Table Of ContentARITHMETIC THEORY OF q-DIFFERENCE EQUATIONS
(G -FUNCTIONS AND q-DIFFERENCE MODULES OF TYPE G,
q
GLOBAL q-GEVREY SERIES)
0
1
0 LUCIADIVIZIO
2
n
a
J
Abstract. InthefirstpartofthepaperwegiveadefinitionofGq-functionandweestablisharegularity
3
result, obtained as a combination of aq-analogue of the André-Chudnovsky Theorem [And89, VI] and
1
KatzTheorem[Kat70,§13]. Inthesecondpartofthepaper,wecombineitwithsomeformalq-analogous
Fouriertransformations,obtainingastatementontheirrationalityofspecialvaluesoftheformalq-Borel
]
T transformation ofaGq-function.
N
Contents
.
h
t 1. Introduction 1
a
m
Part 1. G -functions and q-difference modules of type G 3
[ q
2. Definition and first properties 3
3 3. Role of the “noncyclotomic” places 5
v
4. Main results 6
9
5. Nilpotent reduction at cyclotomic places 7
6
1 6. Proof of Theorem 4.1 9
4 7. Proof of Theorem 4.2 10
.
2
0 Part 2. Global q-Gevrey series 21
9 8. Definition and first properties 21
0 9. Formal Fourier transformations 23
:
v 10. Action of the formal Fourier transformations on the Newton Polygon 24
i 11. Solutions at points of K∗ 26
X
12. Structure theorems 27
r
a 13. An irrationality result for global q-Gevrey series of negative orders 30
References 33
1. Introduction
A G-function, notion introduced by C.L. Siegel in 1929, is a formal power series y = y xn with
n≥0 n
coefficients in the field of algebraic numbers Q, such that:
P
(1) the series y is solution of a linear differential equation with coefficients in Q(x) (condition that
actually ensures that the coefficients of y are contained in a number field K);
(2) there exist a sequence of positive numbers N ∈ N and a positive constant C such that N y is
n n s
an integer of K for any 0≤s≤n and N ≤Cn;
n
(3) for any immersion K ֒→ C, the image of y in C[[x]] is a convergent power series for the usual
norm.
Date:January13,2010.
Institut de Mathématiques de Jussieu, Topologie et géométrie algébriques, Case 7012, 2, place Jussieu, 75251 Paris
Cedex05,France. e-mail: [email protected].
Workpartiallysupported byANR,contract ANR-06-JCJC-0028.
1
2 LUCIADIVIZIO
Roughly speaking, a G-module is a, a posteriori fuchsien, K(x)/K-differential module whose (uniform
part of) solutions are G-functions (cf. [Bom81], [CC85], [And89], [DGS94]). More formally, if Y′(x) =
G(x)Y(x) is the differential system associate with such a connection in a given basis, one can iterate
it obtaining a family of the higher order differential systems 1 dnY(x) = G (x)Y(x). Our differential
n! dxn [n]
module is of type G if there exist a constant C > 0 and a sequence of polynomials P (x) ∈ Z[x], such
n
that
(1) P (x)G (x) is a matrix whose entries are polynomials with coefficients in the ring of integers of
n [s]
K, for any s=1,...,n;
(2) the absolute value of the coefficients of P (x) is smaller that Cn.
n
The unsolved Bombieri-Dwork’s conjecture says that G-modules come from geometry, in the sense that
they are extensions of direct summands of Gauss-Manin connections: the precise conjecture is stated in
[And89,II].Y.AndréprovesthatadifferentialmodulecomingfromgeometryisoftypeG(cf. [And89,V,
App.]). Morerecently,the theoryofG-functions hasbeenthe startingpointfor the papers[And00a]and
[And00b], where the author develops an arithmetic theory of Gevrey series, allowing for a new approach
to some diophantine results, such as the Schidlovskii’s theorem.
The question of the existence of an arithmetic theory of q-difference equations was first asked in
[And00b]. A naive analogue over a number field of the notion above clearly does not work. In fact, let
K be a number field and let q ∈ K, q 6= 0, not be a root of unity. We consider formal power series
y ∈K[[x]]thatsatisfiesconditions2and3ofthedefinitionofG-functiongivenaboveandthatissolution
of a nontrivial q-difference equation with coefficients in K(x), i.e. :
a (x)y(qνx)+a y(qnu−1x)+···+a (x)y(x)=0,
ν ν−1 0
with a (x),...,a (x) ∈ K(x), not all zero. Then the following result by Y. André is the key point of
0 ν
[DV02]:
Proposition 1.1 ([DV02, 8.4.1]). A series y as above is the Taylor expansion at 0 of a rational function
in K(x).
Other unsuccessful suggestions for a q-analogue of a G-function are made in [DV00, App.]. These
considerations may induce to conclude that q-difference equations do not come from geometry over Q.
Here we propose another approach: we consider a finite extension K of the field of rational function
k(q) in q with coefficients in a field k. This is a very natural approach since in the literature, q very
often considered as a parameter. Since K is a global field, we can define a G -function to be a series in
q
K[[x]], solutionofaq-difference equationwithcoefficients in K(x), satisfyinga straightforwardanalogue
of conditions 2 and 3 of the definition above. As far as the definition of q-difference modules of type G
is concerned only the places of K modulo whom q is a root of unity - that we will briefly call cyclotomic
places - comes into the picture (cf. Proposition 3.1 below). In fact, consider a q-difference system
(1.1.1) Y(qx)=A(x)Y(x),
with A(x)∈Gl (K(x)): its solutions can be interpreted as the horizontal vectors of a K(x)-free module
ν
M of rank ν with respect to a semilinear bijective operator Σ verifying Σ (f(x)m) = f(qx)Σ (m) for
q q q
any f(x)∈K(x) and any m∈M. We consider the q-derivation:
f(qx)−f(x)
d (f(x))=
q
(q−1)x
and its iterations:
dn
q , with [0]! =[1]! =1 and [n]! = qn−1[n−1]!.
[n]! q q q q−1 q
q
We can obtain from (1.1.1) a whole family of systems:
dn
q
Y(x)=G (x)Y(x),
[n]! [n]
q
where G (x) = A(x)−1 and qn−1G (x) = G (x)G (qx)+d G (x). The fact that the denom-
1 (q−1)x q−1 [n] [1] [n−1] q [n−1]
inators [n]! of the iterated derivations dnq have positive valuation only at the cyclotomic places has
q [n]!
q
the consequences that “there is no arithmetic growth” at the noncyclotomic places (cf. §3 below for a
ARITHMETIC THEORY OF q-DIFFERENCE EQUATIONS 3
preciseformulation). Moreover,animportantrole in the proofs is playedby the reductionof q-difference
systems modulo a cyclotomic place: this means that we specializes q to a rootof unity and we study the
nilpotencepropertiesoftheobtainedsystem. Incharacteristiczero,oneautomaticallyobtainaniterative
q-difference module, in the sense of C. Hardouin [Har07].
Theroleplayedbythecyclotomicvaluations,andthereforebyrootsofunity,pointsoutsomeanalogies
with other topics:
• The Volume Conjecture predicts a link between the hyperbolic volume of the complement of an
hyperbolicknotandtheasymptoticofthesequenceJ (exp(2iπ/n)),whereJ (q)isaninvariantof
n n
the knotcalledn-thJonespolynomial. TheJonespolynomialsareLaurentpolynomialsinq such
thatthegeneratingseries J (q)xn issolutionofaq-differenceequationswithcoefficientsin
n≥0 n
Q(q)(x)(cf. [GL05]): thesituationisquitesimilartotheoneconsideredinthepresentpaper. The
P
q-differenceequationsappearinginthistopologicalsettinghave,ingeneral,irregularsingularities,
differently from the q-difference operators of type G, that are regular singular. To involve some
irregularsingularoperatorsin the presentframework,one shouldconsider some formal q-Fourier
transformations and develop a global theory of q-Gevrey series, in the wake of [And00a]: this is
the topic of the second part of the paper.
• Asalreadypointout,animportantroleisplayedbythereductionofq-differencesystemsmodulo
the cyclotomic valuations. Conjecturally, the growth at cyclotomic places should be enough to
describe the whole theory (cf. §3). It is natural to ask whether q-difference equations, that
seem not to “come from geometry over Q”, may have some geometric origin, in the sense of the
geometry over F (cf. [Sou04], [CC08]).
1
Noticethatin[Man08],Y.Maninestablishalink betweentheHabiroring,whichisatopologicalalgebra
constructed to deal with quantum invariants of knots, and geometry over F , so that the two remarks
1
above are not orthogonal.
∗∗∗
In the present paper we give a definition of G -functions and q-difference modules of type G. We
q
test those definitions proving that a q-difference module having an injective solution whose entries are
G-functionsisoftypeG: thatistosaythat“theminimalq-differencemodulegeneratedbyaG-function”
is of type G (cf. Theorem 4.2 below). We also prove that q-difference module of type G are regular
singular (cf. Theorem 4.1). These two results are the base for the development of a global theory of
q-Gevrey series.
In part two, we define global q-Gevrey series. Via the study of two q-analogues the formal Fourier
transformation,weestablishsomestructuretheoremsfortheminimalq-differenceequationskillingglobal
q-Gevreyseries(cf. Theorems12.3,12.4and12.6). Weconcludewithanirrationalitytheoremforspecial
values of of global q-Gevrey series of negative orders (cf. Theorem 13.6).
This paper won’t be submitted for publication since the results below can be obtained in a more
direct way. Namely, one can prove that G -functions are all rational (cf. [DVH09]). Nevertheless,
q
the construction of the coefficients of the q-difference module from an injective solution in the proof of
Theorem 4.2 has an interest in itself, since it may be applied to other difference operators.
Acknowledgements. I would like to thank Y. André, J-P. Bézivin and Y. Manin for the interest they
have shown for my work. Of course, I’m the only responsible for the deficiencies of this paper.
Part 1. G -functions and q-difference modules of type G
q
2. Definition and first properties
Let us consider the field of rational function k(q) with coefficients in a fixed field k. We fix d∈(0,1)
and for any irreducible polynomial v =v(q)∈k[q] we set:
|f(q)|v =ddegqv(q)·ordv(q)f(q), ∀f(q)∈k[q].
Thedefinitionof|| extendstok(q)bymultiplicativity. Tothissetofnormsonehastoaddthe q−1-adic
v
one, defined on k[q] by:
|f(q)| =d−degqf(q);
q−1
4 LUCIADIVIZIO
once again this definition extends by multiplicativity to k(q). Then the Product Formula holds:
f(q) f(q) −1
=dPvdegqv(q) (ordv(q)f(q)−ordv(q)g(q)) =ddegqf(q)−degqg(q) = .
g(q) g(q)
v (cid:12) (cid:12)v (cid:12) (cid:12)q−1
Y(cid:12) (cid:12) (cid:12) (cid:12)
Foranyfinitee(cid:12)xtensi(cid:12)onK ofk(q),weconsiderthefamilyP ofultrametricnorms,t(cid:12)hatex(cid:12)tendsthenorms
(cid:12) (cid:12) (cid:12) (cid:12)
defined above, up to equivalence. We suppose that the norms in P are normalized so that the Product
Formula still holds. We consider the following partition of P:
• the set P of places of K such that the associated norms extend, up to equivalence, either | |
∞ q
or | | ;
q−1
• the set P of places of K such that the associated norms extend, up to equivalence, one of the
f
norms | | for an irreducible v =v(q)∈k[q], v(q)6=q.
v
Moreover we consider the set C of places v ∈ P such that v divides a valuation of k(q) having as
f
uniformizer a factor of a cyclotomic polynomial. We will briefly call v ∈C a cyclotomic place.
Definition 2.1. A series y = y xn ∈K[[x]] is a G -function if:
n≥0 n q
(1) Itissolutionofaq-differenceequationswithcoefficientsinK(x),i.e. thereexistsa (x),...,a (x)∈
P 0 ν
K(x) not all zero such that
(2.1.1) a (x)y(x)+a (x)y(qx)+···+a (x)y(qνx)=0.
0 1 ν
(2) The series y has finite size, i.e.
1
σ(y):=limsup log+ sup|y | <∞,
s v
n
n→∞ v∈P (cid:18)s≤n (cid:19)
X
where log+x=sup(0,logx).
Wewillrefertothe invariantσ asthesize, usingthe sameterminologyasinthe classicalcaseofseries
over a number field.
Remark 2.2. (1) One can show that this definition of G -function is equivalent to the one givenin
q
the introduction (cf. [And89, I, 1.3]).
(2) Let k(q) be the algebraic closure of k(q). A formal power series with coefficients in k(q) solution
ofaq-differenceequationswithcoefficientsink(q)(x)isnecessarilydefinedoverafiniteextension
K/k(q).
Proposition 2.3. The set of G -functions is stable with respect to the sum and the Cauchy product1.
q
Moreover, it is independent of the choice of K, in the sense that we can replace K by any finite extension
of K.
Proof. The proof is the same as in the case of classical G-functions (cf. [And89, I, 1.4, Lemma 2]). (cid:3)
The field K(x) is naturally a q-difference algebra, i.e. is equipped with the operator
σ : K(x) −→ K(x)
q .
f(x) 7−→ f(qx)
The field K(x) is also equipped with the q-derivation
f(qx)−f(x)
d (f)(x)= ,
q
(q−1)x
satisfying a q-Leibniz formula:
d (fg)(x)=f(qx)d (g)(x)+d (f)(x)g(x),
q q q
for any f,g ∈ K(x). A q-difference module over K(x) (of rank ν) is a finite dimensional K(x)-vector
space M (of dimension ν) equipped with an invertible σ -semilinear operator, i.e.
q
Σ (f(x)m)=f(qx)Σ (m),
q q
1Itmaybeinteresting toremark, althoughwewon’tneed itinthesequel,that theestimateofthesizeofaproduct of
G-functionsprovedin[And89,I,1.4,Lemma2]holdsalsointhecaseofGq-functions.
ARITHMETIC THEORY OF q-DIFFERENCE EQUATIONS 5
for any f ∈K(x) and m ∈M. A morphism of q-difference modules over K(x) is a morphisms of K(x)-
vector spaces, commuting to the q-difference structure (for more generalities on the topic, cf. [vdPS97],
[DV02, Part I] or [DVRSZ03]).
Let M=(M,Σ ) be a q-difference module overK(x) of rank ν. We fix a basis e of M overK(x) and
q
we set:
Σ e=eA(x),
q
with A(x) ∈ Gl (K(x)). An horizontal vector ~y ∈ K(x)ν with respect to Σ is a vector that verifies
ν q
~y(x)=A(x)~y(qx). Therefore we call
Y(qx)=A (x)Y(x), with A (x)=A(x)−1,
1 1
thesystemassociatedtoMwithrespecttothebasise. Recursivelyweobtainthe familiesofq-difference
systems:
Y(qnx)=A (x)Y(x) and dnY(x)=G (x)Y(x),
n q n
with A (x)∈Gl (K(x)) and G (x)∈M (K(x)). Notice that:
n ν n ν
A (x)−1
1
A (x)=A (qx)A (x), G (x)= and G (x)=G (qx)G (x)+d G (x).
n+1 n 1 1 n+1 n 1 q n
(q−1)x
ItisconvenienttosetA =G =1. Moreoverweset[n] = qn−1 foranyn≥1,[n]! =[n] [n−1] ···[1] ,
0 0 q q−1 q q q q
[0]! =1 and G (x)= Gn(x).
q [n] [n]!
q
Definition 2.4. A q-difference module overK(x) is said to be of type G (or a G-q-difference module) if
the following global q-Galočkin condition is verified:
1
σq(M)=limsup log+ sup G <∞,
C n [s] v,Gauss
n→∞ v∈C (cid:18)s≤n (cid:19)
X (cid:12) (cid:12)
where (cid:12) (cid:12)
a xi sup|a |
i i v
= ,
b xj sup|b |
(cid:12)P j (cid:12)v,Gauss j v
(cid:12) (cid:12)
for all Paixi ∈K(x). (cid:12)P (cid:12)
Pbjxj (cid:12) (cid:12)
Remark 2.5. Notice that the definition of G-q-difference module involves only the cyclotomic places.
Proposition 2.6. The definition of G -module is independent on the choice of the basis and is stable by
q
extension of scalars to K′(x), for a finite extension K′ of K.
Proof. Once again the proof if similar to the classical theory of G-functions and differential modules of
type G. (cid:3)
3. Role of the “noncyclotomic” places
Proposition3.1. Inthe notation introducedabove, for anyq-differencemodule M=(M,Σ )over K(x)
q
we have:
1
σ(q) (M):=limsup log+ sup G <∞.
PfrC n→∞ nv∈XPfrC (cid:18)s≤n(cid:12) [s](cid:12)v(cid:19)
(cid:12) (cid:12)
Proof. We recall that the sequence of matrices G satisfies the recurrence relation:
[n]
G (qx)G (x)+d G (x)
[n] 1 q [n]
G (x)= .
[n+1] [n+1]
q
Since |[n+1] | =1 for any v ∈P rC, we conclude recursively that
q v f
G ≤1,
[n] v,Gauss
foralmostallplacesv ∈PfrC. Forthe rem(cid:12)aini(cid:12)ngfinitely manyplaces v ∈Pf, onecandeduce fromthe
recursive relation there exists a constant C(cid:12)>0 s(cid:12)uch that G ≤Cn. (cid:3)
[n] v,Gauss
We immediately obtain the equivalence of our definitio(cid:12)n of(cid:12)q-difference module of type G with the
(cid:12) (cid:12)
naive analogue of the classical definition of G-module (cf. [And89, IV, 4.1]):
6 LUCIADIVIZIO
Corollary 3.2. A q-difference module is of type G if and only if
1
σ(q)(M):=limsup log+ sup G <∞.
Pf n→∞ nvX∈Pf (cid:18)s≤n(cid:12) [s](cid:12)v(cid:19)
(cid:12) (cid:12)
We expect the same kind of result to be true for G -functions, namely:
q
Conjecture 3.3. Suppose that y = y xn ∈ K[[x]] is solution of a q-difference equations with
n≥0 n
coefficients in K (cf. (2.1.1)). Then:
P
1
σPfrC(y)=limsupn log+ sup|ys|v <∞.
n→∞ v∈XPfrC (cid:18)s≤n (cid:19)
The last statement would immediately imply that one can define G -functions in the following way:
q
Conjectural definition 3.4. We say that the series y = y xn ∈ K[[x]] is a G -function if y is
n≥0 n q
solution of a q-difference equations with coefficients in K and moreover
P
1
σ (y)=limsup log+ sup|y | <∞.
C∪P∞ n s v
n→∞ v∈XC∪P∞ (cid:18)s≤n (cid:19)
Remark 3.5. The fact that for almostall v ∈P rC we have |G (x)| ≤1 for any n≥1 implies
f [n] v,Gauss
that for almost all v ∈ P rC a “solution” y(x) = sup y xn ∈ K[[x]] of a q-difference system with
f n n
coefficient in K(x) is bounded, in the sense that sup |y | < ∞. Unfortunately, one would need some
n n v
uniformity with respect to v and n to conclude something about σPfrC(y).
Notice that if 0 is an ordinary point, the conjecture is trivial since
G (0)xn
[n]
n≥0
X
is a fundamental solution of the linear system Y(qx) = A (x)Y(x). A q-analogue of the techniques
1
developed in [And89, V] (cf. also [DGS94, Chap. VII]) would probably allow to establish the conjecture
undertheassumptionthat0isaregularpoint. Thisisnotsatisfactorybecauseoneofthepurposesofthe
whole theory is the possibility of reading the regularity of a q-difference equation on one single solution
(cf. Theorem 4.1 below), so one does not want to assume regularity a priori.
4. Main results
A q-difference module (M,Σ ) is saidto be regularsingular at 0 if there exists a basis e such that the
q
Taylorexpansionof the matrix A (x) is in Gl (K[[x]]). It is said to be regularsingular tout court if it is
1 ν
regular singular both at 0 and at ∞. We have the following analogue of a well-known differential result
(cf. [Kat70, §13]; cf. also [DV02, §6.2.2] for q-difference modules over a number field):
Theorem 4.1. A G-q-difference module M over K(x) is regular singular.
Let ~y(x) = t(y (x),...,y (x)) ∈ K[[x]]ν be a solution of the q-difference system associated to
0 ν−1
M=(M,Σ ) with respect to the basis e:
q
~y(qx)=A (x)~y(x).
1
We say that ~y(x) is an injective solution if y (x),...,y (x) are lineairly independent over K(x).
1 ν
We have the following q-analogue of the André-Chudnovsky Theorem [And89, VI]:
Theorem 4.2. Let ~y(x) = t(y (x),...,y (x)) ∈ K[[x]]ν be an injective solution of the q-difference
0 ν−1
system associated to M=(M,Σ ) with respect to the basis e.
q
If y (x),...,y (x) are G -functions, then M is a G-q-difference module.
0 ν−1 q
We can immediately state a corollary:
Corollary 4.3. Let ~y(x) = t(y (x),...,y (x)) ∈ K[[x]]ν be an injective solution of the q-difference
0 ν−1
system associated to M=(M,Σ ) with respect to the basis e.
q
If y (x),...,y (x) are G -functions, then M is regular singular.
1 ν q
Thanks to the cyclic vector lemma we can state the following (cf. [Sau00, Annexe B]):
ARITHMETIC THEORY OF q-DIFFERENCE EQUATIONS 7
Corollary 4.4. Let y(x) a G -function and let
q
(4.4.1) a (x)y(x)+a (x)y(qx)+···+a (x)y(qνx)=0.
0 1 ν
a q-difference equation of minimal order ν, having y(x) as a solution.
Then (4.4.1) is fuchsian, i.e. we have ord a ≥ord a =ord a and deg a ≤deg a =deg a , for
x i x 0 x ν x i x 0 x ν
any i=0,...,ν.
The proofs of Theorem 4.1 and Theorem 4.2 are the object of §6 and §7, respectively.
5. Nilpotent reduction at cyclotomic places
We denote by O the ring of integers of K, k the residue field of K with respect to the pace v, ̟
K v v
the uniformizer of v and q the image of q in k , which is defined for all places v ∈P. Notice that q is
v v v
a root of unity for all v ∈C. Let κ ∈N be the order of q , for v ∈C.
v v
Let M = (M,Σ ) be a q-difference module over K(x). We can always choose a lattice M of M over
q
an algebra of the form
f
1 1 1
(5.0.2) A=O x, , , ,... ,
K P(x) P(qx) P(q2x)
(cid:20) (cid:21)
for some P(x) ∈ O [x], such that for almost all v ∈ C we can consider the q -difference module M =
K v v
M⊗ k (x),withthestructureinducedbyΣ . Inthisway,foralmostallv ∈C,weobtainaq -difference
A v q v
module M =(M ,Σ ) over k (x), having the particularity that q is a root of unity. This means that
v v qv v v
σfqκvv =1 and that Σκqvv is a kv(x)-linear operator.
The results in [DV02, §2] apply to this situation: we recall some of them. Since we have:
σκv =1+(q−1)κvxκvdκv
qv qv
and
Σκv =1+(q−1)κvxκv∆κv,
qv qv
where ∆ = Σqv−1 , the following facts are equivalent:
qv (qv−1)x
(1) Σκv is unipotent;
qv
(2) ∆κv is a linear nilpotent operator;
qv
(3) the reduction of A (x)−1 modulo ̟ is a nilpotent matrix;
κv v
(4) the reduction of G (x) modulo ̟ is nilpotent;
κv v
(5) there exists n∈N such that |G (x)| ≤|̟ | .
nκv v,Gauss v v
Definition 5.1. If the conditions aboveare satisfiedwe say thatM has nilpotent reduction (of order n)
modulo v ∈C.
Remark 5.2. If the characteristic of k is 0 and if |G (x)| ≤ |[κ ] | , the module M has a
κv v,Gauss v q v v
structure of iterated q-difference module, in the sense of [Har07, §3]. In particular, if v is a non ramified
place of K/k(q), then |[κ ] | =|̟ | .
v q v v v
The following result is a q-analogue of a well-known differential p-adic estimate (cf. for instance
[DGS94, page 96]). It has already been proved in the case of q-difference equations over a p-adic field
in [DV02, §5.1]. We are only sketching the argument: only the estimate of the q-factorials are slightly
different from the case of mixed characteristic.
Proposition 5.3. If M=(M,Σ ) has nilpotent reduction(of order n) modulo v ∈C then
q
1/m
limsupsup 1, G ≤|̟ |1/nκn|[κ ] |−1/κv.
[m] v,Gauss v v v q v
m→∞
(cid:16) (cid:12) (cid:12) (cid:17)
Proof. The Leibniz formula (cf. [DV0(cid:12)2, Le(cid:12)mma 5.1.2] for a detailed proof in a quite similar situation)
implies that for any s∈N we have:
|G (x)| ≤|̟ |s.
snκv v,Gauss v v
8 LUCIADIVIZIO
Since |G (x)| ≤1, for any m∈N we have:
1 v,Gauss
G (x) ≤ G[nmκv]nκv(x) v,Gauss ≤ |̟v|[vnmκv] ,
[m] v,Gauss (cid:12)(cid:12) |[m]!q|v(cid:12)(cid:12) |[m]!q|v
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where m =max{a∈Z: a≤ m }. The following lemma on the estimate of [m]! allows to conclude.
nκv nκv q
(cid:3)
h i
Lemma 5.4. For v ∈C we have |[m] | =|[κ] | if κ |m and |[m] | =1 otherwise. Therefore:
q v q v v q v
lim |[m]!|1/m =|[κ ] |1/κv.
q v v q v
m→∞
Proof. Let m≥2 and m=sκ +r, with r,s∈Z and 0≤r <κ . If κ does not divide m, i.e. if r >0,
v v v
we have
[m] =1+q+···+qm−1 =[κ ] +qκv[κ ] +···+qsκv(1+q+···+qr−1).
q v q v q
Therefore |[m] | =1. On the other hand, if r =0:
q v
[m] = 1+qκv +···+qκv(s−1) [κ ] .
q v q
(cid:16) (cid:17)
Since qκv ≡1 modulo ̟v, we deduce that 1+qκv +···+qκv(s−1) ≡s modulo ̟v. Therefore
|[m] | =|s| |[κ ] | =|[κ ] | .
q v v v q v v q v
This implies that
[m]!q v =|κv]q|[vκmv] ,
which allows to calculate the limit. (cid:12) (cid:12) (cid:3)
(cid:12) (cid:12)
We obtain the following characterization:
Corollary 5.5. The q-difference module M=(M,Σ ) has nilpotent reduction modulo v ∈C if and only
q
if
1/m
(5.5.1) limsupsup 1, G <|[κ ] |−1/κv.
[m] v,Gauss v q v
m→∞
(cid:16) (cid:12) (cid:12) (cid:17)
Proof. One side of the implication is an im(cid:12)media(cid:12)te consequence of the proposition above. On the other
hand, the assumption (5.5.1) implies that
1/m
limsupsup 1,|G | <1,
m v,Gauss
m→∞
(cid:16) (cid:17)
which clearly implies that there exists n such that |G | ≤|̟ | . (cid:3)
nκv v,Gauss v v
We finally obtain the following proposition, that will be useful in the proof of Theorem 4.1:
Proposition 5.6. Let M be q-difference module over K(x) of type G. Let C be the set of v ∈ C such
0
that M does not have nilpotent reduction modulo v. Then
1
<+∞.
κ
v
vX∈C0
In particular, M has nilpotent reduction modulo v for infinitely many v ∈C.
The proof relies on the following lemma:
Lemma 5.7. The following limit exists:
1
lim log+ sup G (x) .
n→∞n (cid:18)s≤n [s] v,Gauss(cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
ARITHMETIC THEORY OF q-DIFFERENCE EQUATIONS 9
Proof. The proof is essentially the same as the proof of [DV02, 4.2.7], a part from the estimate of the
q-factorials (cf. Lemma 5.4 above). The key point is the following formula:
[n]![s]! dj
G (x)= q q q G (qix) G (x), ∀s,n∈N,
[n+s] [s+n]![j]! [s] [i]
i+j=n q
X (cid:0) (cid:1)
obtained iterating the Leibniz rule. (cid:3)
Proof of Proposition 5.6. The Fatou lemma, together with Lemma 5.7, implies:
1 1
lim log+ sup G (x) ≤liminf log+ sup G (x) ≤σ(q)(M)<∞.
v∈Cn→∞n (cid:18)s≤n [s] v,Gauss(cid:19) n→∞ nv∈C (cid:18)s≤n [s] v,Gauss(cid:19) C
X (cid:12) (cid:12) X (cid:12) (cid:12)
It follows from Corollary(cid:12) 5.5 tha(cid:12)t: (cid:12) (cid:12)
log+|[κ ] |−1
v q v <∞
κ
v
vX∈C0
and hence that
logd−1
<∞,
κ
v
vX∈C0
since only a finite number of places of K/k(q) are ramified. (cid:3)
6. Proof of Theorem 4.1
It is enough to prove that 0 is a regular singular point for M, the proof at ∞ being completely
analogous.
Let r ∈ N be a divisor of ν! and let L be a finite extension of K containing an element q such that
qr = q. We consider the field extension K(x) ֒→ L(t), x 7→tr. The field L(t) has a natural structure of
q-difference algebra extending the q-difference structure of K(x). Remark that: e
e
Lemma 6.1. The q-difference module M is regular singular at x = 0 if and only if the q-difference
e
module M :=(M ⊗ L(t),Σ :=Σ ⊗σ ) is regular singular at t=0.
L(t) K(t) q q q
e e
e
Proof. It is enough to notice that if e is a cyclic basis for M, then e⊗1 is a cyclic basis for M and
L(t)
Σ (e⊗1)=Σ (e)⊗1. (cid:3)
q q
e
The next lemma can be deduced from the formal classification of q-difference modules (cf. [Pra83,
Cor. 9 and §9, 3)], [Sau04, Thm. 3.1.7]):
Lemma 6.2. There exist an extension L(t)/K(x) as above, a basis f of the q-difference module M ,
L(t)
such that Σ f =fB(t), with B(t)∈Gl (L(t)), and an integer ℓ such that
q µ
e
B B e
ℓ ℓ−1
B(t)= + +... ,as an element of Gl (L((t)));
(6.2.1) tℓ tℓ−1 µ
( B is a constant non nilpotent matrix.
ℓ
Proof of Theorem 4.1. Let B ⊂ L(t) be a q-difference algebra over the ring of integers O of L, of the
L
same form as (5.0.2), containing the entries of B(t). Then there exists a B-lattice N of M inheriting
L(t)
the q-difference module structure from MLe(t) and having the following properties:
1. N has nilpotent reduction modulo infinitely many cyclotomic places of L;
2. there exists a basis f of N over B such that Σ f =fB(t) and B(t) verifies (6.2.1).
e q
Iterating the operator Σ we obtain: e
q
e
Bm
Σm(f)=fB(t)B(qt)···B(qm−1t)=f ℓ +h.o.t.
qe qℓm(ℓ2m−1)xmℓ !
We know that for infinitely many cycloetomic pleaces w of L, the matrix B(t) verifies
e
(6.2.2) B(t)B(qt)···B(qκw−1t)−1 n(w) ≡0 mod ̟ ,
w
(cid:0) (cid:1)
e e
10 LUCIADIVIZIO
where ̟ is an uniformizer of the place w, κ is the order q modulo ̟ and n(w) is a convenient
w w w
positive integer. Suppose that ℓ6=0. Then Bκw ≡0 modulo ̟ , for infinitely many w, and hence B is
ℓ w ℓ
a nilpotent matrix, in contradiction with lemma 6.2. So necessarily ℓ=0.
e
Finally we have Σ (f) = f(B +h.o.t). It follows from (6.2.1) that B is actually invertible, which
q 0 0
implies that M iseregular singular at 0. Lemma 6.1 allows to conclude. (cid:3)
L(t)
7. Proof of Theorem 4.2
7.1. Idea of the proof. The hypothesis states that there exists a vector ~y =t(y ,...,y ) ∈K[[x]]ν,
0 ν−1
which is solution of the q-difference system:
(7.0.3) ~y(qx)=A (x)~y(x),
1
and therefore of the systems dn~y = G (x)~y and σn~y = A (x)~y for any n ≥ 1, having the property that
q n q n
y ,...,y are linearly independent over K(x). We recall that
0 ν−1
G (x)=G (qx)G (x)+d G (x)
n+1 n 1 q n
and that
A (x)=A (qx)A (x).
n+1 n 1
Let us consider the operator:
Λ=A (x)−1◦(d −G (x)) .
1 q 1
We know that there exists an extension U of K(x) (for instance the universal Picard-Vessiot ring con-
structed in [vdPS97, §12.1]) such that we can find an invertible matrix Y with coefficient in U solution
of our system d Y =G Y. An explicit calculation shows that:
q 1
d ◦Y−1 =(σ Y)−1(d −G (x))=Y−1A (x)−1(d −G (x))
q q q 1 1 q 1
and therefore that:
(7.0.4) Λn =Y ◦dn◦Y−1, for all integers n≥0.
q
We set n = [n]!q , for any pair of integers n ≥ i ≥ 0. The twisted q-binomial formula shows that
i q [i]![n−1]!
q q
n ≤(cid:0) (cid:1)1 for any v ∈P .
i q v f
(cid:12) Th(cid:12)e proof of Theorem 4.2 is based on the following q-analogue of [And89, VI, §1]:
(cid:12)(cid:0) (cid:1) (cid:12)
(cid:12) (cid:12)
Proposition 7.1. There exist α(n),...,α(n) ∈K such that for all P~ ∈K[x]ν and all n≥0 we have:
0 n
n (−1)i n
(7.1.1) G P~ = α(n)dn−i◦A (x)Λi(P~),
[n] [n]! i i q i
i=0 q (cid:18) (cid:19)q
X
with |α (n)| ≤1, for any v ∈P and n≥i≥0.
i v f
Proof. The iterated twisted Leibniz Formula (cf. for instance [DV02, 1.1.8.1])
n
n
dn(fg)= σj dn−j(f) dj(g), ∀f,g ∈U
q j q q q
j=0(cid:18) (cid:19)q
X (cid:0) (cid:1)
