A Bostan

S Boukraa||

J-M Maillard

J-A Weil

for his 75th birthday. R J Baxter

Diagonals of rational functions and selected differential Galois groups

Keywords: 05.50.+q, 05.10.-a, 02.30.Hq, 02.30.Gp, 02.40.Xx AMS Classification scheme numbers: 34M55, 47E05, 81Qxx, 32G34, 34Lxx, 34Mxx, 14Kxx Diagonals of rational functions, differential Galois groups, Hadamard products, series with integer coefficients, globally bounded series, differential equations

We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics.

In all the examples emerging from physics, the minimal linear differential operators annihilating these diagonals of rational functions have been shown to actually possess orthogonal or symplectic differential Galois groups. In order to understand the emergence of such orthogonal or symplectic groups, we analyze exhaustively three sets of diagonals of rational functions, corresponding respectively to rational functions of three variables, four variables and six variables. We impose the constraints that the degree of the denominators in each variable is at most one, and the coefficients of the monomials are 0 or ±1, so that the analysis can be exhaustive. We find the minimal linear differential operators annihilating the diagonals of these rational functions of three, four, five and six variables. We find that, even for these sets of examples which, at first sight, have no relation with physics, their differential Galois groups are always orthogonal or symplectic groups. We discuss the conditions on the rational functions such that the operators annihilating their diagonals do not correspond to orthogonal or symplectic differential Galois groups, but rather to generic special linear groups.

Introduction

In previous papers [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF] it has been shown that the n-fold integrals χ (n) corresponding to the n-particle contributions of the magnetic susceptibility of the Ising model [START_REF] Zenine | Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ (3) and χ[END_REF][START_REF] Bostan | High order Fuchsian equations for the square lattice Ising model: χ(5)[END_REF][START_REF] Boukraa | High-order Fuchsian equations for the square lattice Ising model: χ (6)[END_REF], as well as various other n-fold integrals of the "Ising class" [START_REF] Boukraa | Singularities of n-fold integrals of the Ising class and the theory of elliptic curves[END_REF][START_REF] Assis | Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations[END_REF], or n-fold integrals from enumerative combinatorics, like lattice Green functions [START_REF] Guttmann | Lattice Green's functions in all dimensions[END_REF][START_REF] Glasser | Lattice Green function (at 0) for the 4D hypercubic lattice[END_REF][START_REF] Boukraa | Differential algebra on lattice Green functions and Calabi-Yau operators[END_REF][START_REF] Zenine | Lattice Green Functions: the seven-dimensional face-centred cubic lattice[END_REF], correspond to a distinguished class of functions generalizing algebraic functions: they are actually diagonals of rational functions [START_REF] Lipshitz | Rational functions, diagonals, automata and arithmetic Number theory[END_REF][START_REF] Lipshitz | The diagonal of a D-finite power series is D-finite[END_REF][START_REF] Christol | Diagonales de fractions rationnelles et équations différentielles Study group on ultrametric analysis[END_REF][START_REF] Christol | Diagonales de fractions rationnelles et équations de Picard-Fuchs Study group on ultrametric analysis[END_REF][START_REF] Christol | Diagonales de fractions rationnelles Séminaire de Théorie des Nombres[END_REF]. As a consequence, the power series expansions of the analytic solutions at x = 0 of these linear differential equations are "Derived From Geometry" [START_REF] Bostan | Globally nilpotent differential operators and the square Ising model[END_REF] and are globally bounded [START_REF] Christol | Globally bounded solutions of differential equations, Analytic number theory[END_REF], which means that, after a rescaling of the expansion variable, they can be cast into series expansions with integer coefficients [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF].

In another paper [START_REF] Boukraa | Differential algebra on lattice Green functions and Calabi-Yau operators[END_REF] revisiting miscellaneous linear differential operators mostly associated with lattice Green functions in arbitrary dimensions, but also Calabi-Yau operators and order-seven linear differential operators corresponding to exceptional differential Galois groups, we showed that these irreducible operators are not only globally nilpotent [START_REF] Bostan | Globally nilpotent differential operators and the square Ising model[END_REF], but are also homomorphic to their (formal) adjoints. Considering these linear differential operators, or, sometimes, equivalent operators †, we showed that, either their symmetric square or their exterior square, have a rational solution [START_REF] Boukraa | Differential algebra on lattice Green functions and Calabi-Yau operators[END_REF]. This is a general result: an irreducible♯ linear differential operator homomorphic to its (formal) adjoint is necessarily such that either its symmetric square, or its exterior square has a rational solution, and this situation corresponds to the occurrence of a special differential Galois group [START_REF] Boukraa | Differential algebra on lattice Green functions and Calabi-Yau operators[END_REF]. We thus defined the notion of being "Special Geometry" for a linear differential operator if it is irreducible, globally nilpotent, and such that it is homomorphic to its (formal) adjoint [START_REF] Boukraa | The Ising model and Special Geometries[END_REF]. Since many "Derived From Geometry" [START_REF] Bostan | Globally nilpotent differential operators and the square Ising model[END_REF] n-fold integrals ("Periods") occurring in physics are seen to be diagonals of rational functions [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF], we address several examples of (minimal order) operators annihilating diagonals of rational functions, and remark that they also seem to be, almost systematically, associated with irreducible factors homomorphic to their adjoints [START_REF] Boukraa | Differential algebra on lattice Green functions and Calabi-Yau operators[END_REF].

Finally in a last paper [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF] revisiting an order-six linear differential operator, already introduced in [START_REF] Boukraa | Differential algebra on lattice Green functions and Calabi-Yau operators[END_REF], having a solution which is the diagonal of a rational function of three variables, we saw that the corresponding linear differential operator is such that its exterior square has a rational solution, indicating that it has a selected differential Galois group, and that it is actually homomorphic to its adjoint. We obtained the two corresponding intertwiners giving this homomorphism to the adjoint relation. We showed that these intertwiners are also homomorphic to their adjoints [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF] and have a simple decomposition, already underlined in a previous paper [START_REF] Boukraa | Differential algebra on lattice Green functions and Calabi-Yau operators[END_REF], in terms of order-two self-adjoint operators. From these results, we deduced a new form of decomposition of operators for this selected order-six linear differential operator in terms of three order-two self-adjoint operators. We generalized this decomposition to decompositions in terms of arbitrary self-adjoint operators of arbitrary orders, provided the orders have the same parity [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF].

This natural emergence in physics of n-fold integrals that are diagonals of rational functions, such that their associated linear differential operators correspond to selected differential Galois groups, SO(n, C) or Sp(n, C) (or subgroups, like exceptional groups [START_REF] Boukraa | Differential algebra on lattice Green functions and Calabi-Yau operators[END_REF]), was illustrated on important problems of lattice statistical mechanics like the n-fold integrals χ (5) and χ (6) of the square Ising model [START_REF] Boukraa | The Ising model and Special Geometries[END_REF], or non-trivial lattice Green functions examples [START_REF] Zenine | Lattice Green Functions: the seven-dimensional face-centred cubic lattice[END_REF].

The occurrence of diagonals of rational functions necessarily yields linear † In the sense of the equivalence of linear differential operators, see [START_REF] Van Der Put | Galois Theory of Linear Differential Equations[END_REF]. ♯ Minimal order operators annihilating diagonals of rational functions may be irreducible, in general. In that case, all their irreducible factors have the above property.

3 F 2 ([1/3, 1/3, 1/3], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 3 6 x) which is the Hadamard product [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF][START_REF] Hadamard | Essai sur l'étude des fonctions données par leur développement de Taylor[END_REF][START_REF] Cameron | Analytic continuation of diagonals and Hadamard compositions of multiple power series Trans[END_REF] of three times the simple algebraic function (1 -3 2 x) -1/3 (Hadamard cube), and is, thus, the diagonal of a rational function [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF], the corresponding order-three linear differential operator having a SL(3, C) differential Galois group. In other words this operator, associated with the diagonal of a rational function, cannot be homomorphic to its adjoint (even with an algebraic extension). One can find many other similar counterexamples of diagonals of a rational function with a SL(3, C) differential Galois group. For instance, 3 F 2 ([1/3, 1/3, 1/5], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 3 4 5 2 x) which is the Hadamard product of two times (1 -3 2 x) -1/3 with (1 -5 2 x) -1/5 , or 3 F 2 ([1/2, 1/2, 1/3], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 2 4 3 2 x) which is the Hadamard product of two times the algebraic function (1 -2 2 x) -1/2 with (1 -3 2 x) -1/3 , are two such examples.

The property, for a linear differential operator, to be homomorphic to its adjoint, can be seen to be "some kind" † † of Poincaré duality [START_REF] Griffiths | Principles of algebraic geometry[END_REF][START_REF] Sabbah | Hodge theory, singularities and D-modules[END_REF]. Actually, considering a diagonal of a rational function, amounts to considering an algebraic variety. If this algebraic variety is "smooth enough" (not too singular), then one will have a Poincaré duality †, which, in some abstract D-module perspective [START_REF] Coutinho | A primer of algebraic D-modules[END_REF], should correspond to the previous "homomorphism to the adjoint" property. A simpler occurence of this phenomenon is shown by Bogner in proposition 3.4, page 5 of [START_REF] Bogner | Algebraic characterization of differential operators of Calabi-Yau type[END_REF]. In simpler words, the emergence of a selected differential Galois group should be natural for the diagonals of a rational function, when the rational function is not "too singular".

For diagonals of rational functions that we know to occur naturally in theoretical physics ‡, understanding the emergence of such a duality yielding selected differential Galois groups, at least in a physicist's perspective ¶ requires the analysis of other well-defined sets of diagonals of rational functions, along the line of the previously mentioned exhaustive analysis [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF] of the 210 explicit linear differential operators arising from mirror symmetries. This paper will provide such an analysis, introducing well-defined sets of diagonals of rational functions of three, four and six variables, showing that all these examples yield selected differential Galois groups, namely orthogonal or symplectic groups. We will first analyze a large set of rational functions of three variables, namely rational functions with denominators with degree bounded by 1. It turns out that all these diagonals of rational functions of three variables correspond to modular forms, so that all can, in principle, be written as 2 F 1 hypergeometric functions with two pullbacks [START_REF] Bostan | The Ising model: from elliptic curves to modular forms and Calabi-Yau equations[END_REF].

We will finally provide miscellaneous examples that are worth keeping in mind when one tries to understand in which case selected differential Galois groups do not occur for diagonals of rational functions, so that one could see if such "exceptional" cases can also occur in a physics framework. † † To be more explicit would require to write a mathematical paper, calling out a Deligne-Steenbrink-Zucker theorem saying that Gauss-Manin connections are "variations of polarized mixed Hodge structures", the associated graded modules being Gauss-Manin connections of smooth projective varieties, and that they are self-adjoint by Poincaré duality. This is far beyond the scope of this "learn-by-example" paper (see for instance [START_REF] Sabbah | Hodge theory, singularities and D-modules[END_REF]). † The Poincaré duality theorem is a result on the structure of the homology and cohomology groups of manifolds (see chapter 4, page 53 of [START_REF] Griffiths | Principles of algebraic geometry[END_REF]). ‡ And we know why, see [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF]. ¶ Who wants to understand why orthogonal, symplectic or exceptional groups occur in his problems.

Diagonals of rational functions: preliminary comments

Let us recall the definition of the diagonal of a rational function of n variables R(x 1 , . . . , x n ) = P(x 1 , . . . , x n )/Q(x 1 , . . . , x n ) where P and Q are polynomials of x 1 , • • • , x n with rational coefficients such that Q(0, . . . , 0) = 0. The diagonal of R is defined from its multi-Taylor expansion

R x 1 , x 2 , . . . , x n = ∞ m1 = 0 • • • ∞ mn = 0 R m1, ..., mn • x m1 1 • • • x mn n , (1)
as the series of one variable

Diag R x 1 , x 2 , . . . , x n = ∞ m = 0 R m, m, ..., m • x m . (2)
In the generic case, denoting d the degree of the denominator of a rational function of n variables, and Ω the order of the minimal order linear differential operator annihilating the diagonal of this rational function, the order Ω grows with n and d like ≃ d n (see e.g. [START_REF] Lairez | Périodes d'intégrales rationnelles[END_REF]):

Ω = 1 d • (d -1) n+1 + (-1) n+1 • (d -1) < d n . (3)
From a theory of singularity perspective [START_REF] Hauser | The Hironaka theorem on the resolution of singularities (or; A proof we always wanted to understand)[END_REF], one can imagine that the analysis of such diagonals of rational functions depends "essentially" on the denominator of the rational function. For that reason we will restrict most of the examples of this paper to rational functions with a numerator normalised to 1, so that the number of cases to analyze will be reasonable. Note, however, that, even in a theory of singularity perspective, this restriction to numerators equal to a constant, is not innocent, as will be seen below (see the Remark in section (7.1)). We impose this restriction, for the simplicity of the calculations, and in order to perform exhaustive analysis of certain sets of examples. Furthermore, in order to be able to find a linear differential operator for the diagonal, using reasonable computer resources, we will, for a given number of variables n, restrict to denominators of the lowest possible degree, imposing, for instance, that the degree, in each variable x i , of the monomials x d1 1 x d2 2 • • • x dn n of the denominator of the rational function is at most 1. Formula (3) gives the upper bound for the order of the linear differential operator, which is actually reached when the polynomial at the denominator, has all its monomials (no monomial has a zero coefficient). In practice, in the denominator of the rational function examples emerging from physics, the polynomial at the denominator is (fortunately) sparse, being the sum of a quite small set of monomials. In such (physical) cases, the order of the corresponding linear differential operator is less than the one given by (3) which grows like ≃ d n and thus becomes quickly too large for any formal calculation.

Imposing a constraint on the degree in each variable, instead of a constraint on the degree of the denominator

d = d 1 + d 2 + • • • + d n ,
reduces the number of monomials and, consequently, reduces quite drastically, the order of the corresponding linear differential operator.

Furthermore, it is clear (by definition of the diagonal of a function) that scaling the variables x i → λ i • x i , amounts to performing a simple scaling on the diagonal:

∆(x) → ∆(λ • x) where λ = λ 1 • λ 2 • • • λ n .
Consequently we will, in this paper, often restrict the coefficients of the monomials to a narrow set of small integer values, namely 0, 1, or 0, ±1, in order to have a reasonably small number of cases to analyze to avoid any explosion of the combinatorics. It is, of course, clear that the coefficients of rational functions cannot, in general, be reduced to 0, ±1.

Diagonals of rational functions of two variables necessarily yield algebraic functions [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF][START_REF] Furstenberg | Algebraic functions over finite fields[END_REF][START_REF] Denef | Algebraic power series and diagonals[END_REF][START_REF] Adamczewski | Diagonalization and Rationalization of algebraic Laurent series[END_REF][START_REF] Pólya | Sur les séries entières, dont la somme est une fonction algébrique Enseignement Math[END_REF]. Therefore, in the next sections, we will consider diagonal of rational functions, of more than two variables.

Diagonals of rational functions of three variables

Let us, first, consider diagonals of rational functions of the form 1/(1 -P (x, y, z)) where P (x, y, z) is a polynomial of three variables x, y, z, sum of monomials x m y n z p where the degrees m, n, p are 0 or 1, and where the coefficients in front of these monomials are restricted to take two values 0, 1. We will say that two rational functions of the form 1/(1 -P ) are in the same class if they have the same diagonal and hence the same linear differential operator annihilating this diagonal. With these two constraints (on the degrees and values of the coefficients), one finds only 20 classes of rational functions (or diagonals). For all these cases, the linear differential operator is of order two, the diagonal being a 2 F 1 hypergeometric function, or a HeunG function [START_REF] Bostan | The Ising model: from elliptic curves to modular forms and Calabi-Yau equations[END_REF] that can be rewritten as a 2 F 1 hypergeometric function with two possible pullbacks, which means that this is, in fact, a modular form [START_REF] Bostan | The Ising model: from elliptic curves to modular forms and Calabi-Yau equations[END_REF].

With the same constraint on the degrees m, n, p to be 0 or 1, restricting the coefficients in front of these monomials to take three values 0, ±1, one obtains 85 classes of linear differential operators.

Let us just give a few examples, the exhaustive list of results being given in a web page of supplementary material [START_REF] Weil | Diagonals of rational functions and selected differential Galois groups[END_REF].

• For the polynomial P (x, y, z) = x + y + z -x z -y z + x y z, the diagonal of 1/(1 -P (x, y, z)), which corresponds to the sequence [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Zenine | Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ (3) and χ[END_REF][START_REF] Lipshitz | The diagonal of a D-finite power series is D-finite[END_REF]63,321, 1683, • • • ] of Central Delannoy numbers (see Sloane's on-line encyclopedia [START_REF] Sloane | The on-line encyclopedia of integer sequences[END_REF] of integer sequences: A001850), is a simple algebraic function (1 -6 x + x 2 ) -1/4 , annihilated by an orderone operator.

• For the polynomial P (x, y, z) = x + y + z -x z -x y z, the diagonal of 1/(1 -P (x, y, z)) is the pullbacked 2 F 1 hypergeometric function

(1 -12 x + 14 x 2 + 12 x 3 + x 4 ) -1/4 • 2 F 1 [ 1 12 , 5 12
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], P 1 (x)

where:

P 1 (x) = 1728 x 5 • (1 -11 x -x 2 ) (1 -12 x + 14 x 2 + 12 x 3 + x 4 ) 3 . (4)
This Hauptmodul [35] P 1 (x) corresponds to the Hauptmodul 12 3 /j ′ 5 (see Table 5 in [START_REF] Maier | On rationally parametrized modular equations[END_REF])

P 1 (x) =
12 3 • z 5 (z 2 + 250 z + 3125) 3 with:

z = 5 3 • x 1 -11 x -x 2 .
(

) 5
This diagonal corresponds to the sequence [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Zenine | Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ (3) and χ[END_REF][START_REF] Van Der Put | Galois Theory of Linear Differential Equations[END_REF]147,1251, 11253, • • •] of Apery numbers (see Sloane's on-line encyclopedia [START_REF] Sloane | The on-line encyclopedia of integer sequences[END_REF] of integer sequences: A005258). The pullbacked 2 F 1 hypergeometric function (4) can also be written with another pullback

(1 + 228 x + 494 x 2 -228 x 3 + x 4 ) -1/4 • 2 F 1 [ 1 12 , 5 12
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], P 2 (x)

where:

P 2 (x) = 1728 x • (1 -11 x -x 2 ) 5 (1 + 228 x + 494 x 2 -228 x 3 + x 4 ) 3 . (6)
This Hauptmodul [START_REF] Bostan | The Ising model: from elliptic curves to modular forms and Calabi-Yau equations[END_REF] P 2 (x) corresponds to the Hauptmodul 12 3 /j 5 (see Table 4 in [START_REF] Maier | On rationally parametrized modular equations[END_REF]) 3 • z (z 2 + 10 z + 5) 3 with:

P 2 (x) = 12
z = 5 3 • x 1 -11 x -x 2 . ( 7
)
Changing P 1 (x) into P 2 (x) amounts to changing z ↔ 5 3 /z. These two pullbacks, Y = P 1 (x) and Z = P 2 (x), are related by an algebraic curve, namely the (genus zero) modular curve:

2 54 • 5 3 • 11 9 • Y 6 Z 6 + 110949 • 2 47 • 11 6 • 5 3 • Y 5 Z 5 • (Y + Z) + 3 • 2 36 • 11 3 • Y 4 Z 4 • (2735484611275 (Y 2 + Z 2 ) -107937074856652 Y Z) + 2 29 • 5 2 • Y 3 Z 3 • (Y + Z) • (4046657341273198 (Y 2 + Z 2 ) + 523793662474799327 • Y Z) + 3 • 2 18 • 5 • Y 2 Z 2 • (13 2 • 647451979 • (Y 4 + Z 4 ) + 5 3 • 3482348755357972227 • Y 2 Z 2 -16391442082714013450 • (Y Z 3 + Y 3 Z)) + 30720 • Y Z • (Y + Z) • (36983 • (Y 4 + Z 4 ) + 2421471845930417 • (Y Z 3 + Y 3 Z) + 506711612589929401008 Y 2 Z 2 ) + (Y 6 + Z 6 ) -246683410950 • (Y Z 5 + Y 5 Z) -441206965512914835246100 • Y 3 Z 3 + 383083609779811215375 • (Y 4 Z 2 + Z 4 Y 2 ) + 2 11 • 3 3 • 5 2 • Y Z • (Y + Z) • (2535689 • (Y 2 + Z 2 ) + 134848657695982 Y Z) -59719680 • Y Z • (227547 • (Y 2 + Z 2 ) -83299968230 • Y Z) + 2 21 • 3 10 • 5 • 31 • Y Z • (Y + Z) -2 24 • 3 12 • Y Z = 0. (8)
The pullbacked 2 F 1 hypergeometric function (4), or equivalently [START_REF] Boukraa | Singularities of n-fold integrals of the Ising class and the theory of elliptic curves[END_REF], is a modular form (see the J 5 of Maier [START_REF] Maier | On rationally parametrized modular equations[END_REF]).

• For polynomial P (x, y, z) = x + y + x z + y z + x y z, the diagonal of 1/(1 -P (x, y, z)) is the pullbacked 2 F 1 hypergeometric function

(1 -20 x + 54 x 2 -20 x 3 + x 4 ) -1/4 • 2 F 1 [ 1 12 , 5 12
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], P 1 (x)

where:

P 1 (x) = 1728 x 4 • (1 -18 x + x 2 ) (x -1) 2 (1 -20 x + 54 x 2 -20 x 3 + x 4 ) 3 . (9)
This Hauptmodul P 1 (x) corresponds to the Hauptmodul 12 3 /j ′ 2 (see Table 5 in [START_REF] Maier | On rationally parametrized modular equations[END_REF])

P 1 (x) = 12 3 • z 2 (z + 256) 3 with: z = 2 12 • x 2 (1 -x) 2 (1 -18 x + x 2 ) . (10)
This diagonal corresponds to the sequence [1, 5, 49, 605, 8281, 120125

• • •] (see Sloane's on-line encyclopedia of integer sequences: A243945).
Similarly, the pullbacked 2 F 1 hypergeometric function ( 9) can also be written

(1 -20 x + 294 x 2 -20 x 3 + x 4 ) -1/4 • 2 F 1 [ 1 12 , 5 12 ], [1], P 2 (x)
where:

P 2 (x) = 1728 x 2 • (1 -18 x + x 2 ) 2 • (1 -x) 4 (1 -20 x + 294 x 2 -20 x 3 + x 4 ) 3 . ( 11
)
This Hauptmodul P 2 (x) corresponds to the Hauptmodul 12 3 /j 2 (see Table 4 in [START_REF] Maier | On rationally parametrized modular equations[END_REF])

P 2 (x) = 12 3 • z (z + 16) 3 with: z = 2 12 • x 2 (1 -x) 2 (1 -18 x + x 2 ) . ( 12
)
Changing P 1 (x) into P 2 (x) amounts to changing z ↔ 2 12 /z. These two pullbacks, Y = P 1 (x) and Z = P 2 (x), are related by an algebraic curve, namely the (genus zero) modular curve:

5 9 Y 3 Z 3 -5 6 • 12 Y 2 Z 2 • (Y + Z) + 375 • Y Z • (16 Y 2 -4027 • Y Z + 16 Z 2 ) -64 • (Y + Z) • (Y 2 + 1487 Y Z + Z 2 ) + 2 12 • 3 3 • Y Z = 0. ( 13
)
The pullbacked 2 F 1 hypergeometric function [START_REF] Glasser | Lattice Green function (at 0) for the 4D hypercubic lattice[END_REF], or equivalently [START_REF] Zenine | Lattice Green Functions: the seven-dimensional face-centred cubic lattice[END_REF], is a modular form (see the J 2 of Maier [START_REF] Maier | On rationally parametrized modular equations[END_REF], or even J 4 of Maier [START_REF] Maier | On rationally parametrized modular equations[END_REF] but with z = 2 12

x • (1 - x) 2 /(1 -18 x + x 2 ) 2 ).
• For the polynomial P (x, y, z)

= xyz + xy + xz + yz + x + y + z, the diagonal of 1/(1 -P (x, y, z)) is the pullbacked ¶ 2 F 1 hypergeometric function 1 1 -x • 2 F 1 [ 1 3 , 2 3
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], P (x) where:

P (x) = 54 x (1 -x) 3 . (14)
This diagonal corresponds to the sequence [1, 13, 409, 16081, 699121, 32193253

• • • ]
(see Sloane's on-line encyclopedia of integer sequences: A126086). This series can also be written as the pullbacked 2 F 1 hypergeometric function

(1 -x) -1/4 • (1 + 429 x + 3 x 2 -x 3 ) -1/4 • 2 F 1 [ 1 12 , 5 12
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], P 1 (x)

where:

P 1 (x) = 3456 x • (1 -57 x + 3 x 2 -x 3 ) 3 (1 -x) 3 (1 + 429 x + 3 x 2 -x 3 ) 3 , (15)
or as the pullbacked 2 F 1 hypergeometric function

(1 -x) -1/4 • (1 -51 x + 3 x 2 -x 3 ) -1/4 • 2 F 1 [ 1 12 , 5 12 ], [1], P 2 (x)
where:

P 2 (x) = 13824 x 3 • (1 -57 x + 3 x 2 -x 3 ) (1 -x) 3 (1 -51 x + 3 x 2 -x 3 ) 3 . (16)
These two pullbacks correspond respectively to the Hauptmodul [35] 12 3 /j 3 of Table 4 in [START_REF] Maier | On rationally parametrized modular equations[END_REF] and 12 3 /j ′ 3 of Table 5 in [START_REF] Maier | On rationally parametrized modular equations[END_REF]:

P 1 (x) = 12 3 • z (z + 27) • (z + 3) 3 , P 2 (x) = 12 3 • z 3 (z + 27) • (z + 243) 3 with: z = 2 • 9 3 • x 1 -57 x + 3 x 2 -x 3 . (17)
Changing P 1 (x) into P 2 (x) amounts to changing z ↔ 3 6 /z. Again, these two pullbacks are related by a (genus zero) modular curve: 16) is a modular form (see the J 3 of Maier [START_REF] Maier | On rationally parametrized modular equations[END_REF]).

2 27 • 5 9 • Y 3 Z 3 • (Y + Z) + 2 18 • 5 6 • Y 2 Z 2 • (27 Y 2 -45946 Y Z + 27 Z 2 ) + 2 9 • 5 3 • 3 5 • Y Z • (Y + Z) • (Y 2 + 241433 Y Z + Z 2 ) + 729 • (Y 4 + Z 4 ) -779997924 • (Y Z 3 ) + Y 3 Z + 31949606 • 3 10 • Y 2 Z 2 + 2 9 • 3 11 • 31 • Y Z • (Y + Z) -2 12 • 3 12 • Y Z = 0. The pullbacked 2 F 1 hypergeometric function (
• For the polynomial P (x, y, z)

= xyz + xy + xz + yz, the diagonal of 1/(1 -P (x, y, z)) is the pullbacked 2 F 1 hypergeometric function 1 1 -x • 2 F 1 [ 1 3 , 2 3
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], P (x) where:

P (x) = 27 x 2 (1 -x) 3 .
(18) ¶ Note that this pullback can be obtained using the Maple program "hypergeomdeg3" of V. J. Kunwar and M. van Hoeij [START_REF] Kunwar | Second order differential equations with hypergeometric solutions of degree three[END_REF][START_REF] Kunwar | [END_REF].

This diagonal corresponds to the sequence [1, 1, 7, 25, 151, 751 • • • ] (see Sloane's on-line encyclopedia of integer sequences: A208425).

In Appendix A we give a set of diagonals of 1/(1 -P (x, y, z)) which can be seen to be modular forms: they can be written as 2 F 1 hypergeometric functions with two different pullbacks.

Most of the other examples of diagonals, given in a web page as well as in the supplementary material [START_REF] Weil | Diagonals of rational functions and selected differential Galois groups[END_REF], are not yet in Sloane's on-line encyclopedia of integer sequences. For instance the sequence [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | High order Fuchsian equations for the square lattice Ising model: χ(5)[END_REF][START_REF] Weil | Diagonals of rational functions and selected differential Galois groups[END_REF]520,7090, 102144, • • • ], which corresponds to the diagonal of the rational function 1/(1 -x -y -x y -x z -y z), can be written as a 2 F 1 hypergeometric function with a rational pullback:

1 1 -16 x -8 x 2 1/4 • 2 F 1 [ 1 12 , 5 12 ], [1], 1728 • x 4 • (1 + x) 2 (2 -34 x -27 x 2 ) (1 -16 x -8 x 2 ) 3 ,
All these results show a close relation between diagonals of our simple examples of rational functions of three variables, and modular forms. This suggests, quite naturally, to perform similar calculations, but, now, with four variables.

Diagonals of rational functions of four variables

Let us now consider diagonal of rational functions of the form 1/(1 -P (x, y, z, w)) where P (x, y, z, w) is a polynomial of four variables x, y, z, w, sum of monomials x m y n z p w q where the degrees m, n, p, q are 0 or 1, and where the coefficients in front of these monomials are restricted to take only two values 0, 1. With these two constraints (on the degrees and on the values of the coefficients), one finds an exhaustive list of only 879 cases ¶.

Remark 1: We have used the package "HolonomicFunctions" written by C. Koutschan [47,[START_REF] Koutschan | HolonomicFunctions (User's Guide)[END_REF]49], based on the method of creative telescoping [START_REF] Zeilberger | The Method of Creative Telescoping[END_REF][START_REF] Koutschan | A Fast Approach to Creative Telescoping[END_REF], which enables to obtain directly, and very efficiently, the linear differential operator annihilating the diagonal of a given rational function, without calculating the series expansion of the corresponding diagonal ‡. From time to time Koutschan's algorithms do not provide the minimal order linear differential operator. It is thus necessary to systematically check whether the linear differential operator obtained by creative telescoping is minimal, and if not, to find the minimal one.

Remark 2:

The command "FindCreativeTelescoping", described in [START_REF] Koutschan | A Fast Approach to Creative Telescoping[END_REF], is (usually) extremely fast, but it uses some heuristics, which means that sometimes it can return a non-minimal result or run forever: it is not an "algorithm" in the strict sense †. Among these 879 cases, the linear differential operators annihilating the diagonals have minimal orders running from 1 to 10, as given in Table 1. ¶ When different rational functions 1/(1 -P (x, y, z, w)) yield the same diagonal (in practice the same first ten coefficients of the series) we select one rational function to represent the diagonal. This is the way we define these 879 different classes of rational functions. ‡ One does not obtain the linear differential operator from a "guessing procedure" on the series of the diagonal. Furthermore this algorithm is "certified": we are sure that the operator annihilates the diagonal. † In contrast, Chyzak's algorithm [START_REF] Chyzak | The ABC of Creative Telescoping Algorithms, Bounds[END_REF] is designed such that it finds the minimal order operator, but it is often much slower than Koutschan's heuristics (in particular for the very large operators emerging in physics [START_REF] Bostan | High order Fuchsian equations for the square lattice Ising model: χ(5)[END_REF][START_REF] Boukraa | High-order Fuchsian equations for the square lattice Ising model: χ (6)[END_REF][START_REF] Boukraa | Experimental mathematics on the magnetic susceptibility of the square lattice Ising model[END_REF][START_REF] Nickel | Square lattice Ising model χ(5) ODE in exact arithmetic[END_REF]). In Koutschan's package, Chyzak's algorithm is implemented in the command CreativeTelescoping (same input/output specification as FindCreativeTelescoping). The order-one linear differential operator corresponds to the diagonal of the rational function 1/(1 -z -wy -xy). It is easily seen that its diagonal is equal to 1, so the order-one operator is just D x .

For the order-two linear differential operators, the corresponding diagonals are two pullbacked 2 F 1 hypergeometric functions, respectively

(1 -32 x + 16 x 2 ) -1/4 • 2 F 1 [ 1 12 , 5 12
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], P (x)

where:

P (x) = 1728 x 3 • (2 -71 x + 16 x 2 ) (1 -32 x + 16 x 2 ) 3 , (19)
for the diagonal of 1/(1 -(y + z + x z + x w + x y w)) which reads 1 + 8 x + 156 x2 + 3800 x3 + 102340 x 4 + 2919168 x 5 + 86427264 x 6

+ 2626557648 x 7 + 81380484900 x 8 + 2559296511200

x 9 + • • • (20)
and

(1 -40 x + 16 x 2 ) -1/4 • 2 F 1 [ 1 12 , 5 12
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF],

P (x) (21)
where:

P (x) = 6912 x 3 • (1 -44 x -16 x 2 ) (1 -40 x + 16 x 2 ) 3 .
for † the diagonal of 1/(1 -(y + z + x w + x z w + x y w)) which reads: 1 + 10 x + 246 x 2 + 7540 x 3 + 255430 x 4 + 9163980 x 5 + 341237820 x 6 + 13042646760 x 7 + 508236930630 x 8 + 20101587623260

x 9 + • • • (22)
Note, however, that the series (20) of the diagonal of 1/(1 -(y + z + x z + x w + x y w)) actually identifies with the diagonal of a rational function of just three variables 1/(1 -x -y -z -x y), already found among the previous 20 cases of section [START_REF] Zenine | Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ (3) and χ[END_REF]. Similarly the series [START_REF] Zenine | The Fuchsian differential equation of the square lattice Ising model χ (3) susceptibility[END_REF] of the diagonal of 1/(1 -(y + z + x w + x z w + x y w)) also identifies with the diagonal of a rational function of three variables 1/(1 -x -yz -x y -y z).

The results for all these 879 cases are given exhaustively in our web page of supplementary material [START_REF] Weil | Diagonals of rational functions and selected differential Galois groups[END_REF]. Let us summarize these results in the following.

For all the twenty cases, corresponding to order-three linear differential operators, we have SO(3, C) differential Galois groups. As a consequence [START_REF] Boukraa | Differential algebra on lattice Green functions and Calabi-Yau operators[END_REF], all these linear differential operators are actually symmetric squares of order-two operators, some with very simple 2 F 1 hypergeometric functions, namely 2 F 1 ([3/8, 1/8], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 256 x 3 ) or decomposition [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF] of the form (A 1 B 3 + 1) • r(x), where A 1 and B 3 are respectively order-one and order-three self adjoint operators, r(x) being a rational function.

Similarly, the 240 order-five linear differential operators are (non-trivially) homomorphic to their adjoints. They have SO(5, C) differential Galois groups and have a canonical decomposition [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF] of the form (A 1 B 1 C 3 +A 1 + C 3 )• r(x), where A 1 , B 1 and C 3 are respectively two order-one and one order-three self adjoint operators, r(x) being a rational function.

The 231 order-six linear differential operators are (non-trivially) homomorphic to their adjoints. They have SO(6, C) differential Galois groups and have a canonical decomposition [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF] of the form (A

1 B 1 C 1 D 3 + A 1 B 1 + A 1 D 3 + C 1 D 3 + 1) • r(x),
where the A 1 , B 1 , C 1 , operators are order-one self-adjoint operators, the rightmost operator D 3 being an order-three self-adjoint operator.

The following 155 order-seven linear differential operators are (non-trivially) homomorphic to their adjoints. They have SO(7, C) differential Galois groups and have a canonical decomposition [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF] of the form (A

1 B 1 C 1 D 1 E 3 + A 1 B 1 E 3 + A 1 D 1 E 3 + A 1 B 1 C 1 +C 1 D 1 E 3 + A 1 + C 1 + E 3 ) • r(x), where the A 1 , B 1 , C 1 , D 1
operators are order-one self-adjoint operators, the rightmost operator E 3 being an order-three self-adjoint operator.

The 54 order-eight linear differential operators are (non-trivially) homomorphic to their adjoints. They have SO(8, C) differential Galois groups and have a canonical decomposition described in [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF], generalization of the previous ones, with, again, five order-one self-adjoint operators, and a rightmost order-three self-adjoint operator.

The 41 order-nine linear differential operators are (non-trivially) homomorphic to their adjoints. They have SO(9, C) differential Galois groups and have a canonical decomposition described in [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF], generalization of the previous ones, with, six orderone self-adjoint operators, and a rightmost order-three self-adjoint operator.

Finally, the seven order-ten linear differential operators are (non-trivially) homomorphic to their adjoints. They have SO(10, C) differential Galois groups and have a canonical decomposition described in [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF], generalization of the previous ones, with, seven order-one self-adjoint operators, and a rightmost order-three self-adjoint operator.

These results are reminiscent of the results obtained on a set of 210 explicit linear differential operators annihilating diagonals of rational functions, arising from mirror symmetries and corresponding to reflexive 4-polytopes [START_REF] Lairez | Computing periods of rational integrals[END_REF], recalled in the introduction. One notes, however, that the symplectic Sp(n, C) differential Galois groups with a canonical decomposition in order-two self-adjoint operators and a rightmost order-four self-adjoint operator, encountered with these reflexive 4polytopes examples, is now replaced by orthogonal SO(n, C) differential Galois groups with a canonical decomposition in order-one self-adjoint operators and a rightmost order-three self-adjoint operator.

The calculations performed here, in order to see that these (quite large) linear differential operators are (non trivially) homomorphic to their adjoints and to find their canonical decompositions [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF], are similar to the ones described in [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF] for the reflexive 4-polytopes examples: in order to find the intertwiner, we introduced a specialized algorithm because the Maple command Homomorphisms(adjoint(L), L) never terminates on these large operators [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF]. We use a fuchsian linear differential system associated to L, the theta-system [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF], a slight generalization of the companion system which has simple poles at each finite singularity. One then finds a rational solution of an associated system with similar coefficients (its second symmetric/exterior power), which gives the intertwiner. An inversion of this intertwiner modulo L gives the intertwiner corresponding to the Maple command Homomorphisms(L, adjoint(L)). Finally, one obtains the canonical decomposition from simple euclidean divisions [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF].

For the last seven order-ten operators even the theta-system calculations were quite massive, and required up to two weeks of CPU time and up to 80 Gigaoctets of memory for one linear differential operator.

Diagonals of rational functions associated with orthogonal as well as symplectic groups

It is tempting to simply, and straightforwardly, generalize † these two sets of results for diagonals of well-defined finite sets of rational functions, namely the symplectic Sp(n, C) differential Galois groups [START_REF] Lairez | Computing periods of rational integrals[END_REF] (with an order-four rightmost self-adjoint operator as for reflexive 4-polytopes [START_REF] Lairez | Computing periods of rational integrals[END_REF]), and the orthogonal SO(n, C) differential Galois groups (with a rightmost order-three self-adjoint operator). The situation can be slightly more involved (and richer) than a straightforward generalization of the previous results. In fact, diagonals are not necessarily solutions of an irreducible linear differential operator.

To see this let us consider the diagonal ¶ of the rational function of four variables with a factorized denominator R = 1/(1 -x -y -u -z)/(1 -u -z -u z), which reads:

Diag(R) = 1 + 42 x + 4878 x 2 + 748020 x 3 + 130916310 x 4 + 24762428460 x 5 + 4929691760532 x 6 + 1017691904736936 x 7 + • • • (23)
It is solution of an order-seven linear differential operator which factorizes § into an order-three and an order-four operator: L 7 = L 3 • L 4 . Note that L 4 does not annihilate this diagonal. This means that the series ( 23) is solution of L 7 = L 3 • L 4 , but not of L 4 which has a solution analytic at x = 0, with a series expansion with integer coefficients♯ different from [START_REF] Boukraa | The diagonal Ising susceptibility[END_REF]:

1 + 214 x + 97278 x 2 + 53983020 x 3 + 32898451110 x 4 + 21172639875156 x 5 + 14121624413802444 x 6 + • • • (24)
The series L 4 (Diag(R)), solution of L 3 , is globally bounded [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF]. If one normalizes L 4 to be an operator with polynomial coefficients (instead of being a monic operator), the series L 4 (Diag(R)), solution of L 3 , is a series with integer coefficients:

17888 + 25769200 x + 17312032256 x 2 + 8722773606816 x 3 + 3775743401539200 x 4

+ 1486619414765913792 x 5 + 548416028673746513280 x 6 + • • • † In particular after a set of other unpublished results we have obtained. ¶ Note that this diagonal is factorized but is not the Hadamard product of two diagonals (see Appendix B.2). § Using the DFactorLCLM command one sees that one does not have a direct sum factorization, just the simple factorization L 7 = L 3 • L 4 . ♯ A natural question corresponds to ask if such series [START_REF] Zenine | Ising model susceptibility: the Fuchsian differential equation for χ (4) and its factorization properties[END_REF] with integer coefficients are necessarily a diagonal of rational function. It is true for ODEs of minimum weight for the monodromy filtration [START_REF] Christol | Globally bounded solutions of differential equations, Analytic number theory[END_REF], but in the general case, it is still a conjecture. It, however, seems that one can prove that such series are "automatic" (i.e. reduce to algebraic functions) modulo powers of primes (G. Christol, private communication).

Note that the order-three operator L 3 has a canonical SO(3, C) orthogonal decomposition [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF]

(A 1 B 1 C 1 + A 1 + C 1 ) • r(x)
, where A 1 , B 1 , C 1 , are orderone self-adjoint operators. In contrast L 4 has the canonical Sp(4, C) symplectic decomposition (A 2 B 2 + 1) • ρ(x), where A 2 , B 2 are order-two self-adjoint operators. In other words, the diagonal ( 23) is associated with both orthogonal and symplectic differential Galois groups! This is, in fact, the situation we expect generically: the diagonal of the rational function will be solution of a non-irreducible linear differential operator, each of its factors corresponding to orthogonal or symplectic differential Galois groups.

We have, for instance, the same results for another rational function such that its denominator is factorised. The diagonal of R = 1/(1 -x -y -z -u)/(1 -u -z -x z), namely 1 + 44 x + 5061 x 2 + 771000 x 3 + 134309890 x 4 + 25316919264 x 5

+ 5026804760628

x 6 + • • • (25)
is solution of an order-seven linear differential operator which factorizes into an orderthree and an order-four operator: L 7 = L 3 • L 4 (but, again there is no direct sum factorization). This diagonal ( 25) is solution of L 7 but not of L 4 . If one normalizes L 4 to be an operator with polynomial coefficients (instead of being a unitary operator), the series L 4 (Diag(R)), solution of L 3 , is a series with integer coefficients:

16 -94464 x -127052100 x 2 -86146838400 x 3 -44244836836200 x 4 -19495756524980736 x 5 -7791904441995369696 x 6 + • • • (26)
The order-three linear differential operator L 3 is MUM (maximal unipotent monodromy [START_REF] Lian | Maximal Unipotent Monodromy for Complete Intersection of CY Manifolds[END_REF]) and has a canonical

SO(3, C) decomposition (A 1 B 1 C 1 + A 1 + C 1 ) • r(x)
, where A 1 , B 1 , C 1 , are order-one self-adjoint operators. It is not the symmetric square of an order-two operator, it is homomorphic to the symmetric square of an order-two operator. In contrast L 4 has the canonical Sp(4, C) decomposition (A 2 B 2 + 1) • ρ(x) where A 2 , B 2 are order-two self-adjoint operators.

Diagonals of rational functions corresponding to n F n-1 hypergeometric functions

Beyond these finite sets of examples of diagonals of rational functions, namely the 210 reflexive 4-polytopes operators [START_REF] Lairez | Computing periods of rational integrals[END_REF] with symplectic Sp(n, C) differential Galois groups, and these 879 operators of section (4) associated with diagonals of rational functions of four variables with orthogonal SO(n, C) differential Galois groups, one can find infinite families of diagonals of rational functions for which exact results can be obtained corresponding to n F n-1 hypergeometric functions. For instance, the diagonal of the rational function of three variables

R = 1/(1 -x -z -y n ), with n being a positive integer, is a 2 n F 2 n -1 hypergeometric function Diag 1 1 -x -z -y n = 2 n F 2 n -1 [ 1 2 n + 1 , 2 2 n + 1 , 3 2 n + 1 , • • • , 2 n 2 n + 1 ], ( 27
) [ 1 n , 1 n , 2 n , 2 n , 3 n , 3 n , • • • , n -1 n , n -1 n , n n ], (2 n + 1) 2 n +1 n 2 n • x n .
The corresponding linear differential operator is an order-2n linear differential operator having a symplectic Sp(2 n, C) differential Galois group. The diagonal of the rational functions of three variables R = 1/(1-x-z -x n y n ), with n being a positive integer, is a n F n -1 hypergeometric function

Diag 1 1 -x -z -x n y n = n F n -1 [ 1 n + 1 , 2 n + 1 , 3 n + 1 , • • • , n n + 1 ], (28)
[ 1 n , 2 n , 3 n , • • • , n -1 n ], (n + 1) n +1 n n • x n .
The corresponding linear differential operator is an order-n linear differential operator having an orthogonal O(n, C) differential Galois group. The diagonal of the rational functions of three variables R = 1/(1-x-z -x y n ), with n being a positive integer, is a n F n -1 hypergeometric function. For n even it reads:

Diag 1 1 -x -z -x y n = n F n -1 [ 1 2 n , 3 2 n , 5 2 n , • • • , 2 n -1 2 n ], (29)
[ 1 n -1 , 2 n -1 , 3 n -1 , • • • , n -2 n -1 , n -1 n -1 ], 4 n • n n (n -1) n -1 • x n .
The corresponding linear differential operator is an order-n linear differential operator having a symplectic Sp(n, C) differential Galois group.

For n odd the same formula (29) holds. Note, however, that the argument 1/2 appears in both the first and second list of arguments of the n F n -1 hypergeometric function, and, hence, can be avoided, which is thus a n-1 F n -2 hypergeometric function. The corresponding linear differential operator is an order-(n -1) operator having a symplectic Sp(n -1, C) differential Galois group.

Diagonals of rational functions associated with operators non homomorphic to their adjoints

After all this accumulation of examples of diagonals of rational functions associated with orthogonal, or symplectic, differential Galois groups, it is tempting to conjecture that counter-examples § like 3 F 2 ([1/3, 1/3, 1/3], [1, 1], 3 6 x), are quite "rare exceptional cases" that one can easily detect, and, hopefully, understand, as situations of algebraic varieties "sufficiently singular" to break the Poincaré duality [START_REF] Griffiths | Principles of algebraic geometry[END_REF]. Along this line of seeking for diagonals of rational functions with a "sufficiently singular" denominator, let us try to provide examples of operators that are not homomorphic to their adjoints.

Appendix B provides two attempts to find examples of diagonals of rational functions such that their corresponding linear differential operators would not be homomorphic to their adjoints. We first study in Appendix B.1 a set of singular denominators for the rational functions, the polynomials of three variables corresponding to classifications of singular varieties performed by V. I. Arnold [START_REF] Arnold | Critical Points of Smooth Functions Proceedings of the International Congress of Mathematicians[END_REF]. § Given in the introduction corresponding to a SL(3, C) differential Galois group, and thus the corresponding operator cannot be homomorphic to its adjoint even with an algebraic extension.

All the corresponding operators yield symplectic Sp(n, C) differential Galois groups: the kind of singular behaviour required to "break the Poincaré duality", considered in V. I. Arnold [START_REF] Arnold | Critical Points of Smooth Functions Proceedings of the International Congress of Mathematicians[END_REF] for three variables, is not "sufficiently singular".

It could be that these "sufficiently singular" situations require much more than three variables †. Therefore we have considered, here, rational functions of six variables, and since the theory of singularities of algebraic varieties [START_REF] Christol | Diagonals of rational fractions[END_REF][START_REF] Hauser | The Hironaka theorem on the resolution of singularities (or; A proof we always wanted to understand)[END_REF] suggests that the situation of algebraic varieties "singular enough" often correspond to denominators that factor † †, we have considered, in Appendix B.2, rational functions of the form 1/ ((1 -P 1 ) (1 -P 2 )) where P 1 and P 2 are two polynomials of two different sets of three variables ‡. Again we found that all♯ the operators that could be analyzed among the 170 operators in Appendix B.2, are homomorphic to their adjoints, and have symplectic or orthogonal differential Galois groups.

We also considered the (minimal order) linear differential operators annihilating large sets ¶ of Hadamard products [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF][START_REF] Hadamard | Essai sur l'étude des fonctions données par leur développement de Taylor[END_REF] of algebraic functions, and, each time, obtained that these linear differential operators are homomorphic to their adjoints. Surprisingly, finding diagonals of rational functions such that their annihilating linear differential operators are not homomorphic to their adjoints, is not so easy, examples like 3 F 2 ([1/3, 1/2, 1/2], [1, 1], 12 2 x), being quite rare. Let us revisit these simple hypergeometric examples yielding operators that are not homomorphic to their adjoints.

Examples of diagonals of rational functions with no homomorphism to the adjoint

The hypergeometric function 3 F 2 ([1/3, 1/2, 1/2], [1, 1], 12 2 x), actually corresponds to a SL(3, C) differential Galois group. It can be seen as the diagonal of a rational function of six variables

Diag 1 -9 x y (1 -3 y -2 x + 3 y 2 + 9 x 2 y) • (1 -u -z) • (1 -v -w) (30)
= 1 + 12 x + 648 x 2 + 50400 x 3 + 4630500 x 4 + 468087984

x 5 + • • •
The corresponding (order-three) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension. This diagonal ( 30) is of the form

Diag(R 1 (x, y) • R 2 (u, z) • R 3 (v, w)
), (where R 1 , R 2 and R 3 are simple rational functions). It is, thus, the Hadamard product [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF][START_REF] Hadamard | Essai sur l'étude des fonctions données par leur développement de Taylor[END_REF] of the three diagonals Diag(R 1 (x, y)), Diag(R 2 (u, z)) and Diag(R 2 (v, w)), which are simple algebraic functions, respectively:

Diag 1 -9 x y 1 -2 x -3 y + 3 y 2 + 9 x 2 y = 1 (1 -9 x) 1/3 , ( 31
) † Counter-examples, like 3 F 2 ([1/3, 1/3, 1/3], [1, 1], 3 6
x), correspond to diagonal of algebraic functions of three variables (which are very simple since they are products of the algebraic functions of one variable), but this means that they are diagonals of rational functions of, at first sight, six variables. † † This is actually the case for the counter-examples, like 3 F 2 ([1/3, 1/3, 1/3], [1, 1], 3 6 x) mentioned in the introduction. ‡ This also corresponds to considering the Hadamard product of the diagonal of 1/(1 -P 1 (x, y, z)) and of the diagonal of 1/(1 -P 2 (u, v, w)). ♯ Even using Koutschan's creative telescoping program [47], two operators among the 170 were difficult to obtain. Among these 170 operators, 71 are so large that they cannot be analyzed using the DEtools commands, and required to switch to a theta-system approach (see [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF]). ¶ Some miscellaneous heuristic examples are given in the supplementary materials [START_REF] Weil | Diagonals of rational functions and selected differential Galois groups[END_REF].

Diag 1 1 -u -z = Diag 1 1 -v -w = 1 (1 -4 x) 1/2 . ( 32
)
The linear differential operators, annihilating the diagonals ( 31) and [START_REF] Griffiths | Principles of algebraic geometry[END_REF], are very simple order-one linear differential operators. There is, of course, no relation between their differential Galois groups and the SL(3, C) differential Galois group of the order-three operator annihilating their Hadamard product (30): the Hadamard product does not preserve algebraic structures, like the differential Galois group.

Remark: Let us consider the following rational function of six variables:

N (x, y, z, u, v, w) (1 -2 x -3 y + 9 x 2 y + 3 y 2 ) • (1 -z -u) • (1 -v -w) (33) = 1 -9 xy 1 -2 x -3 y + 9 x 2 y + 3 y 2 + 1 1 -z -u + 1 1 -v -w .
It is straightforward to see that the diagonal of this rational function [START_REF] Bogner | Algebraic characterization of differential operators of Calabi-Yau type[END_REF] of six variables is nothing but the sum of the three diagonals ( 31) and ( 32), namely:

1 (1 -9 x) 1/3 + 2 (1 -4 x) 1/2 . ( 34
)
This diagonal is solution of an order-two linear differential operator which is (obviously) a direct sum:

D x - 3 1 -9 x ⊕ D x - 2 1 -4 x . (35)
The differential Galois group of the (order-three) operator that annihilates [START_REF] Cameron | Analytic continuation of diagonals and Hadamard compositions of multiple power series Trans[END_REF], and the one of the (order-two) operator ( 35) that annihilates [START_REF] Bogner | Algebraic characterization of differential operators of Calabi-Yau type[END_REF], are quite different, even if the denominators of (30) and ( 33) are the same. In a theory of singularity perspective [START_REF] Hauser | The Hironaka theorem on the resolution of singularities (or; A proof we always wanted to understand)[END_REF][START_REF] Christol | Diagonals of rational fractions[END_REF], the statement that the differential Galois group depends essentially on the denominator ‡ of the rational function that "encodes the singularities", has to be taken "cum grano salis". The restriction we have imposed on our rational functions by imposing the numerators to be equal to one, is far from innocent.

More examples of diagonals of rational functions with no homomorphism to the adjoint

A slight modification of the previous example [START_REF] Cameron | Analytic continuation of diagonals and Hadamard compositions of multiple power series Trans[END_REF] amounts to considering the diagonal of a rational function of six variables

Diag 1 -9 x y (1 -3 y -2 x + 3 y 2 + 8 x 2 y) • (1 -u -z) • (1 -v -w) , (36)
where the coefficient of x 2 y has been modified (9 x 2 y changed into 8 x 2 y). The corresponding (order-five) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(5, C).

Another slight modification of (30) amounts to considering the diagonal of a rational function of five variables

Diag 1 -9 x y (1 -3 y -2 x + 3 y 2 + 9 x 2 y) • (1 -u -v -z -w) . ( 37
)
‡ Mathematicians would say that changing the numerator may change the n-form one integrates, that one must consider the Gauss-Manin Picard-Vessiot module of these differentials. We just try here, heuristically, to make crystal clear that numerators matter.

The corresponding (order-four) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(4, C).

More examples of diagonals of rational functions with linear differential operators, that are not homomorphic to their adjoints, are given in Appendix C.

To sum-up: The cases such that the diagonals of rational functions do not yield linear differential operators homomorphic to their adjoints, are far from being understood, either from a differential algebra viewpoint, or from a theory of singularity perspective. From an experimental mathematics perspective, what we see is that diagonals of rational functions "almost systematically" (but not always !) yield linear differential operators homomorphic to their adjoints, thus giving selected differential Galois groups. Diagonals of rational functions were seen [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF] to naturally emerge in physics. However, in a mathematical framework, not related to physics, diagonals of rational functions seem to "almost systematically" yield orthogonal or symplectic groups, with, at first sight, no obvious "physical interpretation".

Conclusion

We have introduced well-defined sets of diagonals of rational functions of three, four and six variables, showing that all these examples yield selected differential Galois groups, namely orthogonal and symplectic groups.

It has been seen that, in our set of rational functions of three variables with denominators with degree bounded by 1, all diagonals correspond to modular forms that can all be written as 2 F 1 hypergeometric functions with two pull-backs [START_REF] Bostan | The Ising model: from elliptic curves to modular forms and Calabi-Yau equations[END_REF], related by a modular curve.

We have seen that a set of 879 diagonal of rational functions of four variables correspond to orthogonal SO(n, C) differential Galois groups with a remarkable canonical decomposition [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF] with a rightmost self-adjoint operator of order three. These results were obtained using the very powerful package "HolonomicFunctions" written by C. Koutschan [47,[START_REF] Koutschan | HolonomicFunctions (User's Guide)[END_REF]49], based on the method of creative telescoping [START_REF] Zeilberger | The Method of Creative Telescoping[END_REF], which enables to obtain directly, and very efficiently, the linear differential operator annihilating a given diagonal of a rational function, without calculating the series expansion of the corresponding diagonal. In order to find the homomorphisms of these operators to their adjoints, which is the first step towards the analysis of the differential Galois groups of these operators and their "canonical decompositions" [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF], we have, for large linear differential operators, also used a new algorithm that requires to work on the linear theta-system associated with the operators [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF].

We have also seen (Appendix B.2 below) that 170 diagonals of rational functions of six variables of the form 1/ ((1 -P 1 (x, y, z))(1 -P 2 (u, v, w))), actually correspond to a quite rich set of linear differential operators with orthogonal or symplectic differential Galois groups. The systematic analysis performed in this paper of these three sets of diagonals of rational functions of respectively three, four and six variables, suggests that diagonals of rational functions "almost systematically" yield orthogonal or symplectic differential Galois groups.

A contrario, we have provided in section (7) a few miscellaneous examples of diagonals of rational functions where the corresponding linear differential operators are not homomorphic to their adjoints. These cases, such that the diagonals of rational functions do not yield linear differential operators homomorphic to their adjoints, are far from being fully understood. Is it possible that such cases could also emerge with the diagonals of rational functions appearing in physics? This remains an open and challenging question.

varieties "singular enough" to break the Poincaré duality [START_REF] Griffiths | Principles of algebraic geometry[END_REF], it is tempting to study the singular algebraic varieties classified by V. I. Arnold [START_REF] Arnold | Critical Points of Smooth Functions Proceedings of the International Congress of Mathematicians[END_REF]. The classification of the simplest singularities turned out to be related to Lie, Coxeter and Weyl groups, A n , D n , E n , and to the classification of platonic solids in Euclidean three spaces [START_REF] Arnold | Critical Points of Smooth Functions Proceedings of the International Congress of Mathematicians[END_REF]. Arnold's paper gives a set of polynomials of three variables that are obvious candidates to be considered as denominators of rational functions of three variables: P = x 2 z + y 3 + z 5 + 2 y z 4 , x 2 z + y z 2 + a y 3 z, x 3 + y 3 + z 4 + a x y z 2 , ... With these polynomials, we obtained the linear differential operators annihilating the diagonals of the rational functions of the form 1/P . These linear differential operators of various orders (order 22 for Q 10 in [START_REF] Arnold | Critical Points of Smooth Functions Proceedings of the International Congress of Mathematicians[END_REF], order 14 for Q 12 in [START_REF] Arnold | Critical Points of Smooth Functions Proceedings of the International Congress of Mathematicians[END_REF], order 12 for S 11 , order 8 for S 12 , order 10 for U 12 , order 16 for Z 13 , order 18 for W 12 , ...) are all homomorphic to their adjoints, their differential Galois groups being Sp(n, C) symplectic groups, the canonical decomposition [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF] being in terms of only order-two self-adjoint operators.

Appendix B.2. Diagonals of rational functions of six variables

Apparently the kind of singular behaviour required to "break the Poincaré duality", considered in V. I. Arnold [START_REF] Arnold | Critical Points of Smooth Functions Proceedings of the International Congress of Mathematicians[END_REF] for three variables, is not "sufficiently singular". It could be that these "sufficiently singular" situations require much more than three variables. Therefore we have considered in this section, rational functions of six variables (see section (7.1) and (Appendix C) below). Since the theory of singularities of algebraic varieties suggests that the situation of "singular enough" algebraic varieties often correspond to algebraic varieties that factor, we have studied exhaustively the diagonals of rational functions of the form 1/ ((1 -P 1 (x, y, z))(1 -P 2 (u, v, w))), where the degree of the two polynomials in each of their three variables is less than one, and their coefficients are 0 or 1. Note that this also corresponds to considering the Hadamard product of the diagonal of 1/(1 -P 1 (x, y, z)) and of the diagonal of 1/(1 -P 2 (u, v, w)). The number of such classes yielding different diagonals is only 170. We have obtained all the corresponding linear differential operators using the Mathematica "HolonomicFunctions" package [47]. The order of the (minimal order) linear differential operators runs from 2 to 12. The number of linear differential operators corresponding to the various orders is given in Table (B1).

Table B1. Number of operators corresponding to the various orders, number of operators with symplectic and orthogonal differential Galois groups.

Order of Oper. We also give the number of linear differential operators having respectively a symplectic and an orthogonal differential Galois group. Among these 170 linear differential operators, 91 can be analyzed using the DEtools command in order to see that they are homomorphic to their adjoint, find their differential Galois group and their canonical decomposition [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF]. The other linear differential operators are too large to be analyzed that way: they require to switch to a differential theta-system [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF] in order to find the intertwiners to their adjoint, and, then the canonical decomposition from a simple euclidean division [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF]. Note that these calculations (theta-system calculations) are still quite massive. We found that the three operators with a SO(3, C) orthogonal differential Galois group were self-adjoint.

We found that all the seven order-4 linear differential operators with a symplectic differential Galois group (namely Sp(4, C)) are self-adjoint, or conjugated to their adjoints (by a simple rational function), except one with a (L 2 • M 2 + 1) • r(x) canonical decomposition [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF]. We found that the 12 linear differential operators with a SO(4, C) orthogonal differential Galois group had a (L 1 • M 3 + 1) • r(x) canonical decomposition [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF].

All the 13 order-5 linear differential operators which have an SO(5, C) orthogonal differential Galois group have a ( [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF]. The four order-6 linear differential operators with an SO(6, C) [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF].

L 1 • M 1 • N 3 + L 1 + N 3 ) • r(x) canonical decomposition
orthogonal differential Galois group have a (L 1 • M 1 • N 1 • P 3 + L 1 • P 3 + N 1 • P 3 + L 1 • M 1 + 1) • r(x) canonical decomposition
We found that the other 35 order-6 linear differential operators have a Sp(6, C) symplectic differential Galois group. Among these 35 symplectic operators only two have a ( [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF], all the other having a (L 2 • M 4 + 1) • r(x) canonical decomposition [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF].

L 2 • M 2 • N 2 + L 2 + N 4 ) • r(x) canonical decomposition
We found that all the 52 order-8 linear differential operators have a Sp(8, C) symplectic differential Galois group with a (L 2 • M 2 • N 4 + L 2 + N 4 ) • r(x) canonical decomposition [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF].

All the 36 order-10 and 6 order-12 linear differential operators have respectively Sp(10, C) and Sp(12, C) symplectic differential Galois groups, with ( [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF].

L 2 • M 2 • N 2 • P 4 + • • • ) • r(x) and (L 2 • M 2 • N 2 • P 2 • Q 4 + • • • ) • r(x) canonical decomposition
Remark 1: In all the decomposition we have obtained, the rightmost self-adjoint operator [START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF], is always of order three (for orthogonal differential Galois groups) or of order four (for symplectic differential Galois groups), except an order-4 operatoir with a (L 2 • M 2 + 1) • r(x) decomposition and the two order-6 linear differential operators corresponding to the diagonals of the two rational functions

(1 -xy -xz -yz) • (1 -v -w -uv -uw), (B.1) (1 -z -xy -xz -yz -xyz) • (1 -u -v -w -uvw), (B.2) the corresponding operators having a (L 2 • M 2 • N 2 + L 2 + N 2 ) • r(x) canonical decomposition [21].
Remark 2: Using Koutschan's creative telescoping program [47], three linear differential operators were quite difficult to obtain compared to the others. They correspond to the following denominators of the rational functions:

(1 -x -y -z -xz -xyz) • (1 -u -v -w -uw -vw -uvw), (B.3) (1 -x -y -z -xz -xyz) • (1 -u -v -w -uv -uw -vw -uvw), (B.4) (1 -x -y -z -xz -xyz) • (1 -u -v -w -uv -uw -vw). (B.5)
Koutschan's creative telescoping program gives an order-12 linear differential operator for (B.3). Note that one always needs to verify that the operator obtained from this program is actually the minimal order operator annihilating the diagonal †. Actually, † It may not be irreducible: see, for instance, the L 7 = L 3 • L 4 operator associated with the diagonal (23) in section [START_REF] Boukraa | High-order Fuchsian equations for the square lattice Ising model: χ (6)[END_REF].

performing Hadamard products [START_REF] Assis | Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations[END_REF][START_REF] Hadamard | Essai sur l'étude des fonctions données par leur développement de Taylor[END_REF] modulo primes, one can forecast that minimal order for (B.3). We found an order 10 which suggests that the order-12 linear differential operator obtained by Koutschan's creative telescoping program [47], is not the minimal order linear differential operator ‡: this order-12 actually factorises. It is the product of two order-one operators and of the minimal order-10 linear differential operator annihilating the diagonal:

L 12 = L 1 • M 1 • L 10 .
The factorization of such very large linear differential operators♯ is a quite difficult task. In a first step calculations modulo prime have been performed that enable us to find the order of the minimal order operator annihilating all these diagonals. We have, however, been able to perform these factorizations (in characteristic zero, not modulo primes).

Remark 3: When the linear differential operator annihilating the diagonal, obtained from Koutschan's program [47], is not the minimal order linear differential operator, one can try to obtain that minimal order operator by factorization (DFactor in Maple). Unfortunately the factorization of (very) large operators like many of these operators, cannot be obtained using straightforwardly DFactor DEtools command in Maple. The factorization of the largest operators has been obtained, in the following way: one first obtains a large set of coefficients of the series of the corresponding diagonals (that is the non-trivial part of the calculation), and then use it as the input of a "guessing procedure ¶".

Remark 4: Koutschan's creative telescoping program [47] gave us an order-9 linear differential operator corresponding to the diagonal of 1/(

(1 -x -y -z -x z - x y z) (1 -u -v -w -u w -u v w)) (i.e. the Hadamard square of the diagonal of 1/(1 -x -y -z -x z -x y z)).
In fact this order-9 linear differential operator is not minimal, the minimal order operator being of order 8. This program gave us also four order-11 linear differential operators, like, for instance, the operator corresponding to the diagonal of (B.4). Similarly this order-11 linear differential operator is not minimal, the minimal order operator being of order 10 (the degree of the polynomial coefficients is 74). This is also the case for the three other, at first sight, order-11 linear differential operators corresponding to the following denominators of the rational functions:

(1 -z -xz -yz -xy) • (1 -u -v -w -uw -vw -uvw), (B.6) (1 -y -z -xz -xy -yz) • (1 -u -v -w -uv -uw -vw -uvw), (B.7) (1 -x -y -z -xz -xyz) • (1 -u -v -w -uw -vw). (B.8)
They are not minimal order operators, the minimal order linear differential operators, annihilating the corresponding diagonals being of order 10 (the degree of the polynomial coefficients being respectively [START_REF] Weil | Diagonals of rational functions and selected differential Galois groups[END_REF][START_REF] Koutschan | A Fast Approach to Creative Telescoping[END_REF][START_REF] Koutschan | A Fast Approach to Creative Telescoping[END_REF]. Therefore one finds that there is no order-9 or order-11 operators for this set of 170 diagonals.

The linear differential operator annihilating the diagonal of (B.5) was the most difficult to obtain using Koutschan's creative telescoping program♯. This program gives an order-13 linear differential operator. From this exact differential operator one can obtain as long as necessary series of the diagonal, and study, modulo some primes, ‡ One can even forecast the degree of the polynomial coefficients of that order-10 operator: the degree is 74. ♯ Using the DFactor command of DEtools in Maple, or any other method. ¶ Essentially of the same type as in gfun. ♯ The program computes the "telescoper" (the ODE) and the "certificate". It is a general observation that in most examples, the certificate is much larger than the telescoper [START_REF] Bostan | Creative telescoping for rational functions using the Griffiths-Dwork method[END_REF]. We just need the telescoper. An algorithm that computes telescopers without computing the corresponding certificates, has already been built [START_REF] Bostan | Creative telescoping for rational functions using the Griffiths-Dwork method[END_REF][START_REF] Lairez | Computing periods of rational integrals[END_REF].

the linear differential operators annihilating these series. One finds, that way, that the minimal order operator annihilating the diagonal of (B.5) is of order 12 (with degree 85). We actually factorised this order-13 operator, and obtained the minimal order-12 operator that annihilates this diagonal ¶.

Remark 5: Seeking for the (large ...) linear differential operators annihilating the diagonals of (B.4), (B.5), it seems natural to take into account the selected form of all these diagonals of rational functions of six variables, which are, actually, the Hadamard product [START_REF] Hadamard | Essai sur l'étude des fonctions données par leur développement de Taylor[END_REF][START_REF] Assis | Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations[END_REF] of two diagonals of rational functions of three variables. In the case of (B.4), (B.5) (and even (B.3)), the linear differential operators annihilating these various diagonals of rational functions of three variables, are simple order-two operators. One can imagine to obtain these linear differential operators using the gfun command "hadamard product" of two simple order-two operators ‡. Unfortunately, the linear differential operators obtained that way are of order much larger the one obtained from Koutschan's creative telescoping of a diagonal of a function of six variables ! The same remarks apply for the analysis of all the 170 diagonals of Table (B1).

Let us give, here, miscellaneous, simple examples of these diagonal of rational functions of six variables (all the exhaustive results being given in a supplementary material [START_REF] Weil | Diagonals of rational functions and selected differential Galois groups[END_REF]). The corresponding (order-four) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(4, C).

Similarly, the diagonal of the five variable rational function Diag 1 (1 -x + 3 y -27 x y 3 -27 x y 2 -9 x y + 3 y 2 ) • (1 -u -v -u z -v z) , is annihilated by an order-three linear differential operator which is not homomorphic to its adjoint (even with an algebraic extension), and its differential Galois group is SL(3, C).

Similarly, the diagonal of the five variable rational function Diag 1 -9 x y (1 -3 y -2 x + 3 y 2 + 9 x 2 y)

• (1 -u -v -w) , (C.2)
is annihilated by an order-three linear differential operator which is not homomorphic to its adjoint, and its differential Galois group is SL(3, C).

Similarly, the diagonal of the five variable rational function is annihilated by an order-five linear differential operator which is not homomorphic to its adjoint, and its differential Galois group is SL(5, C).

Appendix B. 2 . 1 .x 4 • ( 1 -

 2141 Order-two operators: pullbacked 2 F 1 hypergeometric function • The diagonal of the rational function of six variablesDiag 1 (1 -w -u v -u v w) • (1 -z -x y) (B.9)= 1 + 6 x + 78 x 2 + 1260 x 3 + 22470 x 4 + 424116 x 5 + 8305836 x 6 + • • • is annihilated by an order-two linear differential operator. This diagonal is the Hadamard product[START_REF] Assis | Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations[END_REF][START_REF] Hadamard | Essai sur l'étude des fonctions données par leur développement de Taylor[END_REF] of the diagonal of 1/(1 -z -x y) and of the diagonal of 1/(1 -w -u v -u v w), namely the two simple algebraic functions:(1 -4 x) -1/2 (B.10) = 1 + 2 x + 6 x 2 + 20 x 3 + 70 x 4 + 252 x 5 + 924 x 6 + • • • (1 -6 x + x 2 ) -1/2 (B.11) = 1 + 3 x + 13 x 2 + 63 x 3 + 321 x 4 + 1683 x 5 + 8989 x 6 + • • • This diagonal is actually a pullbacked 2 F 1 hypergeometric function: (1 -24 x + 48 x 2 ) -1/4 • 2 F 1 24 x + 16 x 2 ) (1 -24 x + 48 x 2 ) 3 .(B.12) ¶ After completion of this work we were told by C. Koutschan that giving extra options to "FindCreativeTelescoping" enables the program to find directly the minimal order-12 operator in 84 hours CPU time, instead of the 116 hours CPU time we used to obtain the order-13 operator. With these extra options the telescoper is 232560 bytes when the certificate requires 32743760 bytes, to be compared with 265432 bytes and 38422496 bytes for respectively the telescoper and certificate in the order-13 calculation. ‡ The gfun[hadamardproduct](eq1, eq2, y(z)) command determines the linear differential equation satisfied by the Hadamard product of two holonomic functions, solutions of the linear differential equations eq1 and eq2.

decomposition [ 21 ] 2 x+ (2868 x 2 - 1 ) 1 ( 1 - 1 ( 1 -

 212211111 , L 4 = D x • L 3 + 12, where L 3 is an order-three self-adjoint linear differential operator:L 3 = x 2 • (432 x 2 -1) • D 3 x + 3 x • (864 x 2 -1) • D • D x + 276 x (B.20) • The diagonal of the rational function Diag u v -u w -v w) • (1 -x y -x z -y z) = 1 + 36 x 2 + 8100 x 4 + 2822400 x 6 + • • • (B.21) is annihilated by a linear differential operator of order four, L 4 , which is self-adjoint, its differential Galois group being the symplectic group Sp(4, C).• The diagonal of the rational functionDiag u v -u w -v w -u v w) • (1 -x y -x z -y z) = 1 + 42 x 2 + 13590 x 4 + 7410480 x 6 + • • • (B.22) is annihilated by a linear differential operator of order eight, L 8 , which has a symplectic differential Galois group Sp(8, C), with a simple canonical decomposition[START_REF] Boukraa | Canonical decomposition of linear differential operators with selected differential groups[END_REF],L 8 = (L 2 • M 2 • N 4 + L 2 + N 4 ) • r(x), where L 2 and M 2 are two order-two self-adjoint operators and N 4 is an order-four self-adjoint linear differential operator. Appendix C. Miscellaneous examples of diagonals of rational functions with operators that are not homomorphic to their adjoints Another slight modification of (30) amounts to considering the diagonal of a rational function of six variables Diag 1 -9 x y (1 -3 y -2 x + 3 y 2 + 9 x 2 y) • (1 -z -u -u z) • (1 -v -w) . (C.1)

Diag 1 -

 1 9 x y (1 -3 y -2 x + 3 y 2 + 8 x 2 y) • (1 -u -v -w) , (C.3)

Table 1 .

 1 Number of operators corresponding to the various orders.

	Order of Oper.	1 2 3	4	5	6	7	8	9 10
	Number of Oper. 1 2 20 128 240 231 155 54 41 7

F 1 ([1/3, 1/6], [1], 108 x

), some with, at first sight, more involved HeunG function solutions[START_REF] Bostan | The Ising model: from elliptic curves to modular forms and Calabi-Yau equations[END_REF] which turn out to be pullbacked 2 F 1 hypergeometric functions, with two possible pullbacks, and, in fact, modular forms[START_REF] Bostan | The Ising model: from elliptic curves to modular forms and Calabi-Yau equations[END_REF].The 128 order-four linear differential operators are (non-trivially) homomorphic to their adjoints. They have SO(4, C) differential Galois groups and have a canonical † These two series[START_REF] Boukraa | The Ising model and Special Geometries[END_REF] and[START_REF] Zenine | The Fuchsian differential equation of the square lattice Ising model χ (3) susceptibility[END_REF] are not in Sloane's on-line encyclopedia http://oeis.org.

Acknowledgments. J-M. M. would like to thank R. J. Baxter for so many open, friendly scientific discussions during these last 35 years, since he met him in the 1980 Enschede Summer School. J-M. M. would like to thank L. Dumont and P. Lairez for many enlightening discussions on diagonals of rational functions and C. Koutschan for fruitful comments on his "HolonomicFunctions" package. S. B. would like to thank the LPTMC and the CNRS for kind support. We would like to thank Y. Abdelaziz for checking many of our results. We thank the Research Institute for Symbolic Computation, (Peter Paule) for access to the RISC HolonomicFunctions package. We thank G. Christol for many enlightening discussions on Poincaré duality. As far as physicists authors are concerned, this work has been performed without any support of the ANR, the ERC or the MAE, or any PES of the CNRS.

Appendix A. Diagonals of rational functions of three variables: some modular forms Let us recall the two Hauptmoduls [START_REF] Bostan | The Ising model: from elliptic curves to modular forms and Calabi-Yau equations[END_REF] 12 3 /j 3 and 12 3 /j 4 of Table 4 of Maier [START_REF] Maier | On rationally parametrized modular equations[END_REF]:

12 3 • z (z + 27) • (z + 3) 3 , H 4 (z) = 12 3 • z • (z + 16) (z 2 + 16 z + 16) 3 .

• The diagonal of 1/(1 -x -y -y z -x z) as well as the diagonal of 1/(1 -x -y -z + x z), correspond to the sequence [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Bostan | High order Fuchsian equations for the square lattice Ising model: χ(5)[END_REF][START_REF] Lairez | Périodes d'intégrales rationnelles[END_REF]400, 4900, 63504, • • • ] of the complete elliptic integral K(4 x 1/2 ) (oeis number A002894 in Sloane's on-line encyclopedia) can be written as a pullbacked 2 F 1 hypergeometric function:

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], H 4 (z) with: z = 1 -16 x x . (A.1) Recalling Maier's paper [START_REF] Maier | On rationally parametrized modular equations[END_REF] one knows that (A.1) can alternatively also be written a pullbacked 2 F 1 hypergeometric function with the 12 3 /j ′ 4 of Table 5 of Maier [START_REF] Maier | On rationally parametrized modular equations[END_REF]. The diagonal of 1/(1 -x -y -z + x z) is clearly a modular form. As a byproduct this suggests that the diagonal of the rational function of three variables

is zero. This can be checked directly.

• The diagonal of 1/(1 -x -y -z -x y z), which corresponds to the sequence [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Assis | Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations[END_REF]115,2371,54091, 1307377, • • • ] (oeis number A081798 in Sloane's on-line encyclopedia) can be written as a pullbacked 2 F 1 hypergeometric function:

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], H 3 (z)

with:

or more simply as

(A.4)

• The diagonal of 1/(1 -x -y -z -x y -x z -y z), which corresponds to the sequence [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Lipshitz | Rational functions, diagonals, automata and arithmetic Number theory[END_REF]366,13800,574650, 25335072, • • • ] (no oeis number) can be written as a pullbacked 2 F 1 hypergeometric function:

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], H 3 (z)

with:

• The diagonal of 1/(1 -x-y -x z-y z +x y z), which corresponds to the sequence [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Zenine | Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ (3) and χ[END_REF][START_REF] Bostan | The Ising model: from elliptic curves to modular forms and Calabi-Yau equations[END_REF]243,2601, 29403, • • • ] (oeis number A245925 in Sloane's on-line encyclopedia) can be written as a pullbacked 2 F 1 hypergeometric function:

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], H 4 (z)

with:

• The diagonal of 1/(1 -x -y -z + x y z), which corresponds to the sequence ‡ [1, 5, 67, 1109, 20251, 391355, • • • ] (oeis number A124435 in Sloane's on-line encyclopedia) can be written as a pullbacked 2 F 1 hypergeometric function:

with:

• The diagonal of 1/(1 -x + y + z + x y + x z -y z + x y z), which corresponds to the sequence [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Zenine | Lattice Green Functions: the seven-dimensional face-centred cubic lattice[END_REF]325,11711,465601,19590491, • • • ] (no oeis number) can be written as a pullbacked 2 F 1 hypergeometric function:

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], H 3 (z)

with:

Appendix B. Two attempts to break the Poincaré duality

Along the line, sketched in section [START_REF] Assis | Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations[END_REF], which amounts to seeking for diagonals of rational functions with a "sufficiently singular" denominator, let us try to find examples of linear differential operators that are not homomorphic to their adjoints.

We study here two sets of singular denominators for the rational functions, first polynomials corresponding to classifications of singular varieties performed by V. I. Arnold [START_REF] Arnold | Critical Points of Smooth Functions Proceedings of the International Congress of Mathematicians[END_REF], then denominators that factor into two polynomials.

Appendix B.1. Diagonals of rational functions associated with singular algebraic varieties

If one believes that the situations where the linear differential operator annihilating the diagonal of a rational function should correspond to situations of algebraic ‡ Number of effective multiple alignments of three equal-length sequences.

This Hauptmodul can be seen to be of the form 12 3 /j 2 (see Table 4 of Maier [START_REF] Maier | On rationally parametrized modular equations[END_REF]):

This diagonal (B.9) has another pullbacked 2 F 1 hypergeometric function representation, corresponding to the other pullback P 2 (x) = 12 3 • z 2 /(z + 256) 3 :

where:

The pullbacked 2 F 1 hypergeometric function (B.12) (or equivalently (B.14)) is thus a modular form.

Note that this diagonal (B.9) is, in fact, actually identical to the diagonal of the rational function of three variables 1/(1 -x -y -z -x y + x z).

• The diagonal of the rational function of six variables (1 -36 x + 134 x 2 -36

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], P 1 (x)

where:

It is also be written as pullbacked 2 F 1 hypergeometric function:

(1 + 444 x + 134 x 2 + 444

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], P 2 (x)

where:

(B.17) which shows that this diagonal is a modular form: these two pullbacks can be seen as the Hauptmoduls 12 3 /j 4 , 12 3 /j ′ 4 in Table 4 and 5 of Maier [START_REF] Maier | On rationally parametrized modular equations[END_REF]: 3 where:

Note that this diagonal (B.15) is, in fact, actually identical to the diagonal of the rational function of three variables 1/(1 -x -y -z -x y + x z -y z -x y z). is annihilated by a linear differential operator of order four, L 4 , which has an orthogonal differential Galois group SO(4, C), with a simple canonical