2(−x)n
[n]q!1−q
1−qnα
).
Proof. One verifies directly that the series
Φ(x) = ∑n≥0
(1−q)nxn
(1−qα) · · ·(1−qnα)
is a solution of theq-difference equation
LΦ(x) =[σq−1
][ασq− ((q−1)x+1)
]Φ(x)
= αΦ(q2x)− ((q−1)qx+1+α)Φ(qx)+ (q−1)qx+1= 0 .
Since the roots of the equation (cf. (18.3))
αT2− (α+1)T +1= 0
are exactlyα−1 and 1, any solution ofLy(x) = 0 of the form 1+∑n≥1anxn ∈ K [[x]] must coincide withΦ(x)(cf. (18.3)). Therefore, to conclude the proof of the lemma, we are going to verify that
1−α1−q
(
∑n≥0
xn
[n]q!
)(
∑n≥0
qn(n+1)
2(−x)n
[n]q!1−q
1−qnα
)
is a solution ofLy(x) = 0.Let eq(x) = ∑n≥0
xn
[n]q! . Theneq(x) verifies theq-difference equation
eq(qx) = ((q−1)x+1)eq(x) ,
henceL eq(x) =
[σq−1
]eq(qx)
[ασq−1
]
= eq(x)((q−1)x+1)[((q−1)qx+1)σq−1
][ασq−1
]
= (∗)[((q−1)qx+1)σq−1
][ασq−1
],
42
where we have denote with(∗) a coefficient inK(x), not depending onσq.
Consider the seriesEq(x) = ∑n≥0qn(n−1)
2 xn
[n]q! , which verifies
(1− (q−1)x)Eq(qx) = E(x)
and the series
gα(x) = ∑n≥0
qn(n+1)
2(−x)n
[n]q!1−q
1−qnα.
ThenL eq(x)gα(x) = (∗)
[((q−1)qx+1)σq−1
][ασq−1
]gα(x)
= (∗)[((q−1)qx+1)σq−1
]Eq(−qx)
= (∗)[((q−1)qx+1)Eq(−q2x)−Eq(−qx)
]
= 0 .
It is enough to remark thateq(0)gα(0) =1−q1−α to conclude that1−α
1−qeq(x)gα(x) conicides withΦ(x). ⊓⊔
Corollary 20.2 For anyα ∈ KrqZ, the radius of convergence of
∑n≥0
(1−q)nxn
(1−qα) · · ·(1−qnα)
is |π|typeq(α).
Proof. It follows immediately by the previous results, since the radius of convergence of∑n≥0xn
[n]q! is |π|. ⊓⊔
Acknowledgements.As an undergraduate student in Padova, I attended ProfessorDwork’s course during theacademic year 1993/94: it is an honor to be able to acknowledge his decisive influence on my Ph.D. thesis aswell as on the present work.
I would like to thank Y. Andre, J-P. Ramis and J. Sauloy for the many discussions we have had during thepreparation of this paper (and not only), for sharing their enthusiasm with me and for their constant encourage-ment.
References
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