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regular singular Mahler equations
Marina Poulet
To cite this version:
Marina Poulet. A density theorem for the difference Galois groups of regular singular Mahler equations. 2020. �hal-03084197�
MARINA POULET
Abstract. The difference Galois theory of Mahler equations is an active research area. The present paper aims at developing the analytic aspects of this theory. We first attach a pair of connection matrices to any regular singular Mahler equation. We then show that these connection matrices can be used to produce a Zariski-dense subgroup of the difference Galois group of any regular singular Mahler equation.
Contents
Introduction 3
Notations 4
Part I. Preliminary results 5
1 Difference equations, difference systems and difference modules. . . 5
1.1 Generalities . . . 5
1.1.aThe category of difference modules. . . 5
1.1.bDifference systems and equations . . . 6
1.1.c The category of difference systems and the category of difference mod-ules are equivalent . . . 6
1.2 The category of Mahler systems with different base fields . . . 7
1.2.aThe category E (Kp∞) . . . 7
1.2.bThe category E. . . 7
2 Regular singular Mahler systems . . . 8
2.1 Neighbourhood of the point 0 . . . 8
2.2 Neighbourhood of the point ∞ . . . 10
2.3 Neighbourhood of the point 1 . . . 11
3 Meromorphic functions on (a part of) the universal cover of C?. . . 13
Part II. The categories of Fuchsian systems and regular singular systems at 0, 1 and ∞ 14 4 The category Ers of regular singular systems at 0, 1 and ∞ . . . 14
5 The category Ef of Fuchsian systems at 0, 1 and ∞ . . . 14
Date: December 20, 2020.
2010 Mathematics Subject Classification. 39A06, 12H10, 11R45.
6 The categories Ers and Ef are equivalent. . . 14
Part III. Local categories and local groupoids 16 7 Preliminary results . . . 16
7.1 On morphisms between constant systems . . . 16
7.2 Finite dimensional complex representations of Z. . . 17
8 Localization at 0 . . . 17
8.1 The categories Ers(0) and P(0) . . . 17
8.2 Local Galois groupoid at 0 . . . 18
9 Localization at ∞. . . 18
9.1 The categories Ers(∞) and P(∞). . . 18
9.2 Local Galois groupoid at ∞. . . 18
10Localization at 1 . . . 19
10.1The categories Ers(1) and P(1) . . . 19
10.2Local Galois groupoid at 1 . . . 20
Part IV. Global categories 21 11The category Ers of regular singular systems at 0, 1 and ∞ . . . 21
12The category C of connections . . . 21
13The category S of solutions . . . 22
14Equivalence between the categories . . . 23
15Structure of neutral Tannakian category over C . . . 24
Part V. Mahler analogue of the density theorem of Schlesinger 25 16The Galois groupoid of the category C. . . 25
17Construction of isomorphisms of tensor functors. . . 26
17.1The category CE0,E∞ . . . 26
17.2Construction of isomorphisms of tensor functors. . . 29
18Analogue of the density theorem of Schlesinger for Mahler equations. . . 30
18.1Density criteria. . . 30
18.2Density theorem in the regular singular case . . . 31
18.3Density theorem in the regular case. . . 34
Introduction
A Mahler equation of order n ∈ N? is a functional equation of the form
(1) an(z)f zp n + an−1(z)f zpn−1 + · · · + a0(z)f (z) = 0
with a0(z), . . . , an(z) ∈ C(z), with a0(z)an(z) 6= 0 and with p an integer greater than or egal to 2. The solutions of these equations are called Mahler functions. The study of these functions began with the work of Mahler on the algebraic relations between their special values in [Mah29], [Mah30a], [Mah30b]. This work has been a source of inspiration for many authors e.g. [LP77], [Lox84], [Nis96], [Pel11], [BCZ14], [AF16], [CDDM16], [AB17], [Fer19]. Philippon in [Phi15], then Adamczewski and Faverjon in [AF15], showed that the algebraic relations between special values of Mahler functions come from algebraic relations between the functions; hence the importance of the study of algebraic relations between Mahler functions. Moreover, there is an increased interest in these functions because of their connection with automata theory (see for instance [Cob68], [MF80], [AS92], [Bec94] for more details).
The algebraic relations between Mahler functions are reflected by certain difference Galois groups. This is one of the reasons why the difference Galois theoretic aspects of the theory of Mahler equations has been the subject of many recent works. For more details on this difference Galois theory and applications see for instance [vdPS97], [DHR15], [Phi15], [Roq18]. These articles focus on the algebraic aspects of the difference Galois theory of Mahler equations. The analytic aspects have not yet been studied so much. The aim of the present paper is to start to fill this gap: our main objective is to introduce for any Mahler equation a Zariski-dense subgroup of analytic nature of its difference Galois group. Our result is inspired by a celebrated density theorem due to Schlesinger for differential equations (see [Sch95]) and by its generalizations to (q-)difference equations discovered by Etingof and developed further by Duval, Ramis, Sauloy, Singer, van der Put. More precisely, the density theorem of Schlesinger ensures that the monodromy of a differential equation with regular singular points is Zariski-dense in its differential Galois group. This theorem was transposed to regular q-difference equations by Etingof (see [Eti95]) in 1995 and later to regular singular q-difference equations by Sauloy on the one hand (see [Sau99]) and by van der Put and Singer on the other hand (see [vdPS97]). The key to the extensions of Schlesinger’s theorem to q-difference equations is a connection matrix introduced by Birkhoff playing the role of the monodromy in the differential case. Roughly speaking, this matrix connects 0 to ∞: given a q-difference equation, we associate two bases of solutions, one at 0, the other one at ∞ and the connection matrix reflects the linear relations between these two bases of solutions. Likewise, for a Mahler equation which is regular singular at 0 and ∞ we can attach two bases of solutions, one at 0 and another one at ∞. However, a problem arises for Mahler equations: these two bases of solutions can not be connected in general. Indeed, the basis of solutions at 0 consists of meromorphic functions on the open unit disk while the basis of solutions at ∞ consists of meromorphic functions on the complement of the closed unit disk and, by a theorem due to Rand´e (see [Ran92]), the unit circle is in general a natural boundary for these functions. In order to overcome this problem, our key idea is to use the point 1: we consider three bases of solutions, at 0, 1 and ∞ and we connect 0 to 1 on the one hand and 1 to ∞ on the other hand. In this way, we attach two connection matrices to any regular singular Mahler equation. This allows us to construct particular elements of
the Galois groupoid G of the category of regular singular Mahler equations. We will show that these elements generate a Zariski-dense subgroupoid of G.
We shall now describe more precisely the content of this paper. Part Icontains prereq-uisites and first results about Mahler equations at the points 0, 1 and ∞. Locally at 1, we can use known results about q-difference equations, see for instance [Sau00]. Indeed, locally at 1, a Mahler equation can be seen as a q-difference equation with the change of variables z = exp(u). In part II, we introduce the category of Fuchsian Mahler systems and regular singular Mahler systems at 0, 1 and ∞. We prove that these categories are equivalent categories. In partIII, we construct and describe the local Galois groups and groupoids at 0, 1 and ∞ of the category of regular singular Mahler systems, which are denoted by G0, G1 and G∞respectively. PartIVdeals with the global categories, we prove that these categories are equivalent Tannakian categories. In particular, we introduce the category of connections C constructed from pairs of connection matrices attached to regu-lar singuregu-lar Mahler equations. In partV, we construct elements of the Galois groupoid G of the category of regular singular Mahler equations. A first family of elements come from the local Galois groupoids G0, G1 and G∞. Thanks to the pair of connection matrices, we also build elements which make the link between G0 and G1, denoted by Γ0,ze0 on the one hand, and between G1 and G∞, denoted by Γ∞,zf∞ on the other hand. We prove the main result of this paper:
Theorem18.4The local Galois groupoids G0, G1, G∞ and the Galoisian isomorphisms Γ0,ze0, Γ∞,zf∞ for all ze0 ∈ Σ0\ fE0, zf∞∈ Σ∞\ gE∞ generate a Zariski-dense subgroupoid H of the Galois groupoid G of CE0,E∞.
Moreover, we know Zariski-dense subgroupoids of G0, G∞ (see 7.2,8.2,9.2) and of G1 (see10.2).
Acknowledgements. This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Universit´e de Lyon, within the program “Investisse-ments d’Avenir” (ANR-11-IDEX- 0007) operated by the French National Research Agency (ANR).
Notations
We denote by C[[z]] the ring of formal power series with complex coefficients in the indeterminate z and by C{z} the ring of all power series in C[[z]] that are convergent in some neighborhood of the origin. We denote by C((z)) the fraction field of C[[z]] (which is called the field of formal Laurent series) and by C({z}) the fraction field of C{z}. In a similar way, C1z is the ring of formal power series with complex coefficients in 1/z and C 1z is its fraction field. We denote by C 1
z the convergent power series in 1/z and C 1z its fraction field. Similarly, C{z − 1} is the ring of all power series that are convergent in some neighborhood of 1 and C ({z − 1}) is its fraction field. If E is an open set of a Riemann surface, we denote by M (E) the field of meromorphic functions on E. We consider the universal cover of C?, denoted by fC? and e1 is the point (1, 0) of this Riemann surface.
We denote by Vectf
Cthe finite dimensional C-vector spaces. If M is a matrix, rk(M ) is its rank, that is the dimension of the vector space generated by its columns.
We denote by D(0, 1) the open unit disk and by D(0, 1) the closed unit disk.
If C is a category, Obj (C) and Mor (C) denote respectively the objects and the morphisms of the category C. If A ∈ Obj (C) and B ∈ Obj (C), we denote by MorC(A, B) the class of morphisms in the category C from A to B. If u ∈ MorC(A, B), we write u : A → B. If G is a groupoid and α, β ∈ Obj (C) then G (α, β) denote the morphisms of G from α to β. Part I. Preliminary results
1. Difference equations, difference systems and difference modules For more details, the reader is referred to [vdPS97].
1.1. Generalities
Let K be a field and φ : K → K be an automorphism of K. We extend φ to the vector spaces of matrices with entries in K applying φ on each entry of the matrices.
1.1.a. The category of difference modules
A difference module over the difference field (K, φ) is a pair (M, ΦM) consisting of a finite dimensional K-vector space M equipped with an automorphism
ΦM : M → M which is φ-linear, that is
∀λ ∈ K, ∀x, y ∈ M, ΦM(x + λy) = ΦM(x) + φ (λ) ΦM(y) .
The objects of the category of difference modules are the difference modules (M, ΦM) and the morphisms f from the difference module (M, ΦM) to the difference module (N, ΦN) are the K-linear maps f : M → N such that
ΦN ◦ f = f ◦ ΦM. Let
CK = Kφ= {x ∈ K | φ(x) = x}
be the constant field. According to [vdPS97], the category of difference modules is a CK -linear rigid abelian tensor category. The functor which forgets the difference structure i.e.
(
(M, ΦM); M f ; f
from the category of difference modules to the category of finite dimensional K-vector spaces VectfK is a fibre functor. Therefore, the category of difference modules is a Tan-nakian category over CK (see [DM82]). For a better understanding on what follows, we give here more details on the tensor structure of this Tannakian category over CK. The identity object of this category is the 1-dimensional K-vector space 1 = Ke equipped with the φ-linear automorphism Φ1 : e → e. Let (M, ΦM) and (N, ΦN) be two objects of this category, the internal Hom is
Hom ((M, ΦM) , (N, ΦN)) = HomK(M, N ) , ΦHomK(M,N )
where HomK(M, N ) denotes the K-vector space of K-linear morphisms from M to N and
ΦHomK(M,N ) is the φ-linear automorphism
ΦHomK(M,N ): HomK(M, N ) → HomK(M, N )
The dual of the object (M, ΦM) is the object Hom ((M, ΦM) , (1, Φ1)) that is HomK(M, K) , σ ∈ HomK(M, K) 7→ σ ◦ Φ−1M ∈ HomK(M, K) .
We notice that the category of difference modules is equivalent to its full subcategory whose objects are the K-vector spaces Kn, n ∈ N. This category is therefore a Tannakian category over CK and we denote it by Diff(K, φ).
1.1.b. Difference systems and equations Let
L = φn+ a1φn−1+ · · · + an with a1, . . . , an∈ K, an6= 0. Any difference equation of order n
Lf = φn(f ) + a1φn−1(f ) + · · · + anf = 0 can be transformed into the difference system of rank n
φ (Y ) = ALY with AL= 0 1 0 · · · 0 .. . . .. ... . .. ... .. . . .. . .. 0 0 · · · 0 1 −a0 an · · · −an−2 an −an−1 an ∈ GLn(K) .
Conversely, by the cyclic vector lemma, any system φ(Y ) = BY, B ∈ GLn(K) is equivalent via the equivalence
A ∼ (φ (F ))−1AF, F ∈ GLn(K)
to a system of the form φ(Y ) = ALY . Thus, we can associate a difference equation with a difference system. In this article, we will focus on difference systems called Mahler systems but the results obtained can therefore be applied to Mahler equations.
The objects of the category of difference systems are the pairs (Kn, A) where n ∈ N and A ∈ GLn(K) and the morphisms from (Kn1, A) to (Kn2, B) are the R ∈ Mn2,n1(K) such that φ(R)A = BR. To simplify notations, we will often denote by A the object (Kn, A). This object is identified with the system φ(Y ) = AY .
1.1.c. The category of difference systems and the category of difference modules are equiv-alent
We can associate with the difference system
φ (Y ) = AY, A ∈ GLn(K) the difference module (Kn, ΦM) ∈ Obj (Dif f (K, φ)) where
ΦM : Y ∈ Kn7→ A−1φ(Y ) ∈ Kn
and with the morphism R : A → B in the category of difference systems, introduced in
1.1.b, the morphism
in the category Diff(K, φ). This provides an equivalence of abelian categories between the categories of difference systems and difference modules. We want to obtain an equivalence of tensor categories. In order to do that, we can consider the following tensor product on matrices called Kronecker product: if A = (ai,j) ∈ Mn1,m1(K) and B ∈ Mn2,m2(K) then
A ⊗ B = a1,1B · · · a1,m1B .. . ... an1,1B · · · an1,m1B ∈ Mn1n2,m1m2(K) .
Therefore, we define a tensor product on the category of difference systems as follows: if (Kn1, A) and (Kn2, B) are two objects then
(Kn1, A) ⊗ (Kn2, B) = (Kn1n2, A ⊗ B)
and if R1et R2are two morphisms, their tensor product is R1⊗R2. The above construction provides an equivalence of abelian tensor categories between the category of difference systems and the category of difference modules. Therefore, the category of difference systems is a CK-linear rigid abelian tensor category. It is a Tannakian category over CK with the forgetful functor (Kn, A); Kn, R; R.
As previously for the category of difference modules, let us explain the tensor structure inherited by the category of difference systems. The identity object is 1 ∈ K?. Let (Kn1, A) and (Kn2, B) be two objects of this category, the internal Hom is
Hom ((Kn1, A) , (Kn2, B)) = (Kn1n2, D
A,B)
where DA,B is the inverse of the matrix representing the CK-linear map Y ∈ Mn2,n1(K) 7→ B
−1φ(Y )A in the canonical basis of Mn2,n1(K). The dual of the object (K
n1, A) is Hom ((Kn1, A) , (K, 1)) = Kn1, tA−1 . 1.2. The category of Mahler systems with different base fields 1.2.a. The category E (Kp∞)
The category of Mahler systems with base field Kp∞ :=S
n≥0Kpn with Kpn = C z1/p
n
is the particular case of categories of difference systems introduced in1.1.bwhere K = Kp∞ and the field automorphism of K is φ = φp defined by
φp : f (z) ∈ K 7→ f (zp) ∈ K.
We will denote it by E (Kp∞). In this case, the constant field is CK = Kφp = C. From the previous section, we know that it is a Tannakian category over C.
1.2.b. The category E
We will consider the category of Mahler systems whose base field is K = C(z) equipped with the injective endomorphism φp (which is not anymore surjective) i.e. the objects are the pairs (C(z)n, A) where n ∈ N and A ∈ GL
n(C(z)) and the morphisms from (C(z)n1, A) ∈ Obj (E ) to (C(z)n2, B) ∈ Obj (E ) are the R ∈ M
n2,n1(C(z)) such that φp(R)A = BR. We denote by E this category. It is a subcategory of E (Kp∞).
Proof. The category E is a rigid tensor subcategory of E (Kp∞) because it is stable under tensor product, it contains the unit object, the internal Hom and every object has a dual (see1.1.cfor a description of these elements). To show that it is a Tannakian subcategory of E (Kp∞), the only non trivial point which remains to be proved is the existence of kernels and cokernels in E . Let us prove that every morphism in E has a kernel in E . The existence of cokernels will follow from the fact that duality exchanges kernels with cokernels. Let A, B ∈ Obj (E ) respectively of rank n1 and n2 and R ∈ MorE(A, B). The morphism R has a kernel (C, K : C → A) in the Tannakian category E (Kp∞). There exists n0 ∈ N such that φn0
p (C) , φnp0(K) have entries in C(z). Let
FA= n0 Y i=1 φn0−i p (A) !−1 ∈ GLn1(C(z)) . The following diagram is commutative
A R //B φn0 p (A) φn0p (R) // FA OO φn0 p (B) FB OO φn0 p (C) φn0p (K) OO 0 ;;
and one can check that φn0
p (C) , FAφnp0(K) : φnp0(C) → A is a kernel of R in the category
E.
We will also consider the category of Mahler systems whose base field is K = C ({z}) (respectively K = C ({z − 1}) and K = C 1
z ) equipped with φp, we will denote it by E(0) (respectively by E(1), by E(∞)). These are also Tannakian categories over C. Indeed, for the categories E(0) and E(∞) we can adapt the proof of Proposition 1.1 and for the category E(1), φ
p : z ∈ C ({z − 1}) 7→ zp is an automorphism of C ({z − 1}) so it follows from1.1.c.
2. Regular singular Mahler systems
Definition 2.1. A matrix A is regular at x ∈ C if its entries are analytic functions at x and A(x) ∈ GLn(C). The matrix A is regular at ∞ if A (1/z) is regular at 0.
2.1. Neighbourhood of the point 0 We consider the Mahler system
(2) φp(Y ) = AY
with A ∈ GLn(C ({z})).
Definition 2.3. We say that the system (2) is meromorphically equivalent at 0 to the system
φp(Y ) = BY
with B ∈ GLn(C ({z})) if there exists T ∈ GLn(C ({z})) such that B = φp(T )−1AT.
The motivation behind this definition is the following: if Z is a solution of the Mahler system (2) and T ∈ GLn(C ({z})) then T−1Z is a solution of the system φp(Y ) = BY . Remark 2.4. Two systems of the form (2) are meromorphically equivalent at 0 if and only if they are isomorphic in the category E(0).
Definition 2.5. The system (2) is regular singular at 0 if it is meromorphically equivalent at 0 to a Fuchsian one.
From nom on, we fix a submultiplicative norm ||.|| on Mn(C).
Theorem 2.6. Assume that the system (2) is Fuchsian at 0. There exists a unique F ∈ GLn(C[[z]]) such that F (0) = In and
(3) φp(F )−1AF = A(0).
We have F ∈ GLn(C{z}).
Moreover, if we assume that A ∈ GLn(C(z)) then F ∈ GLn(M (D(0, 1))). Proof. We are looking for a matrix F ∈ GLn(C[[z]]) such that
(4) AF = φp(F ) A(0) F (0) = In We write F (z) = In+ +∞ X k=1 Fkzk= +∞ X k=0 Fkzk with F0 = In and A(z) = +∞ X k=0 Akzk. From (4), we have ∀k ∈ N, p - k, k P j=0 AjFk−j = 0 ∀k ∈ N, p | k, k P j=0 AjFk−j = Fk/pA0 .
Thus, this formal solution is defined by F0 = In ∀k ∈ N, p - k, Fk= −A−10 k P j=1 AjFk−j ! ∀i ∈ N?, F pi= A−10 FiA0− A−10 pi P j=1 AjFpi−j ! .
Now, we have to prove the convergence of the formal solution. We write fk := ||Fk||, ak:= ||Ak|| and b := ||A−10 ||. We introduce the sequence fn
n≥0 defined by f0 = f0 ∀n ∈ N?, f n= b n−1 P i=0 (an−i+ a0) fi .
By induction, we can show that for all k ∈ N, 0 ≤ fk≤ fk. We have S(z) := +∞ X j=0 fjzj = f0+ b +∞ X j=1 j−1 X i=0 (aj−i+ a0) fizj = f0+ b +∞ X i=0 +∞ X j=i+1 (aj−i+ a0) fizj. This implies S(z) = f0+ bS(z) +∞ X j=1 ajzj + a0 +∞ X j=1 zj . Therefore, S(z) = f0 1 − b +∞ P j=1 ajzj + a0 +∞ P j=1 zj
! and S is the expression as a power serie
around the center z = 0 of this holomorphic function. Thus, +∞
P k=0
fkzkhas a non zero radius of convergence.
It remains to prove that if A ∈ GLn(C(z)) then F ∈ GLn(M (D(0, 1))). From the equation (3), for z ∈ D(0, 1), we have
F (z) = A−1(z) · · · A−1zpkFzpk+1Ak+10 .
This is a product of matrices with meromorphic function entries for a large enough k so
the entries of F are meromorphic on D(0, 1).
Corollary 2.7. If the system (2) is regular singular at 0 and A ∈ GLn(C(z)) then there exist F0 ∈ GLn(M (D(0, 1))) and A0∈ GLn(C) such that
φp(F0)−1AF0= A0.
We highlight that we have F0 ∈ GLn(M (D(0, 1))) and, generally, the unit circle is a natural boundary because, from a result of Bernard Rand´e in [Ran92], a solution of the Mahler equation (1) is rational or it has the unit circle as natural boundary.
Remark 2.8. If the system (2) is regular singular at 0, we can reduce the study of its solutions to the study of the solutions of a Mahler system with constant entries. Indeed, Z is a solution of φp(Y ) = A0Y if and only if F0Z is a solution of φp(Y ) = AY .
Remark 2.9. In the category E(0), the system (2) is regular singular at 0 if and only if it is isomorphic to a Mahler system with constant entries.
2.2. Neighbourhood of the point ∞ We consider the Mahler system
(5) φp(Y ) = AY
Definition 2.10. The system (5) is Fuchsian at ∞ if A is regular at ∞.
Definition 2.11. We say that the system (5) is meromorphically equivalent at ∞ to the system
φp(Y ) = BY with B ∈ GLn C 1z if there exists T ∈ GLn C
1
z such that B = φp(T )−1AT.
Remark 2.12. Two systems of the form (5) are meromorphically equivalent at ∞ if and only if they are isomorphic in the category E(∞).
Definition 2.13. The system (5) is regular singular at ∞ if it is meromorphically equiv-alent at ∞ to a Fuchsian one.
Using the change of variables z → 1/z, Theorem 2.6gives the following result.
Theorem 2.14. Assume that the system (5) is Fuchsian at ∞. There exists a unique F ∈ GLn C1z such that F (∞) = In and
(6) φp(F )−1AF = A(∞).
We have F ∈ GLn C1z .
Moreover, if we assume that A ∈ GLn(C(z)) then F ∈ GLn M P1(C) \ D(0, 1). Corollary 2.15. If the system (5) is regular singular at ∞ and A ∈ GLn(C(z)) then there exist F∞∈ GLn M P1(C) \ D(0, 1) and A∞∈ GLn(C) such that
φp(F∞)−1AF∞= A∞.
Remark 2.16. In the category E(∞), the system (5) is regular singular at ∞ if and only if it is isomorphic to a Mahler system with constant entries.
2.3. Neighbourhood of the point 1
We shall now study the regular singular Mahler equations at the point 1. We consider the Mahler system
(7) φp(Y ) = AY
with A ∈ GLn(C ({z − 1})) .
Definition 2.17. The system (7) is Fuchsian at 1 if A is regular at 1.
Definition 2.18. We say that the system (7) is meromorphically equivalent at 1 to the system
φp(Y ) = BY
with B ∈ GLn(C ({z − 1}) if there exists T ∈ GLn(C ({z − 1}) such that B = φp(T )−1AT.
Remark 2.19. Two systems of the form (7) are meromorphically equivalent at 1 if and only if they are isomorphic in the category E(1).
Definition 2.20. The system (7) is regular singular at 1 if it is meromorphically equivalent at 1 to a Fuchsian one.
In order to study the system (7), we transform it into the q-difference system
(8) X(pu) = B(u)X(u) with B(u) = A (exp(u))
with the change of variables z = exp(u). Locally, to transform the q-difference system (8) into the Mahler system (7), we can use the principal value Log because it is biholo-morphic between a neighbourhood of 1 and a neighbourhood of 0. However, for global considerations, to go back to our initial system, we use the following logarithm which is holomorphic on the universal cover fC?:= reib, b | r > 0, b ∈ R of C?:
f
log : Cf? → C
reib, b
7→ log(r) + ib . It is moreover a biholomorphic function.
Notation 2.21. We define
π := exp ◦ flog
and if W is a matrix with meromorphic entries on V ⊂ C, for all ez such that π (z) ∈ V ,e π?W (z) := W (π (e z)) .e
With an abuse of notation, we will also denote by φp the function φp : Cf? → Cf? reib, b 7→ reib, bp where reib, bp := rpeipb, pb. We notice that p flog = flog ◦ φp. We need the following lemma, proved in [Sau99, I.1.3.1].
Lemma 2.22. Let B ∈ GLn(M (C)) be regular at 0. There exist G ∈ GLn(M (C)) and C0 ∈ GLn(C) such that
G(pu)−1B(u)G(u) = C0.
Theorem 2.23. Assume that the system (7) is Fuchsian at 1 and A ∈ GLn(C(z)). There exist fF1 ∈ GLn
MCf?
and C0 ∈ GLn(C) such that
(9) φp f F1 −1 (π?A) fF1= C0.
Proof. We transform the system (7) into the q-difference system (8). From the previous
lemma, we can take fF1 := G ◦ flog.
Corollary 2.24. If the system (7) is regular singular at 1 and A ∈ GLn(C(z)) then there exist fF1 ∈ GLn
MCf?
and A1∈ GLn(C) such that φp f F1 −1 (π?A) fF1 = A1.
Remark 2.25. In the category E(1), the system (7) is regular singular at 1 if and only if it is isomorphic to a Mahler system with constant entries.
3. Meromorphic functions on (a part of) the universal cover of C? One can identify meromorphic functions on C? with meromorphic functions on fC? via
i : M (C?) ,→ M f C? f 7→ f ◦ π . We write π?M (C?) := {f ◦ π | f ∈ M (C?)} the image of i, it consists of the g ∈ MCf?
which are 2π-invariant in b. This identification between M (C?) and π?M (C?) is compatible with z 7→ zp i.e. there is the following commutative diagram (10) M (C?) f (z)7→f (zp) // _ M (C?) _ MCf? f (ez)7→f (ze p) //M f C? . that is φp◦ π = π ◦ φp. We also write π?M (D(0, 1)) := {f ◦ π | f ∈ M (D(0, 1))} and π?M P1(C) \ D(0, 1) := {f ◦ π | f ∈ M P1 (C) \ D(0, 1)}. We can check that
Lemma 3.1. (1) If h ∈ π?M (D(0, 1)) then h ◦ φp ∈ π?M (D(0, 1)). (2) If h ∈ π?M P1(C) \ D(0, 1) then h ◦ φ p ∈ π?M P1(C) \ D(0, 1). Lemma 3.2. We have (11) MCf? ∩ π?M (D(0, 1)) = π?M (C) , (12) MCf? ∩ π?M P1(C) \ D(0, 1) = π? M P1(C) \ {0} and (13) MCf? ∩ π?M (D(0, 1)) ∩ π?M P1(C) \ D(0, 1) = π? M P1(C) = π?C(z). Proof. Let us prove (11). It is clear that
π?M (C) ⊂ MCf?
∩ π?M (D(0, 1)) . It remains to show the reciprocal inclusion. Let f ∈ MCf?
∩π?M (D(0, 1)). There exists g ∈ M (D(0, 1)) such that f = π?g = g ◦ π. As f = g ◦ π = g ◦ exp ◦ flog ∈ MCf?
and f
is a surjective holomorphic function, we deduce that g ∈ M (C?). Therefore, f = g ◦ π with
g ∈ M (D(0, 1)) ∩ M (C?) = M (C) , this proves the equality (11).
Similarly, we can prove the equality (12). The equality (13) is a consequence of (11), (12) and the fact that
M P1(C) = C(z).
Part II. The categories of Fuchsian systems and regular singular systems at
0, 1 and ∞
4. The category Ers of regular singular systems at 0, 1 and ∞
We denote by Ers the category of regular singular Mahler systems at 0, 1 and ∞. It is the full subcategory of E whose objects are the
(C(z)n, A)
where A ∈ GLn(C(z)) is such that the system φp(Y ) = AY is regular singular at 0, 1 and ∞. The integer n is called the rank of the object.
In what follows, to simplify the notations, we will omit C(z)nin the definition of objects of Ers.
5. The category Ef of Fuchsian systems at 0, 1 and ∞
We denote by Ef the category of Fuchsian Mahler systems at 0, 1 and ∞. It is the full subcategory of Ers whose objects are the matrices A ∈ GLn(C(z)) which are regular at 0, 1 and ∞.
We will prove that the categories Ers and Ef are equivalent.
6. The categories Ers and Ef are equivalent
Notation 6.1. We denote by Di,u the identity matrix whose entry in column i and row i is replaced by u. We denote by Ti,l where l = (l1, . . . , ln) the identity matrix whose row i is replaced by l: Di,u= 1 . .. 1 u 1 . .. 1 et Ti,l= 1 . .. 1 l1 l2 · · · li · · · ln−1 ln 1 . .. 1 .
We notice that the inverse matrices of these matrices have the same form. From [Sau99, Corollaire 1, p.53], we have the following result.
Lemma 6.2. If M0 ∈ GLn(C ({z})) then M0 can be written M0= C(0)R(0)
• C(0)∈ GL
n(C(z)) which can be written
C(0) = u−kTi1,l1Di1,v1· · · Tir,lrDir,vr
where u, v1, . . . , vr ∈ C ({z}) have a valuation at 0 which is equal to 1, k ∈ N and l1, . . . , lr ∈ Cn\ {0}.
• R(0)∈ GL
n(C ({z})) regular at 0.
Lemma 6.3. If M1 ∈ GLn(C ({z − 1})) then M1 can be written M1= C(1)R(1)
with
• C(1)∈ GL
n(C(z)) regular at 0 and ∞ which can be written C(1) = u−kTi1,l1Di1,u· · · Tir,lrDir,u where k ∈ N, l1, . . . , lr ∈ Cn\ {0} and u = z−1z+1. • R(1)∈ GL
n(C ({z − 1})) regular at 1.
Proof. To prove this, we adapt the proof of [Sau99, Corollaire 1, p.53]. Let k ∈ N be such that the entries of ukM1 := M10 are analytic at 1. Let v1(det (M10)) be the valuation at 1 of det (M10). If v1(det (M10)) = 0 then M10 is regular at 1 so we can take R(1) := M10 and C(1) := u−kIn. If v1(det (M10)) > 0, we do an induction on v1(det (M10)) : we show that there exists a matrix N10 whose entries are analytic at 1 and such that M10 = Ti1,l1Di1,uN
0 1 with
v1 det N10 < v1 det M10 . If v1(det (M10)) > 0 then M10(1) is not invertible so there exists
l = (l1, . . . , ln) ∈ Cn\ {0}
such that lM10(1) = 0 ∈ Cn. Let i ∈ [|1, n|] be such that li 6= 0. Therefore, the entries of N10 := D−1i,uTi,lM10 are analytic at 1 and v1(det (N10)) = v1(det (M10)) − 1.
From [Sau99, Corollaire 3, p.54], we have the following result.
Lemma 6.4. Let C(0), C(∞)∈ GLn(C(z)). There exists U ∈ GLn(C(z)) such that U C(0) is regular at 0 and U C(∞) is regular at ∞.
Proposition 6.5. If the system φp(Y ) = AY is regular singular at 0, 1 and ∞ then it is rationally equivalent to a Fuchsian system at 0, 1 and ∞ that is there exists R ∈ GLn(C(z)) such that the system
φp(Y ) = BY with B := φp(R)−1AR is Fuchsian at 0, 1 and ∞.
Proof. From Corollaries 2.7,2.15 and Corollary 2.24 (using here the principal value Log instead of flog), there exist F0 ∈ GLn(C ({z})) (respectively F1 ∈ GLn(C ({z − 1})), F∞ ∈ GLn C 1z ) and A0 ∈ GLn(C) (respectively A1, A∞ ∈ GLn(C)) such that for i = 0, 1, ∞,
φp(Fi)−1AFi= Ai.
If we show that there exists R ∈ GLn(C(z)) such that for i = 0, 1, ∞, Ki := R−1Fi is regular at i then
will be regular at 0, 1 and ∞. From Lemma 6.2, F0 can be written C(0)R(0) with C(0) ∈ GLn(C(z)) and R(0) ∈ GLn(C ({z})) regular at 0. Similarly, with the change of variables z 7→ 1/z, we can write F∞ = C(∞)R(∞) with C(∞) ∈ GLn(C(z)) and R(∞) ∈ GLn C 1z regular at ∞. From Lemma 6.4, there exists U ∈ GLn(C(z)) such that U C(0) and U C(∞) are respectively regular at 0 and ∞. The entries of the ma-trix U F1 are meromorphic at 1. From Lemma 6.3, U F1 can be written C(1)R(1) with C(1) ∈ GLn(C(z)) regular at 0, ∞ and R(1) regular at 1. The matrix R := U−1C(1)
satisfies the required properties.
Proposition 6.6. The categories Ers and Ef are equivalent.
Proof. The natural embedding Ef → Ers is an equivalence of categories. Indeed, it is clearly fully faithful and, by Proposition6.5, it is also essentially surjective. Part III. Local categories and local groupoids
7. Preliminary results 7.1. On morphisms between constant systems
Lemma 7.1. Let p ∈ N≥2, A0 ∈ GLn1(C) and B0 ∈ GLn2(C). If T ∈ Mn2,n1(C ((z))) is such that φp(T ) A0 = B0T then T ∈ Mn2,n1(C).
Proof. We write T (z) = P k≥N
Tkzk with Tk ∈ Mn2,n1(C) and N = min{k ∈ Z | Tk 6= 0}. Thereby, we have (14) X k≥N TkA0zpk = X k≥N B0Tkzk.
We obtain that N = 0 because, looking at the lower degree, pN = N namely N = 0. Therefore, by the equality (14), all the Tk such that p does not divide k are equal to zero. Also, if p divides k then B0Tk= Tk/pA0 and we iterate it until we obtain an integer
k pj ∈ N which is not divisible by p so Tk= B
−j
0 Tk/pjAj0 = 0. Hence T (z) = T0 ∈ Mn2,n1(C). We deduce the following result.
Lemma 7.2. Let p ∈ N≥2, A∞∈ GLn1(C) and B∞∈ GLn2(C). If T ∈ Mn2,n1 C
1 z
is such that φp(T ) A∞= B∞T then T ∈ Mn2,n1(C).
Lemma 7.3. Let p ∈ N≥2, A1 ∈ GLn1(C), B1 ∈ GLn2(C) and fS1 be a matrix of size n2 × n1 with meromorphic entries at e1 := (1, 0) ∈ fC?. If φp
f S1
A1 = B1fS1 then the entries of fS1 are Laurent polynomials in flog.
Proof. Let S1 := fS1◦ flog −1
. Then,
∀z ∈ C, S1(pz)A1 = B1S1(z).
Since the entries of S1are meromorphic functions at 0, by [Sau03, 2.1.3.2], the entries of S1 are Laurent polynomials. Therefore, the entries of fS1 = S1◦ flog are Laurent polynomials
in flog.
Notation 7.4. We denote by Chlog, ff log
−1i
7.2. Finite dimensional complex representations of Z
Details and proofs can be found in [Sau99, Annexe A]. Let R be the category of finite dimensional complex representations of Z. A representation ρ of Z is entirely determined by A = ρ(1) ∈ GLn(C). Therefore, the objects of R can be seen as pairs (Cn, A) where A ∈ GLn(C) and the morphisms F ∈ HomR((Cn, A) , (Cp, B)) are the F ∈ Mp,n(C) such that F A = BF . It is a neutral Tannakian category over C with the forgetful functor ω : (Cn, A); Cn, F ; F as fibre functor. Its Galois group is
Zalg := Aut⊗(ω) .
Notation 7.5. Let A ∈ GLn(C) and A = AsAu its multiplicative Dunford decomposition. Let (γ, λ) ∈ Homgr(C∗, C∗) × C. We write A(γ,λ):= γ (As) Aλu= Aλuγ (As) where
• γ acts on the eigenvalues of As: if As= Q.diag(c1, . . . , cn).Q−1 then γ (As) := Q.diag (γ(c1), . . . , γ(cn)) .Q−1. • Aλ u = P k>0 λ k (Au− In) k .
The matrix A(γ,λ)∈ GLn(C) defines an automorphism of Cn= ω (Cn, A). Proposition 7.6. There is a pro-algebraic group isomorphism
Homgr(C?, C?) × C → Zalg (γ, λ) 7→ (Cn, A) 7→ A(γ,λ) 8. Localization at 0
8.1. The categories Ers(0) and P(0)
The category Ers(0) has the same objects as Ers but morphisms from A, of rank n1, to B, of rank n2, are all R ∈ Mn2,n1(C ({z})) such that φp(R) A = BR. From this equation satisfied by R, we deduce that
R ∈ Mn2,n1(M (D(0, 1))) .
We denote by P(0) the full subcategory of Ers(0) whose objects are the A0 ∈ GLn(C). By Lemma 7.1, we deduce that:
Proposition 8.1. If R ∈ HomP(0)(A0, B0) with A0 and B0 respectively of rank n1, n2 then R ∈ Mn2,n1(C).
It is clear that:
Lemma 8.2. The categories R and P(0) are equivalent. The following is an immediate consequence:
Proposition 8.3. The category P(0) is a neutral Tannakian category over C and the functor
ωl,0: P(0)→ VectfC (
A0 ; Cn with n the rank of A0 R; R
is a fibre functor for P(0).
Proof. By definition, this functor is full and faithful. It is also essentially surjective
ac-cording to Corollary2.7.
Corollary 8.5. The category Ers(0) is a Tannakian category over C. 8.2. Local Galois groupoid at 0
In the previous section, we have seen that the categories R, P(0) and Ers(0) are equivalent Tannakian categories over C. Therefore, they have the same Galois group which is Zalg (see7.2).
As a consequence, the local Galois group at 0 is G0= Zalg.
The local Galois groupoid at 0 is a groupoid with one object ωl,0 and such that the mor-phisms from ωl,0 to ωl,0 are
G0(ωl,0, ωl,0) := Zalg. We will also denote it by G0.
9. Localization at ∞ 9.1. The categories Ers(∞) and P(∞)
Similarly, we introduce the categories Ers(∞) and P(∞). The category Ers(∞) has the same objects as Ers but morphisms from A, of rank n1, to B, of rank n2, are the matrices R ∈ Mn2,n1 C
1
z such that φp(R) A = BR. From this equation satisfied by R, we deduce that
R ∈ Mn2,n1 M P
1
(C) \ D(0, 1) .
We denote by P(∞) the full subcategory of Ers(∞) whose objects are the A∞∈ GLn(C). By Lemma 7.2, we deduce that:
Proposition 9.1. If R ∈ HomP(∞)(A∞, B∞) with A∞ and B∞ respectively of rank n1 and n2 then R ∈ Mn2,n1(C).
As before with the localization at 0, the categories R, P(∞) and Ers(∞) are equivalent. These are Tannakian categories over C. In particular, P(∞)is a neutral Tannakian category over C with the fibre functor
ωl,∞: P(∞)→ VectfC (
A∞; Cn with n the rank of A∞ R; R
.
9.2. Local Galois groupoid at ∞
As previouly, localizing at 0, the categories E(∞), P(∞) and R have the same Galois group Zalg.
As a consequence, the local Galois group at ∞ is G∞= Zalg.
The local Galois groupoid at ∞ is a groupoid with one object ωl,∞ and such that the morphisms from ωl,∞ to ωl,∞ are
G∞(ωl,∞, ωl,∞) = Zalg. We will also denote it by G∞.
10. Localization at 1 10.1. The categories Ers(1) and P(1)
Notation 10.1. We write ∆ := reit ∈ C | r > 0, t ∈ −π p, π p and Λ = z ∈ C | =(z) ∈ −π p, π p . We introduce π1 : C \ R−× ]−π, π[ → C \ R− reit, t 7→ reit .
The category Ers(1) has the same objects as Ers but morphisms from A of rank n1 to B of rank n2 are all R ∈ Mn2,n1(C ({z − 1})) such that φp(R) A = BR. From this equation satisfied by R, we deduce that R ∈ Mn2,n1(M (∆)) .
We denote by P(1) the category whose objects are all A1 ∈ GLn(C) and morphisms from A1 of rank n1 to B1 of rank n2 are all matrices eR of size n2× n1 whose entries are meromorphic functions at e1 ∈ fC? and such that
φp
e
RA1= B1R.e By Lemma 7.3, we deduce that:
Proposition 10.2. If R ∈ HomP(1)(A1, B1) with A1 and B1 respectively of rank n1 and n2 then the entries of R are Laurent polynomials in flog.
Proposition 10.3. The categories E(1) and P(1) are equivalent via the functor G : P(1)→ E(1) rs ( A1 ; A1 e R;R ◦ πe 1−1 .
Proof. This functor is full and faithful. It is also essentially surjective according to
Corol-lary2.24.
Lemma 10.4. The category P(1) and the category P(0) introduced in [Sau03, 2.1.2] are isomorphic via the composition with flog:
P(0)→ P(1) (
A1 ; A1 H ; H ◦ flog
.
According to [Sau03], the categoryP(0) is a neutral Tannakian category over C so: Proposition 10.5. The category P(1) is a neutral Tannakian category over C with the fiber functors,
ωl,1(ec): P(1) → Vectf C (
A1 ; Cn with n the rank of A1 e
R;R (e ec)
for allec ∈ fC
?\ {e1}.
10.2. Local Galois groupoid at 1
From Lemma 10.4, we obtain that Aut⊗ω(el,1c) is a subgroup (and Iso⊗ωl,1(ec), ωl,1( ed) a subset) of Zalg. We also have the following result.
Theorem 10.6. With the identification of Zalg with Homgr(C?, C?) given in Proposition 7.6, for all ec, ed ∈ fC?\ {e1}, Iso⊗ω(el,1c), ωl,1( ed)=n(γ, λ) ∈ Zalg γ(p) flog (ec) = flog e d o.
Notation 10.7. We denote by G1 the local Galois groupoid at 1. Its objects are the ωl,1(ec),ec ∈ fC?\ {e1}
and its morphisms are the G1 ω(ec) l,1, ω ( ed) l,1 = Iso⊗ω(ec) l,1, ω ( ed) l,1 .
It is a transitive groupoid because there exists a morphism between each object. Therefore, we can take as local Galois group at 1 a group G1
ω1,l(ec), ω1,l(ec)for a ec ∈ fC?\ {e1}.
Corollary 10.8. With the same identification, for ec ∈ fC?\e1, the local Galois group at 1 is G1:= G1 ω(ec) l,1, ω (ec) l,1 = n (γ, λ) ∈ Zalg| γ(p) = 1o. Therefore, G1 ω(ec) l,1, ω (ec) l,1 ' HomgrC?/pZ, C?× C := G 1,s× G1,u.
The local Galois group at 1 obtained is the same group as the one obtained by Jacques Sauloy in [Sau03, 2.2.2.2]. Therefore, from [Sau03, 2.2.3], we know a Zariski-dense sub-group of this sub-group:
Lemma 10.9 (elements of the unipotent component G1,u of G1). Let A1 ∈ Obj P(1). Let A1 = A1,uA1,s be its Dunford decomposition. The element 1 ∈ C which corresponds to A1 ; A1,u ∈ Aut⊗
ω(ec)
l,1
generates a Zariski-dense subgroup of the unipotent component G1,u.
Notation 10.10. Let p ∈ N≥2. We recall that all z ∈ C? can be uniquely written z = upx where u belongs to the unit circle and x ∈ R. We define
γ1: C? → C? upx 7→ u and
γ2: C? → C? upx 7→ e2iπx .
Lemma 10.11 (elements of the semi-simple component G1,s of G1). We denote by γ1, γ2 ∈ Homgr
C?/pZ, C?
the morphisms induced by γ1 and γ2 respectively. They generate a Zariski-dense subgroup of G1,s.
Theorem 10.12. The subgroup of Homgr C?/pZ, C? × C whose unipotent component is Z ⊂ C and whose semi-simple component is generated by γ1 and γ2 is Zariski-dense in G1, the local Galois group at 1.
Part IV. Global categories
11. The category Ers of regular singular systems at 0, 1 and ∞
We will show that Ers is a Tannakian category over C. From 6, it will also provide Ef a structure of Tannakian category over C.
Proposition 11.1. The category Ers is a Tannakian category over C.
Proof. From1.2.b, we know that E is a Tannakian category over C. The category Ers is a rigid tensor subcategory of E because it is closed under tensor operations, it contains the unit object 1 = 1 ∈ C?, the internal Hom and every object has a dual. To show that it is a Tannakian subcategory of E , the only non trivial point which remains to be proved is the existence of kernels and cokernels in Ers. This follows from the lemma herebelow and the fact that duality in the Tannakian category E exchanges kernels with cokernels. Lemma 11.2. All morphisms in the category Ers have kernels.
Proof. We recall that the categories E (introduced in1.2.b), E(0)(introduced in1.2.b) and Ers(0) (introduced in 8.1) are Tannakian categories. We consider the following embeddings of categories Ers _ // E _ Ers(0) //E(0) .
Let R ∈ Mn2,n1(C(z)) be a morphism in the category Ers from A ∈ GLn1(C(z)) to B ∈ GLn2(C(z)). We know that the morphism R has a kernel in the category E, we denote it by
(15) (C, K : C → A) .
To show that this kernel is also a kernel of R in the category Ers it remains to prove that the system φp(Y ) = CY is regular singular at 0, 1 and ∞. The morphism R has also a kernel in the category Ers(0), we denote it by
(16) (C0, K0: C0 → A)
so the system φp(Y ) = C0Y is regular singular at 0. The kernels (15) and (16) are kernels of the morphism R in the category E(0) so they are isomorphic that is there exists an invertible morphism u1: C0 → C in the category E(0) such that Ku1= K. Because of the fact that u1 ∈ MorE(0)(C0, C), the entries of u1 are meromorphic functions at 0 and u1 satisfies
φp(u1) C0 = Cu1
thus the system φp(Y ) = CY is also regular singular at 0. Similarly, we can show that the system φp(Y ) = CY is regular singular at 1 and at ∞ so C ∈ Obj (Ers) and (C, K : C → A)
is also a kernel of R in the category Ers.
12. The category C of connections We define
Σ0 := n
and Σ∞:= n reib, b | r > 1, b ∈ Ro⊂ fC?. We introduce the category C of connections. Its objects are the
(A0, A1, A∞, fM0, gM∞) ∈ GLn(C)3× GLn(M (Σ0)) × GLn(M (Σ∞)) such that φp f M0 = A1Mf0A−10 φp g M∞ = A1Mg∞A−1∞ and such that there exist
f W1 ∈ GLn MCf? , W0 ∈ GLn(M (D(0, 1))), W∞∈ GLn M P1(C) \ D(0, 1) such that ( f W1Mf0 = π?W0 f W1Mg∞= π?W∞ . The integer n is called the rank of the object.
Morphisms from (A0, A1, A∞, fM0, gM∞) ∈ Obj (C) to (B0, B1, B∞, fN0, gN∞) ∈ Obj (C) respectively of rank n1 and n2 are the triples
S0, fS1, S∞ ∈ Mn2,n1(C) × Mn2,n1Chlog, ff log −1i × Mn2,n1(C) such that S0A0 = B0S0 φp f S1 A1 = B1Sf1 S∞A∞= B∞S∞ f S1Mf0= fN0S0 f S1Mg∞= gN∞S∞ .
Remark 12.1. From Lemmas7.1,7.2and 7.3, this amounts to assuming that the entries of S0, fS1 and S∞ are meromorphic functions at 0, e1 and ∞ respectively.
13. The category S of solutions The objects of the category S of solutions are the
(A0, A1, A∞, F0, fF1, F∞) ∈ GLn(C)3× GLn(M (D(0, 1))) × GLn MCf? × GLn M P1(C) \ D(0, 1) . such that π?A := φp(fF1)A1Ff1 −1 , A = φp(F0)A0F0−1, and A = φp(F∞)A∞F∞−1. The integer n is called the rank of the object.
Morphisms from (A0, A1, A∞, F0, fF1, F∞) to (B0, B1, B∞, G0, fG1, G∞), two objects of S respectively of rank n1 and n2, are the
(R, S0, fS1, S∞) ∈ Mn2,n1(C(z)) × Mn2,n1(C) × Mn2,n1 C h f log, flog−1i× Mn2,n1(C)
such that S0A0= B0S0 φp f S1 A1 = B1fS1 S∞A∞= B∞S∞ RFi = GiSi pour i = 0, ∞ e RfF1 = fG1Sf1 where we define eR := π?R.
14. Equivalence between the categories Our goal is to obtain an equivalence between the categories C and Ers. Proposition 14.1. The categories S and Ers are equivalent by the functor
F : S → Ers
((A0, A1, A∞, F0, fF1, F∞); A defined in the category S (in 13)
R, S0, fS1, S∞ ; R
.
Proof. We check thatF is well-defined on objects. By the equalities A = φp(F0) A0F0−1, A = φp(F∞) A∞F∞−1 and π?A = φp f F1 A1Ff1 −1
, we obtain respectively that the entries of π?A belong to π?M (D(0, 1)), π?M P1(C) \ D(0, 1) and M f C? . By Lemma 3.2, π?A ∈ GLn(π?C(z)) so A ∈ GLn(C(z)) .
Moreover, the system φp(Y ) = AY is regular singular at 0, 1, ∞. The functor F is also well-defined on morphisms because R ∈ Mn2,n1(C(z)) and it satisfies
φp(R)A = φp(R)φp(F0) | {z } =φp(G0)S0 A0F0−1 = φp(G0)B0S0F0−1 | {z } =G−10 R = BR.
The functor F is essentially surjective because of Corollaries 2.7, 2.15, 2.24. It is also fully faithful. Indeed, keeping the notations introduced in 13, if R ∈ HomErs(A, B), R, G−10 RF0, fG1 −1 e RfF1, G−1∞RF∞
is the unique morphism of S such that F (R, S0, fS1, S∞) = R
because it is uniquely determined and it satisfies • for i = 0, ∞, φp(Si) Ai= φp(Gi)−1φp(R) φp(Fi) Ai = BiG−1i B −1φ p(R) A | {z } =R Fi = BiSi
where we used the fact that φp(Gi) = BGiB−1i and φp(Fi) = AFiA−1i for the second equality. By Lemmas 7.1and 7.2, we obtain that Si has constant entries. • Similarly, φp
f S1
A1 = B1fS1 and by Lemma 7.3, the entries of fS1 are Laurent polynomials in flog.
Proposition 14.2. The categories S and C are equivalent by the functor G : S → C A0, A1, A∞, F0, fF1, F∞ ; A0, A1, A∞, fF1 −1 π?F0, fF1 −1 π?F∞ R, S0, fS1, S∞ ; S0, fS1, S∞ .
Proof. We write eFi := π?Fi for i = 0, ∞ and eR := π?R. The functor G is clearly well-defined on objects. It is also well-well-defined on morphisms because if
(R, S0, fS1, S∞) ∈ MorS (A0, A1, A∞, F0, fF1, F∞), (B0, B1, B∞, G0, fG1, G∞) , for i = 0, ∞, introducing fMi = fF1 −1 e Fi and fNi= fG1 −1 f Gi, f S1Mfi = fS1Ff1 −1 | {z } = fG1 −1 e R e Fi = fNiSi.
The functorG is an equivalence of categories because it is:
• essentially surjective. Here, each object (A0, A1, A∞, fM0, gM∞) of C is of the form G(A0, A1, A∞, F0, fF1, F∞). Indeed, we set fF1 := fW1 and F0 := W0, F∞ := W∞ (notations introduced in the category of connections, 12).
• fully faithful. Let (S0, fS1, S∞) ∈ HomC
(A0, A1, A∞, fM0, gM∞), (B0, B1, B∞, fN0, gN∞)
. There exists exactly one morphism (R, S0, fS1, S∞) ∈ Mor (S) such that
G (R, S0, fS1, S∞) = (S0, fS1, S∞). It is clear because R = GiSiFi−1 for i = 0, ∞ and
π?R = fGiSiFei −1
= fG1Sf1Ff1 −1
. Therefore, the entries of π?R belong to
MCf?
∩ π?M (D(0, 1)) ∩ π?M P1(C) \ D(0, 1) .
By Lemma 3.2, the entries of π?R belons to π?C(z) that is the entries of R are rational functions.
From the two previous propositions, we have the following result.
Corollary 14.3. The categories C and Ers are equivalent.
15. Structure of neutral Tannakian category over C
As is the case with Ers in 4 (and Ef in 5), we equip the categories C and S with a tensor product on objects and morphisms. It is defined componentwise using the tensor product on matrices defined in1.1.c. The equivalences defined in the previous section are compatible with these tensor structures. Since Ers is a rigid abelian tensor category (see 11), the categories C and S are also rigid abelian tensor categories.
Proposition 15.1. For allea ∈ fC
?\ {e1}, we consider the functors ω0, ω∞, ω(ea)
1 : C → Vect f C defined by:
• if X ∈ Obj (C) is of rank n then for i = 0, ∞,
ωi(X ) = Cn and ω1(ea)(X ) = Cn, • if S0, fS1, S∞
∈ Mor (C) then for i = 0, ∞, ωi S0, fS1, S∞ = Si and ω1(ea) S0, fS1, S∞ = fS1(ea).
These are fibre functors and the category of connections C is a neutral Tannakian category over C.
Proof. The only non immediate point is the faithfulness of the functors ω(ea)
1 ,ea ∈ fC
?\ {e1}.
To prove it, we assume that fS1(ea) = 0 and we want to show that fS1 = 0. From the equality φp f S1 = B1fS1A−11 it follows that fS1 ea pk
= 0 for all k ∈ Z. Sinceea ∈ fC
?\ {e1},
the entries of fS1 are Laurent polynomials in flog with an infinity of roots, thus they are
equal to 0.
We can also show that S is a Tannakian category over C.
Part V. Mahler analogue of the density theorem of Schlesinger 16. The Galois groupoid of the category C
Definition 16.1. The Galois groupoid of the category of connections C, denoted by G, is such that:
• the objects are the fiber functors ω0, ω∞, ω1(ea) : C → VectfC defined in Proposition 15.1,
• the morphisms between ω, ω0 ∈ {ω
0} ∪ {ω∞} ∪ n
ω1(ea)|ea ∈ fC?\ {e1} o
are the iso-morphisms of tensor functors from ω to ω0:
G ω, ω0 = Iso⊗ ω, ω0 .
Proposition 16.2. The local Galois groupoids G0, G1 and G∞ can be identified with subgroupoids of the Galois groupoid G of the category C.
Proof. Let us prove that G0 can be identified with a subgroupoid of G. We can see that ω0= ωl,0P with P : C → P(0) A0, A1, A∞, fM0, gM∞ ∈ Obj (C) ; A0 S0, fS1, S∞ ; S0
the functor “projection”. The restriction of the elements of Aut⊗(ωl,0) to C gives a mor-phism of proalgebraic groups
P? : Aut⊗
(ωl,0) → Aut⊗(ω0)
We also denote by G0, G1, G∞ the subgroupoids of G which are identified with G0, G1, G∞ by the previous proposition.
17. Construction of isomorphisms of tensor functors Let
A0, A1, A∞, fM0, gM∞
∈ Obj (C). In order to construct elements of the Galois groupoid of C, we need to evaluate the matrices fM0 and gM∞. It is why we introduce the following definition.
Definition 17.1. The singular locus of a matrix M is
S(M ) := {poles of M } ∪ {poles of M−1} = {poles of M } ∪ {zeros of det M }. 17.1. The category CE0,E∞
Notation 17.2. Let E0 (respectively E∞) be a finite subset of D(0, 1) \ {0} (respec-tively C \ D(0, 1)). We consider the full subcategory CE0,E∞ of C whose objects are the
A0, A1, A∞, fM0, gM∞
∈ Obj (C) such that for i = 0, ∞, S f Mi ⊂ fEi:= EipZ, arg EipZ + 2πZ
where for z ∈ C∗, arg(z) denotes an argument of the complex number z and for k ∈ Z<0, Eipk = {z ∈ C | zp−k∈ Ei}.
Lemma 17.3. Every morphism in the category CE0,E∞ has a kernel. Proof. Let S0, fS1, S∞ be a morphism from X := A0, A1, A∞, fM0, gM∞ of rank n1 to Y := B0, B1, B∞, fN0, gN∞
of rank n2 in the category CE0,E∞. By the equivalence between the categories Ers and C (see Corollary14.3) and Lemma11.2, there exist
X0 := A00, A01, A0∞, fM0 0 , gM∞ 0 ∈ Obj (C) of rank n1− r with r = rk(S0) = rk(fS1) = rk(S∞) and
K0, fK1, K∞ ∈ Mor (C) such that (17) X0,K0, fK1, K∞ : X0 → X is a kernel of the morphism
S0, fS1, S∞
in the category C with rk(K0) = rk f K1 = rk(K∞) = n1− r.
It remains to show that X0 is an object of CE0,E∞. Let us study the singular locus of fM0
0 . Given thatK0, fK1, K∞ ∈ HomC(X0, X ), we have fK1Mf0 0 = fM0K0 that is (18) K0Mf0 0−1 = fM0 −1 f K1. There exists a permutation matrix T ∈ GLn1(C) such that
T K0= L1
L2
with L1∈ GLn1−r(C) and L2 ∈ Mr,n1−r(C) because rk(K0) = n1− r. The equality (18) implies
f M0 0−1 = L−11 0 TMf0 −1 f K1 so (19) polesMf0 0−1 ⊂ fE0. In order to finish the study of the singular locus of fM0 0
, we show that
(20) zerosdetMf0
0−1
⊂ fE0. Let U ∈ GLn1(C) be a permutation matrix such that
U fK1= f L1 f L2 ! with fL1∈ GLn1−r C h f
log, flog−1i and L2 ∈ Mr,n1−r
C h
f
log, flog−1i. By contradiction, assume that there exists
e a ∈ zero det f M0 0−1 ∩Σ0\ fE0 . From (19), we know that ea is not a pole of fM0
0−1
so we can consider fM0 0−1
(ea) and there exists v ∈ Cn1−r\ {0} such that
f M0
0−1
(ea) v = 0 ∈ Cn1−r. The equality (18) implies
0 = K0Mf0 0−1 (ea) v = fM0 −1 (ea) U −1 Lf1(ea) f L2(ea) ! v so f L1(ea) f L2(ea) ! v = 0 because fM0 −1 (ea) U
−1 is an invertible matrix. In particular, e a ∈ zero det f L1 . From the equality φp f K1 A01= A1Kf1, we have f L1(ea p) f L2(ea p) ! A01v = 0 soeap ∈ zerodet f L1
. By induction, it follows that for all k ∈ N, ea pk ∈ zerodetLf1 . Asea 6= e1, detLf1
has an infinity of roots, it is a Laurent polynomial in flog so it is equal to 0. This contradicts the fact that fL1 is an invertible matrix.
Therefore, from (19) and (20), S f M0 0 ⊂ fE0 and, similarly, we can show that
SMg∞
0
⊂ gE∞.
The category CE0,E∞ is a subcategory of C which is stable by tensor and abelian con-structions, by the lemma hereabove it becomes clear that:
Theorem 17.4. The category CE0,E∞ is a neutral Tannakian category over C.
In what follows, we keep the same notations for the restrictions to CE0,E∞ of the fibre functors ω0, ω∞, ω(1ea),ea ∈ fC ?\ {e1} defined in Proposition15.1. Proposition 17.5. If X = A0, A1, A∞, fM0, gM∞
∈ Obj(C) then there exists a category CE0,E∞ such that X ∈ Obj (CE0,E∞).
Proof. Using the notations of the category C, in12, the matrix
(21) A := φp(W0) A0W0−1 satisfies (22) A = φp(W∞) A∞W∞−1 and (23) φp f W1 = π?A. fW1.A−11 .
Therefore, from Lemma 3.2, A ∈ GLn(C(z)). From the equality fM0 = fW1 −1 π?W0, we have SMf0 ⊂SWf1 ∪ S (π?W 0) ∩ Σ0. From the equality (23), we have
f W1 = φ−1p (π?A) · · · φ−np (π?A) φ−np f W1 A−n1 . For a n ∈ N big enough and using that fW1∈ GLn
MCf? we obtain S f W1 ∩ Σ0 ⊂ ∆pZ, arg ∆pZ + 2πZ
with ∆ := S(A) ∩ (D(0, 1) \ {0}). Similarly, from the equality (21) we can show that S (π?W0) ⊂ ∆pZ, arg∆pZ+ 2πZ. Therefore, S f M0 ⊂∆pZ, arg ∆pZ + 2πZ
and ∆ is by construction a finite subset of D(0, 1) \ {0}. The singular locus of gM∞ satisfy
17.2. Construction of isomorphisms of tensor functors Proposition 17.6. Letea ∈ Σ0\ fE0. The natural transformation
Γ0,ea: A0, A1, A∞, fM0, gM∞ ∈ Obj (CE0,E∞);Mf0(ea) is an element of Iso⊗ω0, ω(e1a) . Proof. To prove that Γ0,ea ∈ Iso
⊗ω
0, ω(e1a)
, we have to verify the two following assump-tions:
• Condition to be an isomorphism of functors. Let X :=A0, A1, A∞, fM0, gM∞
and Y :=B0, B1, B∞, fN0, gN∞
be two objects of CE0,E∞. The following diagram is commutative because in the category of connections fN0S0 = fS1Mf0: ω0 A0, A1, A∞, fM0, gM∞ S 0 // g M0(ea) ω0 B0, B1, B∞, fN0, gN∞ f N0(ea) ω(ea) 1 A0, A1, A∞, fM0, gM∞ f S1(ea) //ω(ea) 1 B0, B1, B∞, fN0, gN∞
This shows that it is a morphism of functors. Moreover, it is an isomorphism because the fM0(ea) are isomorphisms.
• Condition of tensor compatibility. For all X , Y ∈ Obj (CE0,E∞), Γ0,ea(X ) ⊗ Γ0,ea(Y) = fM0(ea) ⊗ fN0(ea) = f M0⊗ fN0 (ea) = Γ0,ea(X ⊗ Y) . Similarly, we have that:
Proposition 17.7. Let eb ∈ Σ∞\ gE∞. The natural transformation Γ0,eb : A0, A1, A∞, fM0, gM∞ ∈ Obj (CE0,E∞);Mg∞(eb) is an element of Iso⊗ ω∞, ω( e b) 1 . We also have that:
Proposition 17.8. The natural transformation
A0, A1, A∞, fM0, gM∞
∈ Obj (CE0,E∞); A0 is an element of Aut⊗(ω0) .
The natural transformation
A0, A1, A∞, fM0, gM∞
∈ Obj (CE0,E∞); A∞ is an element of Aut⊗(ω∞).
Let ea ∈ fC?\
f E0∪ gE∞
. The natural transformation A0, A1, A∞, fM0, gM∞ ∈ Obj (CE0,E∞); A1 is an element of Iso⊗ω(ea) 1 , ω (ea p) 1 .
We will keep the same notations for the restriction of G to the Galois groupoid of CE0,E∞.
Corollary 17.9. The local Galois groupoids G0, G1, G∞, the Γ0,ae, ea ∈ Σ0\ fE0 (defined in Proposition17.6) and the Γ0,eb, eb ∈ Σ∞\ gE∞ (defined in Proposition17.7) are elements of the Galois groupoid G of CE0,E∞.
In the next section, we will prove that they generate a Zariski-dense subgroupoid of the Galois groupoid G.
18. Analogue of the density theorem of Schlesinger for Mahler equations 18.1. Density criteria
We use the following results (see [Sau99, II.3.2.1]):
Theorem 18.1. Let C be a Tannakian category and ω a fibre functor for C. Let hX i be the Tannakian subcategory generated by the object X of C. Then the morphism
Aut⊗ ω|hX i
→ Aut (ω (X )) g 7→ g (X )
is injective. It identifies the Galois group G (X ) = Aut⊗ ω|hX i of X with an algebraic subgroup of Aut (ω (X )) ' GLn(C) (n is the rank of X namely the dimension of the C-vector space ω (X )).
The following result is a density criterion, we keep the notations introduced in the previous theorem:
Theorem 18.2. Let H be a subgroup of G = Aut⊗(ω). If X is an object of C, we denote by H (X ) the image of H in G (X ).
We assume that for all objects Y of C, for all y ∈ ω (Y), if the line Cy is stable by H (Y) then it is also stable by G (Y).
Then, for all objects X of C, H (X ) is Zariski-dense in G (X ).
Herebelow, the analogue of the previous density criterion for groupoids:
Theorem 18.3. We consider a transitive groupoid G whose objects are fiber functors ωi for the Tannakian category C and whose morphisms from the object ωi to the object ωj are the elements of G (ωi, ωj) := Iso⊗(ωi, ωj). Let H be a subgroupoid of G which is transitive. We assume that for all objects Y of C, for all yi ∈ ωi(Y), if the lines Cyi are globally stable by H (Y) then they are also globally stable by G (Y).
18.2. Density theorem in the regular singular case
We consider the following regular singular system at 0, 1 and ∞ φp(Y ) = AY
with A(z) ∈ GLn(C(z)).
We recall that we restrict to the full subcategory CE0,E∞ of C.
Theorem 18.4. The local Galois groupoids G0, G1, G∞ and the Galoisian isomorphisms Γ0,ze0, Γ∞,zf∞ for all ze0 ∈ Σ0\ fE0, zf∞∈ Σ∞\ gE∞ generate a Zariski-dense subgroupoid H of the Galois groupoid G of CE0,E∞.
Proof. To prove this theorem, we use the density criterion given in Theorem18.3. Let X =A0, A1, A∞, fM0, gM∞
be an object of CE0,E∞. For each object ω0, ω
(ec) 1 , ω∞,ec ∈ fC ?\Ef 0∪ gE∞∪ {e1} of G, we choose a line Di ⊂ ωi(X ) = Cn, i = 0, ∞ and fD1
(ec)
⊂ ω(ec)
1 (X ) = Cn. We assume that this family of lines is globally stable by H. We have to show that this family of lines is also globally stable by G.
We denote by ρ : G → GL (ω (X )) the representation of G corresponding to X by Tannaka duality.
The restriction ρ|G1 : G1 → GL (ω (X )) is the representation corresponding to A1 by Tannaka duality for the category P(1). The corestriction ρ
|GL f D1 (c)e |G1 is a subrepresentation of rank 1 of ρ|G1. Therefore, by Tannaka duality, it comes from a subobject of rank 1 of A1 in P(1). This subobject is an object a1of P(1) of rank 1 and a monomorphism ed1 : a1 → A1 of P(1). In short, a1∈ C? and ed1is a column matrix whose entries are Laurent polynomials in flog such that
(24) φp e d1 a1 = A1de1 and ω(e1c)de1 C = ed1(ec) C = fD1 (ec) .
The restriction ρ|G0 : G0→ GL (ω (X )) is a representation which corresponds to A0 by Tannaka duality for the category P(0). The corestriction ρ|GL|G (Df0)
0 is a subrepresentation of rank 1 of ρ|G0. Therefore, it comes from a subobject of rank 1 of A0 in P
(0) namely an object a0 of P(0) of rank 1 and a monomorphism d0 : a0 → A0 of P(0). In short, a0 ∈ C? and d0 ∈ Cn is such that
(25) d0a0 = A0d0
and
Cd0 = D0.
Similarly, for the category P(∞) and the groupoid G∞, we have that
(26) d∞a∞= A∞d∞
with a∞∈ C?, d∞∈ Cn and
For allea ∈ Σ0\ fE0, the stability under Γ0,ea of the family of lines means that f
M0(ea) D0 = fD1
(ea)
.
Consequently, for all ea ∈ Σ0 \ fE0, there exists mf0(ea) ∈ C such that fM0(ea) d0 = f
m0(ea) ed1(ea). Therefore, we obtain a functionmf0 : Σ0→ C such that
(27) Mf0d0 =mf0de1.
Similarly, for all eb ∈ Σ∞\ gE∞, D∞ = gM∞
eb −1
D1(eb) and there exists mg∞ : Σ∞ → C such that
(28) Mg∞d∞=mg∞de1.
Thus, we have: • X0 := (a
0, a1, a∞,mf0,mg∞) is an object of CE0,E∞. Indeed, the equality (27) implies that mf0 is a meromorphic function on Σ0. From the equality
φp f M0 = A1Mf0A−10 , given in 12, we have φp f M0 d0= A1Mf0A−10 d0 and the equalities (24), (25) imply
φp(mf0) = a1mf0a −1
0 .
The same argument shows thatmg∞is a meromorphic function on Σ∞and satisfies φp(mg∞) = a1mg∞a
−1
∞.
Moreover, the singular locus are such that S (mfi) ⊂ fEi, i = 0, ∞ (for more details see the following remark).
In order to prove that X0 is an object of CE0,E∞, it remains to prove that there exists a non-zerowf1 ∈ M f C? , w0 ∈ M (D(0, 1)) and w∞∈ M P1(C) \ D(0, 1) such that f w1mf0 = π ?w 0 f w1mg∞= π ?w ∞ AsA0, A1, A∞, fM0, gM∞
is an object of CE0,E∞, there exists f W1 ∈ GLn MCf? , W0 ∈ GLn(M (D(0, 1))), W∞∈ GLn M P1(C) \ D(0, 1)
such that fW1Mf0= π?W0 and fW1Mg∞= π?W∞. Therefore, (
π?W0d0 = fW1Mf0d0 =mf0Wf1de1 on Σ0, π?W∞d∞= fW1Mg∞d∞=mg∞Wf1de1 on Σ∞.
A non-zero entry of the column matrix fW1de1 satisfies the conditions to be f w1.
• m = d0, ed1, d∞
: X0 ; X is a morphism of CE0,E∞. This is due to the fact that d0, d∞∈ Cn, the entries of ed1 are Laurent polynomials in flog and
( e d1mf0 = fM0d0 e d1mg∞= gM∞d∞ .
The family of lines, fixed by the subgroupoid H, are globally stable by the groupoid G because:
• for all gi∈ Aut⊗ ωi|hX i, i = 0, ∞, the following diagram is commutative ωi(X0) gi(X0)// ωi(m)=di ωi(X0) ωi(m)=di ωi(X ) gi(X ) //ωi(X ) so, gi(X ) di = digi X0 | {z } ∈C? = gi(X0) di and gi(X ) Di = Di.
• for all g1∈ Iso⊗ ω(ce1) 1 |hX i, ω (ce2) 1 |hX i
, the following diagram is commutative ω(ce1) 1 (X 0)g1(X0)// ω(c1)f 1 (m)=fd1(ce1) ω(ce2) 1 (X 0) ω(fc2) 1 (m)=fd1(ce2) ω(ce1) 1 (X ) g 1(X ) //ω(ce2) 1 (X ) so, g1(X ) fD1 (ce1) = fD1 (ce2) . • similarly, for all h0∈ Iso⊗ ω0|hX i, ω1(ea)|hX i and h∞∈ Iso⊗ ω∞|hX i, ω( eb) 1 |hX i , we can prove that h0(X ) D0 = fD1
(ea)
and h∞(X ) D∞= fD1 (eb)
.
Remark 18.5. Let us look at the singular locus ofmf0. The equality (27),
f m0−1d0 = fM0 −1 e d1, implies that polesmf0−1 ⊂polesMf0 −1 ∪ polesde1 ∩ Σ0⊂ fE0. Also, zerosmf0−1 ⊂ fE0 because if α ∈ zerose f m0−1 ∩ Σ0\ fE0 then fM0(α)e −1 is well-defined and invertible but 0 = fM0(α)e
−1 e
d1(α) and ee d1(α) 6= 0. This implies thate VectCde1(
e
α) = D(α)e
1 = 0, which is absurd. Therefore, S (mf0) ⊂ fE0. Similarly,
