Y Abdelaziz

S Boukraa

C Koutschan

J-M Maillard

Heun functions and diagonals of rational functions (unabridged version)

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, Heun functions, creative telescoping, extremal rational elliptic surfaces, pullbacked hypergeometric functions, Schwarz map, Schwarz function, modular forms, modular equations, Hauptmoduls, Belyi coverings, Shimura curves, automorphic forms

We provide a set of diagonals of simple rational functions of four variables that are seen to be squares of Heun functions. Each time, these Heun functions, obtained by creative telescoping, turn out to be pullbacked 2 F 1 hypergeometric functions and in fact classical modular forms. We even obtained Heun functions that are automorphic forms associated with Shimura curves as solutions of telescopers of rational functions.

Introduction

Diagonals of rational functions naturally emerge in lattice statistical mechanics, enumerative combinatorics or more generally for n-fold integrals of theoretical physics [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: integrality versus modularity Preprint[END_REF]. In previous papers [START_REF] Bostan | Diagonal of rational functions and selected differential Galois groups[END_REF][START_REF] Abdelaziz | Diagonals of rational functions, pullbacked 2 F 1 hypergeometric functions and modular forms (unabridged version)[END_REF][START_REF] Abdelaziz | Diagonals of rational functions, pullbacked 2 F 1 hypergeometric functions and modular forms[END_REF] we have seen † that many diagonals of simple rational functions were pullbacked 2 F 1 hypergeometric functions that turn out to be related to classical modular forms. Sticking with diagonals of simple rational functions that are solutions of linear differential operators of order two, it is natural to study diagonals of simple rational functions that are Heun functions.

Heun functions emerge in different areas of physics [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF][START_REF] Takemura | The Heun equation and the Calogero-Moser-Sutherland system I: the Bethe Ansatz method[END_REF][START_REF] Valent | Difference Equations, Special Functions and Orthogonal Polynomials[END_REF][START_REF] Smirnov | Elliptic solitons and Heun equations[END_REF] (see also page 60 of [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF]) and enumerative combinatorics: the simple cubic lattice Green functions [START_REF] Glasser | Lattice Green function (at 0) for the 4D hypercubic lattice[END_REF] can be written as a Heun function. Experimentally the Heun functions emerging in physics often † correspond to globally bounded series [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: integrality versus modularity Preprint[END_REF], i.e. series that can be recast, after some rescaling, into series with integer coefficients. Most of the time they turn out to be pullbacked 2 F 1 hypergeometric functions and in fact classical modular forms. This suggests to study the class of Heun functions that are diagonals of rational functions † †, and thus, globally bounded series [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF]. We will discard the case where the Heun functions are almost trivial, their order-two linear differential operators factorising into two order-one linear differential operators. Such rather trivial cases are recalled in Appendix A. In this paper we examine non trivial Heun functions, which happen to be diagonals of simple rational functions [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF] of (mostly) four variables or solutions of telescopers of rational functions of (mostly) four variables. We see that they happen to fall into one of three categories:

• Heun functions that are diagonals of rational functions, having globally bounded series expansions, and can be rewritten as pullbacked hypergeometric functions that are classical modular forms. • Heun functions that are diagonals of rational functions, having globally bounded series expansions, and can be rewritten as pullbacked hypergeometric functions that are derivatives of classical modular forms. • Heun functions that are solutions of telescopers of rational functions that have series expansions that are not globally bounded ‡. They will be seen to correspond to Shimura automorphic forms or derivatives of automorphic forms.

The Heun function Heun(a, q, α, β, γ, δ, x) is solution of the order-two Heun linear differential operator with four singularities (D x denotes d/dx)

H 2 = D 2 x + γ x + δ x -1 + x -a • D x + α β x -q x • (x -1) • (x -a) , (1)
where one has the Fuchsian constraint = α + β -γ -δ + 1, where α, β, γ, δ need to be rational numbers, and a is an algebraic number. The parameter q is called the accessory parameter and the ratio q/α/β is called the normalised accessory parameter.

In the first two sections, we examine the Heun functions emerging from diagonals of simple rational functions that fall into the first or second category above, and show how they happen to be related to classical modular forms, or derivatives of classical modular forms, corresponding to pullbacked 2 F 1 hypergeometric functions. These Heun functions have integer coefficient series, (or can be recast as series with integer coefficients [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF] after a rescaling of the variable), and are solutions of globally nilpotent [START_REF] Bostan | Globally nilpotent differential operators and the square Ising model[END_REF] linear differential operators: the critical exponents of all the singularities are rational numbers. This leads to define a criterion in Appendix F, that allows to draw up a list of parameters of the Gauss hypergeometric function 2 F 1 ([a, b], [c], x), for which it corresponds to a classical modular form (see section 2). Furthermore, we will find that some of these Heun functions turned out to be periods of extremal rational surfaces (see section 2.4). We do this while avoiding trivial cases ¶.

In the third section, we examine first the solutions of the telescoper of a rational function, corresponding to a Heun function with a non globally bounded series expansion, and we show that this Heun function is related to a specific Shimura curve [START_REF] Elkies | Shimura curve computations via K3 surfaces of Néron-Severi rank at least 19[END_REF][START_REF] Voight | Shimura curves of genus at most two[END_REF][START_REF] Voight | Three lectures on Shimura curves[END_REF][START_REF] Shaska | Genus 2 fields with degree 3 elliptic subfields[END_REF][START_REF] Hallouin | Computation of a cover of Shimura curves using a Hurwitz space[END_REF][START_REF] Kurihara | On some examples of equations defining Shimura curves and the Mumford uniformization[END_REF]. We then examine a larger class of Heun functions listed in [START_REF] Van Hoeij | Belyi functions for hyperbolic Hypergeometric-to-Heun transformations[END_REF], and show that they are linked to Shimura curves. We are able to show the link between these Heun functions and Shimura curves thanks to a result by K.Takeuchi [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF].

Recalls on lattice Green functions as diagonals of rational functions

The simple cubic lattice Green function [START_REF] Joyce | On the Simple Cubic Lattice Green Function[END_REF] 1

(2 π) 3 • 2 π 0 2 π 0 2 π 0 dθ 1 dθ 2 dθ 3 1 -x • (cos(θ 1 ) + cos(θ 2 ) + cos(θ 3 )) , (2)
is nothing but † † the diagonal of the rational function in four variables x, z 1 , z 2 , z 3 :

1 1 -x • z 1 z 2 z 3 • ((1 + z 2 1 )/z 1 /2 + (1 + z 2 2 )/z 2 /2 + (1 + z 2 3 )/z 3 /2) = 2 2 -x • z 1 z 2 z 3 • (z 1 + 1/z 1 + z 2 + 1/z 2 + z 3 + 1/z 3 ) . (3)
The linear differential operator annihilating the diagonal of this rational function in four variables has order three and is the symmetric square of a linear differential operator of order two (θ is the homogeneous derivative x • d/dx):

9 x 4 • (2 θ + 3) • (2 θ + 1) -4 x 2 • (10 θ 2 + 10 θ + 3) + 4 θ 2 , (4)
having a Heun function as a solution. Consequently, the simple cubic lattice Green function [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF], or equivalently the diagonal of (3) reads: (

Heun 1 9 , 1
) 5
The Heun function on the RHS of (5) happens to be a period of an extremal rational curve as can be seen in the work of Doran and Malmendier [START_REF] Doran | Calabi-Yau Manifols Realizing Symplectically Rigid Monodromy Tuples[END_REF]. The diagonal [START_REF] Abdelaziz | Diagonals of rational functions, pullbacked 2 F 1 hypergeometric functions and modular forms[END_REF] can also be written as a Hadamard product † of a simple algebraic function and a Heun function:

Heun 1 9 , 1 12 , 1 4 , 3 4 , 1, 1 2 , x 2 2
= (1 -

4 x 2 ) -1/2 Heun 1 9 , 1 3 , 1, 1, 1, 1, x 2 4 . (6)
¶ (See Appendix A) corresponding to factorizations of the order-two linear differential operator of the Heun function (see Appendix A.1), or corresponding to situations where the fourth singularity is in fact an apparent singularity (see Appendix A.2), a situation which often corresponds to the previous factorization of the order-two linear differential operator. † † Cooking recipe: change cos(θ i ) = (1 + z 2 i )/2/z i (i.e. z i = exp(i θ i )) and x → x • z 1 z 2 z 3 . † Denoted here by a star (*).

Similarly, considering pencils of K3-surfaces, Peters and Stienstra introduced [START_REF] Peters | A pencil of K3-surfaces related to Apéry's recurrence for ζ(3) and Fermi surfaces for potential zero[END_REF] the integral §

I(x) = - 1 2 i π 3 • |z1|=1 |z2|=1 |z3|=1 dz 1 dz 2 dz 3 z 1 z 2 z 3 × x 1 -x • (z 1 + 1/z 1 + z 2 + 1/z 2 + z 3 + 1/z 3 ) , (7)
is annihilated by the linear differential operator of order three that is the symmetric square of the order-two linear differential operator:

L 2 = 576 • x 4 • θ • (θ + 1) -8 • x 2 • (20 θ 2 + 1) + (2 θ -1) 2 , ( 8
)
where θ denotes the homogeneous derivative θ = x • d/dx. Its solution, analytic at x = 0, reads:

x 1/2 • Heun 1 9 , 1 12 , 1 4 , 3 4 , 1, 1 2 , 4 x 2 , (9)
the other solution having a formal series expansion with a logarithm. Note that this square of a Heun function can be recast into a series with integer coefficients: + 936369720 z 7 + 27770358330

Heun 9, 3
z 8 + • • • (10)
Alternative forms of Heun functions like Heun 9, 3,

can be introduced for the simple cubic lattice Green function. They are displayed in Appendix B. They all reduce to pullbacked 2 F 1 hypergeometric functions ‡ which turn out to correspond to classical modular forms †.

Diagonals of rational functions of three and four variables yielding Heun functions corresponding to classical modular forms

We are going to provide a set of exact expressions for diagonals [23] of simple rational functions of three and four variables yielding Heun functions. These exact expressions are obtained using the creative telescoping approch and, more specifically, the program of C. Koutschan [6], these diagonals being analytic at x = 0 globally bounded series, solutions of a telescoper obtained with this creative telescoping program ¶.

Diagonals of rational functions of four variables yielding Heun functions

• Example 1. The diagonal of the rational function

R(x, y, z, w) = 1 1 -(w x y + w x z + w y z + x y z + w x + y z) , (11)
§ In section 3 of [START_REF] Peters | A pencil of K3-surfaces related to Apéry's recurrence for ζ(3) and Fermi surfaces for potential zero[END_REF], the variable x is denoted t = 1/s. The correspondence between this x and the x in the lattice Green function (2) which corresponds to I(x)/x, is x → x/2. Thus (5) becomes, once divided by x, the square of the Heun function [START_REF] Smirnov | Elliptic solitons and Heun equations[END_REF]. ‡ See also Appendix D below for [START_REF] Smirnov | Elliptic solitons and Heun equations[END_REF] or [START_REF] Glasser | Lattice Green function (at 0) for the 4D hypercubic lattice[END_REF]. † The emergence, for this fibration into K3 surfaces, of modular functions, cusp forms of weight two, via Dedekind's η-functions, can be found in section 4 of [START_REF] Peters | A pencil of K3-surfaces related to Apéry's recurrence for ζ(3) and Fermi surfaces for potential zero[END_REF].

¶ This program also provides other expressions called the "certificates" that we do not use here.

reads:

Diag R(x, y, z, w)

= 1 + 2 x + 18 x 2 + 164 x 3 + 1810 x 4 + • • • (12)
A creative telescoping program [6] gives the order-three linear differential operator annihilating the diagonal [START_REF] Elkies | Shimura curve computations via K3 surfaces of Néron-Severi rank at least 19[END_REF] of the previous rational function [START_REF] Bostan | Globally nilpotent differential operators and the square Ising model[END_REF]:

L 3 = 2 + 60 x -(1 -40x -444 x 2 ) • D x -3 x • (1 -18 x -128 x 2 ) • D 2 x -x 2 • (1 + 4 x) • (1 -16 x) • D 3 x . (13)
This order-three linear differential operator corresponds to the symmetric square of the order-two linear differential operator:

L 2 = x 2 • (8 θ + 5) • (8 θ + 3) + x • (12 θ 2 + 6 θ + 1) -θ 2 . ( 14
)
Thus the solution corresponding to the diagonal of ( 11) is given by the square of a Heun function:

Heun - 1 4 , 1 16 , 3 8 , 5 8 , 1, 1 2 , -4 x 2 = 1 + 2 x + 18 x 2 + 164 x 3 + 1810 x 4 + 21252 x 5 + 263844 x 6 + 3395016 x 7 + 44916498 x 8 + • • • (15)
This Heun function can be written as a pullbacked 2 F 1 hypergeometric function :

Heun - 1 4 , 1 16 , 3 8 , 5 8 , 1, 1 2 ,
-4 x = A • 2 F 1 [ 1 8 , 3 8 ], [1], H , (16)
where A and the Hauptmodul H are algebraic functions expressed with square roots:

H ± = -128 x • 1 -20 x + 50 x 2 + 400 x 3 -224 x 4 -512 x 5 (1 -88 x -112 x 2 -256 x 3 ) 2 (17)
± 128 x • (1 + 2 x) (1 -12 x) (1 -4 x) • (1 + 4 x) 1/2 • (1 -16 x) 1/2 (1 -88 x -112 x 2 -256 x 3 ) 2 .
These Hauptmoduls [START_REF] Kurihara | On some examples of equations defining Shimura curves and the Mumford uniformization[END_REF] are also given by the genus-zero quadratic relation

(256 x 3 + 112 x 2 + 88 x -1) 2 • H 2 ± -256 • x • (512 x 5 + 224 x 4 -400 x 3 -50 x 2 + 20 x -1) • H ± + 65536 x 6 = 0, (18)
and have the series expansions:

H -= -256 x -39936 x 2 -5116416 x 3 -595357696 x 4 -65525931776 x 5 -6954923846656 x 6 -719583708750336 x 7 + • • • H + = -256 x 5 -5120 x 6 -89600 x 7 -1433600 x 8 -22201600 x 9
-337755136 x 10 -5094679040

x 11 + • • • (19)
The relation between these two Hauptmoduls corresponds to a genus-zero q ↔ q 5 modular equation (q denotes the nome of the order-two operator).

This Heun function can also be written alternatively as:

Heun - 1 4 , 1 16 , 3 8 , 5 8 , 1, 1 2 , -4 x = A 1 • 2 F 1 [ 1 12 , 5 12 ], [1], H , (20)
using the identity

2 F 1 [ 1 8 , 3 8 ], [1], x = A 2 • 2 F 1 [ 1 12 , 5 12 ], [1], H , (21)
where A 1 , A 2 denote some algebraic functions and where:

H = 27 • (27 x 2 -414 x + 512) • x (9 x + 16) 3 - 54 • (81 x -256) • x (9 x + 16) 3 • (1 -x) 1/2 . (22)
• Example 2. Let us consider the rational function in four variables:

R(x, y, z, w) = 1 1 -(w x y + w x z + w y z + x y + x z + y + z) . ( 23
)
The diagonal of this rational function (23) reads:

Diag R(x, y, z, w) = 1 + 4 x + 48 x 2 + 760 x 3 + 13840 x 4 + 273504 x 5 + 5703096 x 6 + 123519792 x 7 + • • • (24)
The linear differential operator annihilating the diagonal of this rational function is the third order linear differential operator:

x 2 • (1 + x) • (1 -27 x) • D 3 x + 3 x • (1 -39 x -54 x 2 ) • D 2 x + (1 -86 x -186 x 2 ) • D x -4 • (1 + 6 x) (25)
This third order linear differential operator [START_REF] Maier | On rationally parametrized modular equations[END_REF] is the symmetric square of an ordertwo linear differential operator, having as solution a (square of a) Heun function given as series expansion with integer coefficients:

Heun - 1 27 , 2 27 , 1 3 , 2 3 , 1, 1 2 , -x 2
= 1 + 4 x + 48 x 2 + 760 x 3 + 13840 x 4 + 273504 x 5 + 5703096 x 6 + 123519792 x 7 + • • • (26)
We also have the following series expansion with integer coefficients:

Heun - 1 27 , 2 27 , 1 3 , 2 3 , 1, 1 2
, -x = 1 + 2 x + 22 x 2 + 336 x 3 + 6006 x 4

+ 117348 x 5 + 2428272 x 6 + 52303680

x 7 + • • • (27)
This Heun function [START_REF] Musty | A database of Belyi maps[END_REF] can be written as a pullbacked 2 F 1 hypergeometric function

Heun - 1 27 , 2 27 , 1 3 , 2 3 , 1, 1 2 , -x = (28) 25 -80 x -24 • (1 + x) 1/2 • (1 -27 x) 1/2 -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], H + ,

where the Hauptmodul H reads:

H ± = 864 • x • (1 -21 x + 8 x 2 ) • (1 -42 x + 454 x 2 -1008 x 3 -1280 x 4 ) (1 + 224 x + 448 x 2 ) 3 ± 864 • x • (1 -8 x) • (1 -2 x) • (1 -24 x) • (1 -16 x -8 x 2 ) × (1 + x) 1/2 • (1 -27 x) 1/2 (1 + 224 x + 448 x 2 ) 3 . ( 29
)
The series expansions of these two Hauptmoduls [START_REF] Tu | Algebraic Transformations of hypergeometric functions arising from theory of Shimura curves[END_REF] read respectively

H + = 1728 x -1270080 x 2 + 593381376 x 3 -226343666304 x 4 + 76907095308288 x 5 -24246668175851520 x 6 + 7253781581324351808 x 7 -2087529169324932180288 x 8 + • • • (30)
and:

H -= 1728 x 7 + 108864 x 8 + 4536000 x 9 + 158251968 x 10 (31)
+ 5017070016 x 11 + 150134378688 x 12 + 4328271255168 x 13 + • • • These two Hauptmoduls are the two solutions of the quadratic genus-zero relation:

1728 2 • x 8 + 1728 • (1 -21 x + 8 x 2 ) (1280 x 4 + 1008 x 3 -454 x 2 + 42 x -1) • x • H ± + (1 + 224 x + 448 x 2 ) 3 • H 2 ± = 0, (32)
and the two j-invariants (H ± = 1728/j ± ) are solution of the quadratic relation:

x 8 • j 2 ± + (1 -21 x + 8 x 2 ) (1280 x 4 + 1008 x 3 -454 x 2 + 42 x -1) • x • j ± + (1 + 224 x + 448 x 2 ) 3 = 0. ( 33
)
Denoting A = H + and B = H -and considering the two (identical) quadratic relations (32) Q(x, A) = 0 and Q(x, B) = 0, one easily gets by elimination of x (performing the resultant between Q(x, A) = 0 and Q(x, B) = 0 in x), the modular equation P (A, B) = 0. One gets a quite large modular equation (corresponding to q ↔ q 7 in the nome q, see ( 30) and ( 31)):

81600 9 • A 6 B 6 • (343 A 2 + 286 A B + 343 B 2 ) + • • • -2 36 3 18 • A B = 0. (34)
Note that this (symmetric) algebraic curve is a genus-zero curve.

Also note that the previous Heun function can be written alternatively with another algebraic Hauptmodul H (and another algebraic function A)

Heun - 1 27 , 2 27 , 1 3 , 2 3 , 1, 1 2 ,
-x = A • 2 F 1 [ 1 12 , 5 12 ], [1], H , (35)
where this alternative Hauptmodul is solution of a degree six equation

p 6 (x) 3 • (1 -2 x) 6 • H 6 + 3 • 1728 • x 4 • p 20 (x) • (1 -2 x) 3 • H 5 -1728 2 • x • p 23 (x) • H 4 + 1728 3 • x 3 • p 21 (x) • H 3 + 1728 4 • x 8 • p 16 (x) • H 2 -1728 5 • x 10 • p 14 (x) • H + 1728 6 • x 24 = 0, ( 36
)
where the polynomials p 6 (x), p

corresponding to q ↔ q 7 in the nome q.

Denoting A and B two Hauptmoduls solutions of the two identical degree six relations [START_REF] Kontsevich | Periods[END_REF], Q 6 (x, A) = 0 and Q 6 (x, B) = 0, one easily gets † the modular equation P (A, B) = 0. This modular curve is also a genus-one curve.

• Example 3. The rational function in four variables

R(x, y, z, w) = 1 1 -(y + z + w z + x y + x z + w x y) , ( 39
) † By elimination of x performing a resultant of Q 6 (x, A) and Q 6 (x, B) in x.
has a diagonal whose series expansion with integer ocoefficients reads:

Diag R(x, y, z, w) = 1 + 4 x + 60 x 2 + 1120 x 3 + 24220 x 4 + 567504 x 5

+ 14030016 x 6 + 360222720

x 7 + • • • (40)
The linear differential operator annihilating the diagonal of this rational function [START_REF] Hounkonnou | Generalized Heun and Lamé equations: factorization[END_REF] has order three:

4 + 96 • x -(1 -92 • x -864 • x 2 ) • D x -3 x • (1 -42 • x -256 • x 2 ) • D 2 x -x 2 • (1 + 4 x) • (1 -32 x) • D 3 x . (41)
This order-three linear differential operator is the symmetric square of an order-two linear differential operator:

L 2 = 8 x 2 • (4 θ + 3) • (4 θ + 1) + 2 • (14 θ 2 + 7 θ + 1) -θ 2 . ( 42
)
The solution of the linear differential operator [START_REF] Denef | Algebraic power series and diagonals[END_REF], analytic at x = 0, is thus given by the square of a Heun function which has a series expansion with integer coefficients:

Heun - 1 8 , 1 16 , 1 4 , 3 4 , 1, 1 2 , -4 x 2 = 1 + 4 x + 60 x 2 + 1120 x 3 + 24220 x 4 + 567504 x 5 + • • • (43)
The linear differential operator operator [START_REF] Denef | Algebraic power series and diagonals[END_REF] is the symmetric square of a linear differential operator of order two, such that one of its solutions can be written as a pullbacked 2 F 1 hypergeometric function:

Heun - 1 8 , 1 16 , 1 4 , 3 4 , 1, 1 2
, -4 x = 1 + 2 x + 28 x 2 + 504 x 3 + 10710 x 4

+ 248220 x 5 + 6091680 x 6 + 155580000 x 7 + 4092325500

x 8 + • • • = A (1) ± • 2 F 1 [ 1 6 , 2 3
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], H

± = A (2) ± • 2 F 1 [ 1 8 , (1)
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], H

± , (2)
where A

(1)

± , A (2)
± and the two Hauptmoduls H

± are square root algebraic functions:

H (1) ± = -54 x • 1 -19 x -200 x 2 (1 + 4 x) • (1 -50 x) 2 ± 54 • x • (1 -32 x) 1/2 • 1 -5 x (1 + 4 x) • (1 -50 x) 2 . ( 45
)
The two Hauptmoduls H

± are solutions of the quadratic relation:

(1 + 4 x) • (1 -50 x) 2 • (H (1)
± ) 2 -108 x • (200 x 2 + 19 x -1) • H (1) ± + 11664 x 3 = 0. ( 46
)
The two Hauptmoduls H

± in [START_REF] Barkatou | Apparent Singularities of Differential Systems with Rational Function Coefficients[END_REF] are also square root algebraic functions:

H (2) ± = -28 • x • 1 -30 x + 64 x 2 (1 -96 x) 2 ± 28 • x • (1 -16 x) • (1 + 4 x) 1/2 • (1 -32 x) 1/2 (1 -96 x) 2 , ( 47
)
solutions of the quadratic relation

(1 -96 x) 2 • (H (2)
± ) 2 + 256 x • (64 x 2 -30 x + 1) • H (2) ± + 65536 x 4 = 0, (48)
the algebraic function A

(1)

± being solution of 512 -27 • (1 -20 x) • (19 -312 x -6000 x 2 -80000 x 3 ) • Y + (1 + 4 x) 3 • (1 -50 x) 6 • Y 2 = 0, (49)
where Y = (A

± ) 18 , the algebraic function A

(2) 16 • Y 2 = 0, where:

± being solution of 1 + 2 • q 8 (x) • Y + 3 32 • (1 -96 x)
q 8 (x) = 92393273930231100473344 x 8 -182396792383587915661312 x 7

+ 7442201965961886564352 x 6 + 10564527655702470066176 x 5

-1994146206485388984320

x 4 + 154408466296830427136 x 3 (51)
-6048257896412868608

x 2 + 118593292086518528 x -926510094425921, where Y = (A (2)
± ) 64 . The series expansions of the Hauptmoduls H

± read:

H (1) -= -108 x -8208 x 2 -547776 x 3 -34193664 x 4 -2048523264 x 5 -119335292928 x 6 -6811411267584 x 7 -382782182326272 x 8 + • • • (52)
and:

H (1) + = -108 x 2 -2160 x 3 -56592 x 4 -1475712 x 5 -39711168 x 6 -1088716032 x 7 -30317739264 x 8 -854924599296 x 9 + • • • (53)
The relation between these two Hauptmoduls corresponds to the genus-zero modular equation:

625 A 3 B 3 -525 A 2 B 2 • (A + B) -96 A B • (A 2 + B 2 ) -3 A 2 B 2 -4 • (A 3 + B 3 ) + 528 • A B • (A + B) -432 • A B = 0, (54)
which can (for instance) be rationally parametrised as follows:

A(v) = 108 • v • (1 + v) 2 (16 + 15 v) • (2 + 3 v) 2 , B(v) = - 108 • (1 + v) • v 2 (4 + 3 v) • (32 + 33 v) 2 , ( 55
)
where A(v) and B(v) are related by an involution:

B(v) = A - 64 • (1 + v) 63 v + 64 , A(v) = B - 64 • (1 + v) 63 v + 64 . ( 56
)
The series expansions of the Hauptmoduls H

± read:

H (2) -= -56 x -9072 x 2 -1229256 x 3 -152418672 x 4 -17935321320 x 5 -2038883437584 x 6 -226173478925520 x 7 + • • • (57)
and

H (2) + = -56 x 3 -1680 x 4 -46872 x 5 -1291248 x 6 -35752752 x 7 -998627616 x 8 -28151491032 x 9 -800518405680 x 10 + • • • (58)
The relation between these last two Hauptmoduls H

(2)

± corresponds to the (genus-zero) modular equation:

640000 • A 2 B 2 • (9 A 2 + 14 A B + 9 B 2 ) + 4800 A B • (A + B) • (A 2 -1954 A B + B 2 ) + A 4 + B 4 -56196 A B • (A 2 + B 2 ) + 3512070 A 2 B 2 + 116736 • A B • (A + B) -65536 • A B = 0. ( 59
)
• Example 4. The rational function in four variables

R(x, y, z, w) = 1 1 -(w x z + w y + w z + x y + x z + y + z) , (60)
has a diagonal that reads:

Diag R(x, y, z, w) = 1 + 6 x + 114 x 2 + 2940 x 3 + 87570

x 4 + • • • (61)
The telescoper † of the diagonal (61) of this rational function of four variables (60) reads:

6 + 12 • x -(1 -144 • x -108 • x 2 ) • D x -x • (3 -198 • x -96 • x 2 ) • D 2 x -x 2 • (1 -44 • x -16 • x 2 ) • D 3 x , (62)
It is the symmetric square of an order-two linear differential operator which has a Heun solution analytic at x = 0. Consequently the order-three telescoper (62) has a square of a Heun solution. It has a series expansion with integer coefficients: + 3446781624 x 7 + 126047377170

Heun - 123 2 + 55 2 • 5 1/2 , - 33 8
x 8 + • • • (63)
The square of the Heun funcion (63), solution of (62), can be rewritten as a pullbacked 2 F 1 hypergeometric function:

A • 2 F 1 [ 1 12 , 7 12 ], [1], H 2 (64)
where A is an algebraic function and where the Hauptmodul H reads:

H = -864 • 7776 x 4 -12600 x 3 + 1890 x 2 -80 x + 1 (6480 x 2 + 540 x -1) 2 • x (65) + 864 • (1 -4 x) • (1 -18 x) • (1 -36 x) • (1 -44 x -16 x 2 ) 1/2 (1 -540 x -6480 x 2 ) 2 • x = -1728 x 5 -138240 x 6 -7793280 x 7 -383961600 x 8 -17716017600 x 9 + • • •
The pullback H is solution of the genus-zero quadratic relation:

(6480 x 2 + 540 x -1) 2 • H 2 + 1728 • (7776 x 4 -12600 x 3 + 1890 x 2 -80 x + 1) • x • H + 2985984 x 6 = 0. ( 66
)
Note that changing the sign of the square root in (65) (Galois conjugate) yields the alternative expansion:

-1728 x -1728000 x 2 -1388016000 x 3 -1005452352000 x 4

-686965980744000 x 5 -451977565258368000 x 6 + • • • (67) † By abuse of terminology we will call, everywhere in this paper, "telescoper" of a rational function R(x, y, z) the output of the creative telescoping program [6]. For instance, for a rational function of three variables R(x, y, z), we will call "telescoper" of a rational function R(x, y, z), what is stricto sensu, the telescoper of the rational function R(x/y, y/z, z)/(yz).

These two Hauptmoduls series (65) and (67) are related by the genus-zero modular equation:

383093207587837762627239936 • A 4 B 4 • (25 A 2 + 14 A B + 25 B 2 ) -331453065290799513600 • A 3 B 3 • (A + B) • (15047 A 2 + 31514658 A B + 15047 B 2 ) + 4480842240 A 2 B 2 • 144903770079 • (A 4 + B 4 ) -7730345599747300 • A B • (A 2 + B 2 ) + 401951713284567050 • A 2 B 2 + 3386880 • A B • (A + B) • 15047 • (A 4 + B 4 ) -419175723722072 • A B • (A 2 + B 2 ) -4206296569303686878 • A 2 B 2 + A 6 + B 6 -25 A B • 72243325686 • (A 4 + B 4 ) -38887039753371909735 • A B • (A 2 + B 2 ) + 17585442099134941585204 • A 2 B 2 + 1382400 • A B • (A + B) • 8142703 (A 2 + B 2 ) -149242947792862 • A B -1373552640 A B • 18909 (A 2 + B 2 ) -3621715210 • A B + 25386119331840 • A B • (A + B) -8916100448256 • A B = 0. ( 68
)
The square of the Heun solution (63) can also be rewritten as a pullbacked 2 F 1 hypergeometric function:

A 5/12 (x) • 2 F 1 [ 1 12 , 5 12 ]
The two Hauptmoduls read

H ± = 864 x • (3456 x 5 + 7776 x 4 -12600 x 3 + 1890 x 2 -80 x + 1) (144 x 2 + 216 x + 1) 3 ± 864 (1 -36 x) • (1 -18 x) (1 -4 x) x (144 x 2 + 216 x + 1) 3 • (1 -44 x -16 x 2 ) 1/2 , (71)
which expands respectively as:

H + = 1728 x -1257984 x 2 + 575828352 x 3 -214274336256 x 4 + • • • H -= 1728 x 5 + 138240 x 6 + 7793280 x 7 + 383961600 x 8 + • • • (72)
These two Hauptmoduls series (71) are related by a genus-zero modular equation which can be parametrized rationally ‡ as:

H + = 1728 z (z 2 + 10 z + 5) 3 , H -= 728 z 5 (z 2 + 250 z + 3125) 3 .
(73) ‡ It corresponds to N = 5 in Table 4 and Table 5 of [START_REF] Maier | On rationally parametrized modular equations[END_REF].

Note: The Heun function (63) resembles the Heun function associated with extreme rational surfaces [START_REF] Doran | Calabi-Yau Manifols Realizing Symplectically Rigid Monodromy Tuples[END_REF][START_REF] Malmendier | Higher Genus Curves in Mathematical Physics and Arithmetic Geometry[END_REF] (φ denotes the golden number (1 + √ 5)/2):

Heun 8 -5 φ 3 + 5 φ , 816 + 165 φ (3 + 5 φ) 3 , 1, 1, 1, 1, t (74)
= Heun -

123 2 + 55 2 • 5 1/2 , 3 2 • (145185 • 5 1/2 -324643), 1, 1, 1, 1, t .
This Heun function ( 63) is also reminiscent of the Heun function solution of the Apéry operator [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: integrality versus modularity Preprint[END_REF]:

Heun - 123 2 + 55 2 • 5 1/2 , - 33 2 + 15 2 • 5 1/2 , 1, 1, 1, 1, 1 2 • (11 -5 • 5 1/2 ) • x = 1 + 3 x + 19 x 2 + 147 x 3 + 1251 x 4 + 11253 x 5 + 104959 x 6 + • • • (75)
This Heun function (75) can be rewritten as a pullbacked hypergeometric function:

1 (1 -12 x + 14 x 2 + 12 x 3 + x 4 ) 1/4 × 2 F 1 [ 1 12 , 5 12 ], [1], 1728 • x 5 • (1 -11 x -x 2 ) (1 -12 x + 14 x 2 + 12 x 3 + x 4 ) 3 , ( 76
) or 1 (1 + 228 x + 494 x 2 -228 x 3 + x 4 ) 1/4 × 2 F 1 [ 1 12 , 5 12 ], [1], 1728 • x • (1 -11 x -x 2 ) 5 (1 + 228 x + 494 x 2 -228 x 3 + x 4 ) 3 , (77)
where the Hauptmodul in (76) can be written

1728 x (x 2 + 10 x + 5) 3 • 1 -11 x -x 2 x = 1728 x 5 (x 2 + 250 x + 3125) 3 • 125 x 1 -11 x -x 2 , ( 78
)
when the Hauptmodul in (77) can be written:

1728 x 5 (x 2 + 250 x + 3125) 3 • 1 -11 x -x 2 x = 1728 x (x 2 + 10 x + 5) 3 • 125 x 1 -11 x -x 2 .
(79)

• Example 5. The rational function in four variables

R(x, y, z, w) = 1 1 -(y + z + w z + x y + x z + w x y + w x y z) , (80)
has a diagonal that reads:

Diag R(x, y, z, w) = 1 + 5 x + 73 x 2 + 1445 x 3 + 33001 x 4 + • • • (81)
The telescoper of the diagonal of this rational function (80) of four variables reads:

L 3 = x 2 • (1 -34 x + x 2 ) • D 3 x + 3 x • (1 -51 x + 2 x 2 ) • D 2 x + (1 -112 x + 7 x 2 ) • D x + x -5. ( 82
)
It is the symmetric square of an order-two linear differential operator with a Heun solution, analytic at x = 0. Consequently the diagonal of (80), solution of (82), can be written in terms of the square of two (Galois conjugate) Heun functions which have a series expansion with integer coefficients:

(1 -34 x + x 2 ) × Heun 577 + 408 • 2 1/2 , 663 2 + 234 • 2 1/2 , 3 2 , 3 2 , 1, 3 2 , ( 17
+ 12 • 2 1/2 ) • x 2 = (1 -34 x + x 2 ) × Heun 577 -408 • 2 1/2 , 663 2 -234 • 2 1/2 , 3 2 , 3 2 , 1, 3 2 , ( 17
-12 • 2 1/2 ) • x 2 = 1 + 5 x + 73 x 2 + 1445 x 3 + 33001 x 4 + 819005 x 5 + 21460825 x 6 + 584307365 x 7 + • • • (83)
It can also be written as a pullbacked 2 F 1 hypergeometric function

A -• 2 F 1 [ 1 3 , 2 3 ], [1], H - 2 , ( 84
)
where the Hauptmodul H ± reads

H ± = 1 -24 x + 30 x 2 + x 3 2 • (1 + x) 3 ± 1 -7 x + x 2 2 • (1 + x) 3 • (1 -34 x + x 2 ) 1/2 , ( 85
)
with the expansions:

H -= 27 x 2 + 648 x 3 + 15471 x 4 + 389016 x 5 + 10234107 x 6 + 278861616 x 7 + 7808397759 x 8 + 223397228880 x 9 + • • • (86) 1 -H + = 27 x -81 x 2 + 891 x 3 + 15039 x 4 + 389691 x 5 + • • • (87)
and where the algebraic factor A -reads:

A -= 3 2 • 1 -x (1 + x) 2 - (1 -34 x + x 2 ) 1/2 2 • (1 + x) 2 (88) = 1 + 5 x + 61 x 2 + 1097 x 3 + 23737 x 4 + 569549 x 5 + • • •
The two series (86) and (87) are related by the (symmetric) modular equation

8 A 3 B 3 -12 B 2 A 2 • (A + B) + 3 AB • (2 A 2 + 13 AB + 2 B 2 ) -(A + B) • (A 2 + 29 AB + B 2 ) + 27 AB = 0. (89)
• Example 6. Let us consider the following rational function in four variables x, y, z and w

R(x, y, z, w) = 1 1 -(y + z + w y + x z + w x y + w x z) , (90)
or the rational function:

R(x, y, z, w) = 1 1 + x y + y z + z w + w x + y w + x z . ( 91
)
The diagonals of these two rational functions (90), (91) give the same series expansion with integer coefficients:

Diag R(x, y, z, w) = 1 + 6 x + 90 x 2 + 1860 x 3 + 44730 x 4 + 1172556 x 5

+ 32496156 x 6 + 936369720

x 7 + • • • (92)
The order-three linear differential operator annihilating this series (92) is the symmetric square of the linear differential operator of order two, and is given by:

L 3 = x 2 • (1 -36 x) • (1 -4 x) • D 3 x + 3 x • (1 -60 x + 288 x 2 ) • D 2 x + (1 -132 x + 972 x 2 ) • D x -6 • (1 -18 x). (93)
The solution of this order-three telescoper L 3 reads:

Heun 1 9 , 1 12 , 1 4 , 3 4 , 1, 1 2 , 4 x 2 = (1 -4 x) • Heun 1 9 , 5 36 , 3 4 , 5 4 , 1, 3 2 , 4 x 2 = 1 + 6 x + 90 x 2 + 1860 x 3 + 44730 x 4 + 1172556 x 5 + • • • (94)
This series expansion (94) already occurred in [START_REF] Glasser | Lattice Green function (at 0) for the 4D hypercubic lattice[END_REF] for the simple cubic lattice Green function. This Heun function ( 94) is quite simply related ‡ to the Heun function of example 3:

Heun - 1 8 , 1 16 , 1 4 , 3 4 , 1, 1 2 , -4 x 2 = (1 + 4 x) -1/2 • Heun 1 9 , 1 12 , 1 4 , 3 4 , 1, 1 2 , 4 x 1 + 4 x 2 . ( 95
)
This Heun function in (94) can also be written as a pullbacked 2 F 1 hypergeometric function

Heun 1 9 , 1 12 , 1 4 , 3 4 , 1, 1 2 , 4 x (96)
= 1 + 3 x + 81 2 x 2 + 1617 2 x 3 + 152955 8 x 4 + 3969405 8 x 5 + • • • = A (1)
± • 2 F 1 [ 1 6 , 2 3
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], H

± = A (1)
± • 2 F 1 [ 1 8 , (2)
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], H

± , (2)
where H

± , H

± are algebraic functions expressed in terms of square roots. The detailed calculations which are similar to the ones of example 3 are given in Appendix D. As a consequence of identity (95), there is a close relation between example 3 and example 6 and of course the simple cubic lattice Green functions (see [START_REF] Glasser | Lattice Green function (at 0) for the 4D hypercubic lattice[END_REF]).

A few comments on example 6

Considering the second form for the rational function of example 6, namely (91), it is straightforward to see that the one-parameter family of rational functions

R λ (x, y, z, w) = 1 1 + λ • (x y + y z + z w + w x + y w + x z) , (98)
has a diagonal † deduced from (94)

Heun 1 9 , 1 12 , 1 4 , 3 4 , 1, 1 2 , 4 λ 2 x 2 = 1 + 6 λ 2 x + 90 λ 4 x 2 + • • • , (99)
the telescoper of (98) being the pullback of the order-three linear differential operator (93) by x → λ 2 • x. ‡ This corresponds to the fact that the order-three linear differential operator of example 3 by ( 41)

is equal to (1 + 4 x) -1/2 • pullback(L 3 , x/(1 + 4 x)) • (1 + 4 x) 1/2
where L 3 is the order-three linear differential operator given by (93). † Deduced, without any calculation, for the scaling transformation (x, y, z, u)

→ (λ 1/2 • x, λ 1/2 • y, λ 1/2 • z, λ 1/2 • u).
Let us now consider the rational function

R(x, y, z, w) = 1 1 + x y z w • (x y + y z + z w + w x + y w + x z) , (100)
deduced from (91) by the following monomial ‡ transformation:

(x, y, z, w) -→ (yzw, xzw, xyw, xyz). (101)
It is straightforward to see, from its definition, that the diagonal of the rational function (100), actually corresponds to the diagonal of the rational function (98) where the parameter λ is taken to be equal to the product λ = xyzu, thus reading the formula (99) with λ = x:

Heun 1 9 , 1 12 , 1 4 , 3 4 , 1, 1 2 , 4 x 3 2 = 1 + 6 x 3 + 90 x 6 + 1860 x 9 + • • • (102)
One verifies that the telescoper of the rational function ( 100) is actually the pullback ¶ of the order-three linear differential operator (93) by x → x 3 . In contrast let us now consider the rational function

R(x, y, z, w) = x wx 3 y + x 3 yz + wxz + x 2 y + x + z + w , (103)
deduced from (91) by the following (involutive) birational monomial transformation:

(x, y, z, w) -→ 1 x , x 2 y, z, w . (104)
The telescoper of ( 103) is actually the same telescoper as the one for (91), namely the order-three linear differential operator L 3 given by ( 93). This telescoper has, thus, the Heun function (94) as a solution. However (94) does not coincide with the diagonal † † of the rational function ( 103). The solutions of the telescoper of a rational function and the diagonal of a rational function are two concepts which do not necessarily coincide.

We will underline this point several times in this paper (see section 3, Appendix L.1 and Appendix L.2).

Comment on the simplicity of the rational functions yielding Heun functions

Let us consider the rational function (39) of example 3, and perform the simple transformation (x, y, z, w) → (1 + x, 1 + y, 1 + z, 1 + w) on [START_REF] Hounkonnou | Generalized Heun and Lamé equations: factorization[END_REF]. One obtains that way the following new rational function:

- 1 wxy + wx + wy + wz + 2 xy + xz + 2 w + 3 x + 3 y + 3 z + 5 . ( 105
)
The telescoper of this quite simple rational function ( 105) is a very large order-nine linear differential operator of degree 48 in x. Along this line let us consider the rational function

R(x, y, z, w) = 1 1 -(y + 2 z + w z + x y + x z + w x y) , (106)
‡ Note a typo in the footnote of section 4.2 in [START_REF] Abdelaziz | Diagonals of rational functions, pullbacked 2 F 1 hypergeometric functions and modular forms (unabridged version)[END_REF][START_REF] Abdelaziz | Diagonals of rational functions, pullbacked 2 F 1 hypergeometric functions and modular forms[END_REF]. A determinant equal ± 1 condition is missing to have a birational transformation. A monomial transformation like (101) is not birational (the determinant is -3 here).

¶ Performing the same monomial change of variable (101) on the rational function (90) instead of (91), one gets, as it should, the same telescoper pullback of the order-three linear differential operator (93) by x → x 3 , with the same diagonal series (102). † † Which is not well-defined (depending on the ordering of the four variables).

which corresponds to a very simple modification † of the rational function ( 39) of example 3. The telescoper of this new simple rational function ( 106) is an irreducible order-five linear differential operator ¶. Again we are far from having diagonals of rational functions and solutions of telescopers that can be simply expressed as Heun functions (or even solutions of order-two linear differential operators). Considering simple rational functions of the form 1/P (x, y, z, w) with a polynomial P (x, y, z, w) of degree at most one in x, y, z, w is far from being sufficient to get Heun functions.

Conversely let us consider the slightly more involved rational function

R(x, y, z, w) = (107) 1 + x 1 + x -2 y -2 z -3 xy -3 xz -uxy -uxz -x 2 y -x 2 z -ux 2 y .
The diagonal of the rational function (107) reads:

2 F 1 [ 1 2 , 1 2 ], [1], 32 x = 1 + 8 x + 144 x 2 + 3200 x 3 + 78400 x 4 + • • • (108)
The telescoper of the rational function ( 107) is an order-six linear differential operator which is the direct sum of two order-three linear differential operators

L 6 = L 3 ⊕ M 3 ,
where L 3 factorises into the product of an order-one and an order-two linear differential operator

L 3 = L 1 • L 2 ,
and where M 3 is exactly ‡ the order-three linear differential operator [START_REF] Denef | Algebraic power series and diagonals[END_REF] which has the Heun solution [START_REF] Chen | Desingularization of Ore Operators[END_REF], namely:

Heun - 1 8 , 1 16 , 1 4 , 3 4 , 1, 1 2 , -4 x 2
= Heun -8, 1 2 , 1 4 , 3 4 , 1, 1 2 , -32 x 2 . (109)
The diagonal of the rational function ( 107) is, in fact, solution of the order-two linear differential operator L 2 , which is thus, the miminal order linear differential operator operator annihilating the diagonal (108). The creative telescoping method yields a higher order telescoper which provides a "companion" order-three linear differential operator M 3 with the same square of Heun solution † † (109) as example 3.

Periods of extremal rational surfaces

Let us now introduce the rational function in just three variables:

R(x, y, z) = 1 1 + x + y + z + x y + y z -x 3 y z . ( 110
)
The diagonal of this rational function (110) has the following series expansion:

Diag R(x, y, z) = 1 -2 x + 6 x 2 -11 x 3 -10 x 4 + 273 x 5 -1875 x 6 + 9210 x 7 -34218 x 8 + 78721 x 9 + 108581 x 10 + • • • (111)
In order to find the diagonal of this rational function of three variables, one gets the telescoper annihilating this diagonal using creative telescoping [6]. This telescoper is actually an order-four linear differential operator L 4 which, not only factorizes into two order-two linear differential operators, but is actually the direct sum (LCLM) of † We have just changed one coefficient in the denominator of (39): the z term becomes a 2 z term.

¶ This order-five linear differential operator is homomorphic to its adjoint, its differential Galois group being SO(5, C). ‡ It is thus is the symmetric square of the order-two linear differential operator [START_REF] Maier | identities[END_REF] which has the Heun solution [START_REF] Barkatou | Apparent Singularities of Differential Systems with Rational Function Coefficients[END_REF]. † † Which corresponds to a "Period" over another cycle than the evanescent cycle of the diagonal.

two † order-two linear differential operators L 4 = L 2 ⊕ M 2 . These two order-two linear differential operators read respectively

L 2 = 27 x 2 • (θ + 1) 2 + 3 x • (3 θ 2 + 3 θ + 1) + θ 2 (112) = (1 + 9 x + 27 x 2 ) • x 2 • D 2 x + (1 + 9 x) 2 • x • D x + 3 x • (1 + 9 x)
, and:

M 2 = (1 + 9 x + 27 x 2 ) • (5 + 18 x) • (1 -2 x) • x 2 • D 2 x + (5 + 70 x + 261 x 2 -756 x 3 -2916 x 4 ) • x • D x + x • (1 -9 x) • (5 + 60 x + 108 x 2 ). ( 113
)
Note that L 2 and M 2 share exactly the same singularities x = 0, ∞ and 1 + 9 x + 27 x 2 = 0. In contrast, the factor (5 + 18 x) in (113) corresponds to an apparent singularity, when the factor (1 -2 x) corresponds to a true singularity. One can get rid of the 5 + 18 x = 0 apparent singularity performing the following desingularization

L 2 → L 3 = L 1 • L 2
, changing the order-two operator L 2 into an order-three linear differential operator L 3 , the order-one operator L 1 reading:

L 1 = D x + A (x) A(x)
where: (114)

A(x) = (5 + 18 x) • (1 -2 x) • (1 + 9 x + 27 x 2 ) • x 2/7 .
The solution of the order-two linear differential operator L 2 has the following Heun function ‡ solution, analytic at x = 0:

S 1 = Heun 1 2 - i √ 3 2 , 1 2 - i √ 3 6 , 1, 1, 1, 1, 3 2 • -3 + i √ 3 • x (115) = 1 -3 x + 9 x 2 -21 x 3 + 9 x 4 + 297 x 5 -2421 x 6 + 12933 x 7
-52407 x 8 + 145293 x 9 -35091 x 10 -2954097

x 11 + • • • (116)
This Heun function (115) can also be written alternatively in terms of other 2 F 1 hypergeometric functions: 3 , † These two order-two linear differential operators L 2 and M 2 are not homomorphic. ‡ This Heun function Heun(a, q, α, β, γ, δ, ρ x) is such that q = a/(1 + a), q/ρ = -1/9, a/ρ 2 = 1/27, 1/ρ and a/ρ being complex conjugate.

Heun 1 2 - i √ 3 2 , 1 2 - i √ 3 6 , 1, 1, 1, 1, 3 2 • -3 + i √ 3 • x = 1 1 + 3 x • 2 F 1 [ 1 3 , 2 3 ], [1], 27 • x 3 (1 + 3 x) 3 ) = 1 1 + 9 x + 27 x 2 -27 x 3 1/3 • 2 F 1 [ 1 6 , 2 3 ], [1], - 108 • x 3 • (1 + 9 x + 27 x 2 ) (1 + 9 x + 27 x 2 -27 x 3 ) 2 = 1 1 + 3 x 1/4 • 1 1 + 9 x + 27 x 2 + 3 x 3 1/4 (117) × 2 F 1 [ 1 12 , 5 12 ], [1], 1728 • x 9 • (1 + 9 x + 27 x 2 ) (1 + 3 x) 3 • (1 + 9 x + 27 x 2 + 3 x 3 ) 3 = (1 + 9 x) -1/4 • (1 + 3 x) -1/4 • (1 + 27 x 2 ) -1/4 × 2 F 1 [ 1 12 , 5 12 ], [1], 1728 • x 3 • (1 + 9 x + 27 x 2 ) 3 (1 + 3 x) 3 • (1 + 9 x) 3 • (1 + 27 x 2 )
= (1 + 9 x) -1/4 • (1 + 243 x + 2187 x 2 + 6561 x 3 ) -1/4 × 2 F 1 [ 1 12 , 5 12 ], [1], 1728 • x • (1 + 9 x + 27 x 2 ) (1 + 9 x) 3 • (1 + 243 x + 2187 x 2 + 6561 x 3 ) 3 .
Note that the Hauptmoduls in (117) can be rewritten as the composition of two pullbacks:

1728 • x 9 • (1 + 9 x + 27 x 2 ) (1 + 3 x) 3 • (1 + 9 x + 27 x 2 + 3 x 3 ) 3 = 1728z 3 (z + 27) (z + 243) 3 • 729 x 3 1 + 9 x + 27 x 2 , ( 118
) 1728 • x 3 • (1 + 9 x + 27 x 2 ) 3 (1 + 3 x) 3 • (1 + 9 x) 3 • (1 + 27 x 2 ) 3 = 1728z (z + 27) (z + 3) 3 • 729 x 3 1 + 9 x + 27 x 2 , ( 119
) 1728 • x • (1 + 9 x + 27 x 2 ) (1 + 9 x) 3 • (1 + 243 x + 2187 x 2 + 6561 x 3 ) 3 = 1728z (z + 27) (z + 3) 3 • 729 x • (1 + 9 x + 27 x 2 ) . ( 120
)
The modular equation relating the Hauptmodul (118) with the Hauptmodul (120) corresponds to q ↔ q 9 in the nome q (see also Table 4 and Table 5 in [START_REF] Maier | On rationally parametrized modular equations[END_REF]). This Heun function ( 117) is in fact the period of an extremal rational surface [START_REF] Doran | Calabi-Yau Manifols Realizing Symplectically Rigid Monodromy Tuples[END_REF], and was shown to be related ¶ to classical modular forms in table 15 in [START_REF] Maier | On rationally parametrized modular equations[END_REF] for N = 9:

Heun -9 ∓ 3 √ 3 i -9 ± 3 √ 3 i , 9 ± 3 √ 3 i 18 , 1, 1, 1, 1, 2 x -9 ± 3 √ 3 i = Heun 1 ± √ 3 i 2 , 3 ± √ 3 i 6 , 1, 1, 1, 1, -3 ∓ √ 3 i 18 • x . (121)
The other order-two linear differential operator M 2 has the following (classical modular form, see Appendix G) pullbacked 2 F 1 hypergeometric solution ‡ analytic at x = 0:

S 2 = 1 (1 + 4 x -2 x 2 -36 x 3 + 81 x 4 ) 1/4 • × 2 F 1 [ 1 12 , 5 12 ], [1], 1728 • x 5 • (1 + 9 x + 27 x 2 ) • (1 -2 x) 2 (1 + 4 x -2 x 2 -36 x 3 + 81 x 4 ) 3 = 1 -x + 3 x 2 -x 3 -29 x 4 + 249 x 5 -1329 x 6 + 5487 x 7 -16029 x 8 + 12149 x 9 + 252253 x 10 + • • • (122)
This second order-two linear differential operator M 2 is not homomorphic to the previous one † †. The Hauptmoduls H 1 for L 2 (see ( 118)) and H 2 for M 2 (see ( 122))

H 1 = 1728 • x • (1 + 9 x + 27 x 2 ) (1 + 9 x) 3 • (1 + 243 x + 2187 x 2 + 6561 x 3 ) 3 , ( 123
) ¶ Change x → x/27 to match S 1 ,
given by (115), with (121). ‡ It seems that this pullbacked 2 F 1 hypergeometric (122) cannot be seen as a (simple) Heun function: see Appendix G. † † They cannot be homomorphic: they do not have exactly the same singularities. The order-two linear differential operator M 2 has the extra x = 1/2 singularity.

H 2 = 1728 • x 5 • (1 + 9 x + 27 x 2 ) • (1 -2 x) 2 (1 + 4 x -2 x 2 -36 x 3 + 81 x 4 ) 3 , (124)
are not simply related †. They just both vanish at 1 + 9 x + 27 x 2 = 0. A rational parametrisation is introduced in Appendix G for these two order-two linear differential operators making clear the differences and similarities of these two linear differential operators.

One finds that the diagonal of ( 110) is actually the half-sum of the two series (115) and (122):

Diag R(x, y, z) = S 1 + S 2 2 . ( 125
)
The order-four linear differential operator L 4 = L 2 ⊕ M 2 is thus the minimal order telescoper. The diagonal of the rational function ( 110) is the sum of two classical modular forms.

Remark 1: The previous results can also be understood as follows. The telescoper of the rational function

R(x, y, z) = x 1 + x + y + z + x y + y z -x 3 y z , (126)
similar to (110) (where the numerator of the rational function has been changed from 1 to x) is actually the same as the one for (110), namely L 4 = L 2 ⊕M 2 . The diagonal of (127) reads:

Diag R(x, y, z) = S 2 -S 1 2 = x -3 x 2 + 10 x 3 -19 x 4 -24 x 5 + 546 x 6 -3723 x 7 + 18189 x 8 -66572 x 9 + 143672 x 10 + • • • (127)
The telescoper of the rational function of three variables

R(x, y, z) = 1 -x 1 + x + y + z + x y + y z -x 3 y z , (128)
is the order-two linear differential operator L 2 with the hypergeometric solution S 1 .

The telescoper of the rational function of three variables

R(x, y, z) = 1 + x 1 + x + y + z + x y + y z -x 3 y z , (129)
is the order-two linear differential operator M 2 with the hypergeometric solution S 2 . Note however that the telescoper of the rational function

R(x, y, z) = x y 1 + x + y + z + x y + y z -x 3 y z , (130)
is an order-five linear differential operator L 5 which is the direct sum of the order-two operator L 2 and an order-three linear differential operator

L 3 = N 2 • D x , namely L 5 = L 2 ⊕ (N 2 • D x )
where N 2 is non-trivially homomorphic to M 2 (with new apparent singularities 36 x 2 -6 x + 1 = 0 and (1 + 3 x) = 0). The series expansion of the diagonal of this last rational function is non trivial and reads:

Diag R(x, y, z) = -x + 3 x 2 -10 x 3 + 23 x 4 -6 x 5 -378 x 6

+ 3009 x 7 -15993 x 8 + 64394 x 9 -175102 x 10 + • • • (131) † Their associated nomes are not simply related. However, one can imagine that these Hauptmoduls (123), (124) are Igusa invariants of an algebraic surface (for instance a split Jacobian of a genus-two algebraic curve).

Remark 2: All the previous Heun functions occurring as diagonals of simple rational functions can all be rewritten in terms of pullbacked 2 F 1 hypergeometric functions which turn out to correspond to classical modular curves. These 2 F 1 hypergeometric functions are not arbitrary, they are "special" 2 F 1 's corresponding to selected parameters, namely 2 F 1 's related to classical modular curves. Appendix F gives a simple condition on the nome of these 2 F 1 's to be related to classical modular curves. In Appendix H we give the exhaustive list of these 28 hypergeometric 2 F 1 's related to classical modular curves.

Derivatives of classical modular forms

Let us recall example 6, and let us consider, instead of the rational function (91), its homomogeous partial derivative with respect to one of its four variables:

x • ∂R(x, y, z, w) ∂x = x • (y + z + w) (1 + x y + y z + z w + w x + y w + x z) 2 . ( 132
)
The telescoper of this rational function ( 132) is an order-three linear differential operator M 3 which is homomorphic to the order-three operator L 3 given by (93) which was the telescoper of the rational function (91). This homomorphism reads:

M 3 • θ = L 1 • L 3
where:

L 1 = (1 -18 x) • θ + 18 x, ( 133
)
where θ is the homogeneous derivative θ = x • D x . Consequently the solutions of the order-three linear differential operator M 3 are simply obtained by taking the homogeneous derivative θ = x • D x of the solutions of the order-three linear differential operator L 3 . In particular, the diagonal of the rational function ( 132) is the homogeneous derivative of the diagonal of the rational function (91):

Diag x • ∂R(x, y, z, w) ∂x = x • d dx Diag R(x, y, z, w) , (134)
The diagonal of (132) will thus be the homogeneous derivative of the classical modular form (94). This is a general result on diagonals of rational functions. We have the following identity valid for any order-N linear differential operator

L Diag L R(x, y, z, w) = L Diag R(x, y, z, w) , (135)
where:

L = N n=0 P n (x) • θ n , L = N n=0 P n (x y z w) • Θ n , (136)
with:

θ = x • d dx , • • • Θ = w • ∂ ∂w , (137)
where the P n 's are polynomials. This identity can, of course be generalised to the diagonal of rational functions of an arbitrary number of variables. For any Heun function or classical modular form of this paper, obtained as a diagonal of a rational function, we can use these identities (134), (135) to get other rational functions that will not be Heun functions or classical modular forms, but derivatives of Heun functions or classical modular forms.

Note that the derivative of a classical modular form, or more generally an orderone linear differential operator like (136) acting on a classical modular form, is no longer a classical modular form. With this example we see that a Heun function which has a series expansion with integer coefficients (or more generally is globally bounded series), is not necessarily a classical modular form, but can be an order-one linear differential operator acting on a classical modular form.

A simple example of diagonals of rational functions of three variables, corresponding to derivatives of Heun functions, is given in Appendix I. Along this line see also Appendix H.2.

Heun function solutions of telescopers of rational functions related to Shimura curves

The rational function of four variables

R(x, y, z, u) = x y z 1 -x y z u + x y z • (x + y + z) + x y + y z + x z , (138)
has a telescoper that is a linear differential operator of order three:

L 3 = 8 x • (1 -x) • (1 -4 x) • D 3 x + 12 • (1 -10 x + 12 x 2 ) • D 2 x -6 • (7 -17 • x) • D x + 3, ( 139
)
which corresponds to the symmetric square of an order-two linear differential operator reading in terms of the homogeneous derivative θ:

x 2 • (8 θ + 3) • (8 θ + 1) -x • (80 θ 2 + 1) + 8 • θ • (2 θ -1). ( 140
)
The solutions of order-three linear differential operator L 3 are, thus, expressed in terms of the following Heun functions

Heun 1 4 , 1 64 , 1 8 , 3 8 , 1 2 , 1 2 , x 2
, x • Heun 1 4 , 21 64 , 5 8 , 7 8 , 3 2 , 1 2 , x 2 , (141)
or: it is different from this solution of the telescoper (139). The two concepts, namely being the diagonal of a rational function and being the solution of the telescoper of that rational function do not necessarily identify. The solutions of the telescoper are n-fold integrals of that rational function integrand over all possible cycles: a solution like (141) is thus a "Period" of an algebraic variety corresponding to a particular nonevanescent cycle. It is different from the diagonal of that rational function which is a "Period" over evanescent cycles.

x 1/2
In contrast with all the other (square of) Heun functions of this paper, which are associated with classical modular forms, the series expansion (143) is not globally bounded: it cannot be recast ¶ into a series with integer coefficients. It cannot be a diagonal of a rational function: it is only a solution of the telescoper of a rational ¶ After a rescaling of the variable.

function. The order-three linear differential operator (139), is the symmetric square of the linear differential operator of order two L 2 :

L 2 = D 2 x + 1 -10 x + 12 x 2 2 x • (1 -4 x) • (1 -x) • D x - 1 -3 x 16 • x • (1 -4 x) • (1 -x) , (144)
whose (formal) series expansions at 0, 1, and ∞ do not contain ‡ logarithms. This order-two linear differential operator L 2 admits the solutions:

x 1/2 • (1 -x) -7/8 • 2 F 1 [ 7 24 , 11 24 ], [ 5 4
], 27 4

• x 2 (1 -x) 3 , ( 145
) (1 -x) -1/8 • 2 F 1 [ 1 24 , 5 24 ], [ 3 4 ], 27 4 • x 2 (1 -x) 3 .
The precise correspondence with the Heun functions in (141) reads:

Heun 1 4 , 1 64 , 1 8 , 3 8 , 1 2 , 1 2 , x = (1 -x) -1/8 • 2 F 1 [ 1 24 , 5 24 ], [ 3 4 ], 27 4 • x 2 (1 -x) 3 , (146)
Heun 1 4 , 21 64 , 5 8 , 7 8 , 3 2 , 1 2 , x = (1 -x) -7/8 • 2 F 1 [ 7 24 , 11 24 ], [ 5 4 ], 27 4
• x 2 (1 -x) 3 . (147)
The pullbacks in all the 2 F 1 hypergeometric functions of this paper are special rational (or algebraic) functions: they correspond to the concept of Belyi maps † †. In this case which does not correspond to a classical modular form but a (Shimura) automorphic form †, the pullback 27 4 • x 2 (1 -x) 3 in (145) being also "special" (see Appendix J.1). The two solutions of the linear differential operator (144) can be used to construct a basis for space of automorphic forms, which can then be used to construct Hecke operators relative to this basis ¶. The second solution in (145) corresponds to an automorphic form associated with a Shimura curve with signature (0, 4, 2, 6) which appears in Table 1 in [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF]. More details on Heun, or 2 F 1 , automorphic forms associated with Shimura curves [START_REF] Tu | Algebraic Transformations of hypergeometric functions arising from theory of Shimura curves[END_REF][START_REF] Tu | Algebraic Transformations of Hypergeometric Functions and automorphic Forms on Shimura Curves[END_REF][START_REF] Vidunas | A Classification of coverings yielding Heun-to-hypergeometric Reductions[END_REF] are given in Appendix J. Note in particular the fact that there exists an algebraic series y(x) corresponding to a modular equation, such that the two 2 F 1 hypergeometric functions (146), (147) actually verify the following identity/symmetry:

w 3/8 • ρ • y (x) 1/2 • x 3/8 • (1 -x) 1/4 • 2 F 1 [ 1 24 , 5 24 ], [ 3 4 ], x = y(x) 3/8 • (1 -y(x)) 1/4 • 2 F 1 [ 1 24 , 5 24 ], [ 3 4 ], y(x) , (148)
where the two complex numbers w and ρ are on the unit circle. More details are given in Appendix J. Such an identity is reminiscent of the hypergeometric identities we studied in [START_REF] Abdelaziz | Modular forms, Schwarzian conditions, and symmetries of differential equations in physics[END_REF] for classical modular forms. ‡ We have three elliptic points. † † An important area where Belyi functions [START_REF] Van Hoeij | Belyi functions for hyperbolic Hypergeometric-to-Heun transformations[END_REF][START_REF] Slisling | On computing Belyi maps[END_REF][START_REF] Musty | A database of Belyi maps[END_REF] appear is precisely Shimura curves. Any Belyi covering gives a modular curve with respect to some (not necessarily congruence) subgroup.

† They lie at the crossroads of many areas of mathematics. They have played an important role in the proof of Fermat's last theorem. A Shimura curve is simply a Riemann surface which is uniformized by an arithmetic Fuchsian group. ¶ See example 9 in [START_REF] Yang | Schwarzian differential equations and Hecke eigenforms on Shimura curves[END_REF] for more details.

This "Shimura" 2 F 1 hypergeometric function can be seen to correspond to other Heun functions than (146) or (147). Using the general identity [START_REF] Valent | Difference Equations, Special Functions and Orthogonal Polynomials[END_REF][START_REF] Maier | On reducing the Heun equation to the hypergeometric equation[END_REF] Heun(2, 16 a b, 4 a, 4 b, a + b + 1/2, 2 (a + b), x)

= 2 F 1 [a, b], [a + b + 1 2 ], 4 • x • (2 -x) • (1 -x) 2 , (149)
one deduces the identity

Heun 2, 5 36 , 1 6 , 5 6 , 3 4 , 1 2 , x = 2 F 1 [ 1 24 , 5 24 ], [ 3 4 ], 4 • x • (2 -x) • (1 -x) 2 , ( 150
)
as well as the identity:

Heun 2, 77 36 , 7 6 , 11 6 , 5 4 , 3 2 , x
= 2 F 1 [ 7 24 , 11 24 ], [ 5 4
], 4

• x • (2 -x) • (1 -x) 2 . ( 151
)
More generally, Heun functions, related to Shimura curves, often emerge in the context of Belyi maps where Heun functions with four singularities, are expressed as pullbacked 2 F 1 hypergeometric functions. For example Table 3.4.4 of [START_REF] Van Hoeij | Belyi functions for hyperbolic Hypergeometric-to-Heun transformations[END_REF] (see also [34]), corresponds to the 2

F 1 hypergeometric function 2 F 1 ([ 1 3 , 1 12 ], [ 3 4 ], x) with three different pullbacks 2 F 1 [ 1 3 , 1 12 ], [ 3 4
],

x 4 • (x 2 -3) 1 -3 x 2 = (1 -3 x 2 ) 1/3 • Heun 1 9 , 1 6 , 1 2 , 1, 1 2 , 3 4 , x 2 3 , (152)
namely to the pullback B3 in the Table 3.4.4 in [START_REF] Van Hoeij | Belyi functions for hyperbolic Hypergeometric-to-Heun transformations[END_REF]. In the paper [START_REF] Van Hoeij | Belyi functions for hyperbolic Hypergeometric-to-Heun transformations[END_REF] most of the 2 F 1 hypergeometric functions are in fact associated with Shimura curves: all tables ‡ except 2.3.13, 2.3.14, 2.5.7, correspond to Heun functions corresponding to pullbacked 2 F 1 hypergeometric functions, that are automorphic forms associated with Shimura curves †.

On a related issue we found the transformation :

2 F 1 [ 1 3 , 1 12 ], [ 3 4 ]
, x = (1 -x) -1/12 • 2 F 1 [ 1 24 , 5 24 ], [ 3 4 ], - 4 x (1 -x) 2 . (153)
Relations ( 152) and (153) show that there is a relation between several Heun functions corresponding to automorphic forms associated with Shimura curves, namely Heun 1 9 , 1 6 , 1 2 , 1, 1 2 , 3 4 , x) in (152), and the Heun functions (141) which emerged as solutions of telescoper of the rational function (138). We actually have the relation [START_REF] Van Hoeij | Belyi functions for hyperbolic Hypergeometric-to-Heun transformations[END_REF] the table number, e.g. 3.4.4, means that the elliptic points are 3, 4 and 4. For an example on how to obtain the hypergeometric function associated to these elliptic points, see for example, paragraph 2.5 in [START_REF] Tu | Algebraic Transformations of hypergeometric functions arising from theory of Shimura curves[END_REF]. † Because they do not appear in Takeuchi's [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF] table 1, which gives a complete list of hypergeometric functions that are associated with Shimura curves.

(1 -3 x 2 ) 3/8 • 1 - x 2 3 1/8 • Heun 1 9 , 1 6 , 1 2 , 1, 1 2 , 3 4 , x 2 3 = Heun 1 4 , 1 64 , 1 8 , 3 8 , 1 2 , 1 2 , 4 x 2 (1 -3 x 2 ) • (x 2 -3) , ( 154
) ‡ In
Which might be in the literature, yet we have not seen it.

which is a consequence of the identity on Belyi maps:

-4 x (1 -x) 2 • x 4 • (x 2 -3) 1 -3 x 2 = 27 4 • x 2 (1 -x) 3 • 4 x 2 (1 -3 x 2 ) • (x 2 -3)
.

(155)

Other Heun functions solutions of telescopers of rational functions related to Shimura curves

The rational function of four variables

R(x, y, z, u) = x y z u ux 2 y 2 z 2 + ux 2 yz + uxy 2 z + uxyz 2 + uxy + uxz + uyz -xyz , (156)
has a telescoper that is a linear differential operator of order three:

M 3 = -3 + 6 • (1 -7 x + 4 x 2 ) • x • D x + 12 • (4 -10 x + 3 x 2 ) • x 2 • D 2 x + 8 • (x -1) • (x -4) • x 3 • D 3 x , (157)
which actually corresponds to the symmetric square of an order-two linear differential operator readings in terms of the homogeneous derivative θ:

M 2 = x 2 • θ 2 -2 x • (14 θ 2 -7 θ + 1) -8 • (4 θ -1) • (4 θ -3). ( 158
)
The solutions of order three operator M 3 are, thus, expressed in terms of the following (square and product of) Heun functions:

x 1/4 • Heun 4, 9 64 , 1 8 , 5 8 , 3 4 , 1 2 , x 2
, x 3/4 • Heun 4, 49 64 , 3 8 , 7 8 , 5 4 , 1 2 , x 2 , and
This order-two linear differential operator M 2 has the pullbacked 2 F 1 solutions:

x 3/8 • (1 -x) -7/8 • 2 F 1 [ 7 24 , 11 24 ], [ 5 4
], -27

4 • x (1 -x) 3 , (160)
x 1/8 • (1 -x) -1/8 • 2 F 1 [ 1 24 , 5 24 ], [ 3 4 ]
, -27 4 • x (1 -x) 3 .
reminiscent of (145). One recovers the same (Shimura) 2 F 1 hypergeometric function as the one in (145), but with another selected pullback. Similarly to the pullback in (145), this last pullback -27 4 •

x (1 -x) 3 is also "special" as can be seen in Appendix J.1 with equations (J.4) and (J.5).

These 2 F 1 solutions (160) can also be rewritten † as

A 1 (x) • 2 F 1 [ 1 3 , 2 3 ], [ 5 4
],

H 1 (x) , A 2 (x) • 2 F 1 [ 1 12 , 5 12 ], [ 3 4 ], H 2 (x) , (161)
or

A 3 (x) • 2 F 1 [ 1 12 , 1 4 ], [ 1 2 ], H 3 (x) , A 4 (x) • 2 F 1 [ 7 12 , 3 4 ], [ 3 2 ], H 4 (x) , (162)
where the A i (x)'s are algebraic functions, and where the H i (x)'s are algebraic functions that can be simply expressed with square roots.

As far as, not Heun, but 2 F 1 hypergeometric functions related with Shimura curves are concerned, several identities also appear in the litterature (see also Appendix J.6). Note that the set of Gauss hypergeometric functions, or Heun functions, that are associated with Shimura curves is a finite set [START_REF] Voight | Shimura curves of genus at most two[END_REF][START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF].

Remark 1: There exist "true" Heun functions (that cannot be reduced to pullbacked 2 F 1 hypergeometric functions) which correspond to automorphic forms associated with a Shimura curve. One example comes from the order-two linear differential operator

L 2 = D 2 x + 12 x 4 -238 x 3 + 3157 x 2 -3648 x + 2592 16 • x 2 • (x -1) 2 • (2 x -27) 2 , ( 163
)
which has the two Heun solutions

x 1/3 • (1 -x) 1/4 • (27 -2 x)
x 2/3 • (1 -x) 1/4 • (27 -2 x)
We have not (yet ...) been able to see such "true" Heun functions as solutions of telescopers of rational functions. This Heun example (164) has a genus-zero level three modular equation given in [START_REF] Elkies | Shimura curve computations[END_REF][START_REF] Yang | Schwarzian differential equations and Hecke eigenforms on Shimura curves[END_REF] and in Appendix J.7.

Remark 2:

The Heun functions we found as diagonals of rational functions or solutions of telescopers of rational functions, were pullbacked 2 F 1 hypergeometric functions which turn out to correspond to classical modular forms or (Shimura) automorphic forms. In both cases this means that the Heun function can be rewritten as a 2 F 1 hypergeometric function with, not just one pullback, but an infinite number of pullbacks (generated by the modular equations, see for instance (148)). Is it possible for a Heun function to correspond to a globally bounded series and to reduce to pullbacked 2 F 1 hypergeometric function with a rational or algebraic pullback, without being automatically a classical modular forms ? In Appendix K we show that a Heun function can actually correspond to a globally bounded series, being reducible to a pullbacked 2 F 1 hypergeometric function, without necessarily corresponding to a classical modular form. We have not yet been able to find such Heun functions as diagonal of rational functions, or even, as solutions of telescopers of rational functions.

Remark 3: More rational functions yielding Heun functions for their diagonals can be obtained using ‡ the (x, y, z, u) → (x n , y n , z n , u n ) transformation for positive integers n. The case where the integer n is negative, in particular n = -1, is different and sketched in Appendix L.

Conclusion

The examples of diagonals of rational functions in three or four variables, that we presented here, illustrate cases where the diagonal of the rational functions are given by Heun functions with integer coefficients series, and can be expressed either in terms of pullbacked hypergeometric functions that are classical modular forms, or derivatives of classical modular forms. Furthermore, we constructed in subsection 2.4, a rational function whose diagonal is given by a Heun function that has already ‡ Or, more generally monomial transformations.

been identified as a "Period" of an extremal rational elliptic surface [START_REF] Doran | Calabi-Yau Manifols Realizing Symplectically Rigid Monodromy Tuples[END_REF], and that has also emerged in the context of pullbacked 2 F 1 hypergeometric functions [START_REF] Maier | On rationally parametrized modular equations[END_REF]. The emergence of squares of Heun functions for most of the diagonals of rational functions of this paper, suggests a "Period" of algebraic surfaces (possibly product of elliptic curves) interpretation. The exact expressions of the diagonal of rational functions in this paper, or in previous papers [START_REF] Bostan | Diagonal of rational functions and selected differential Galois groups[END_REF][START_REF] Abdelaziz | Diagonals of rational functions, pullbacked 2 F 1 hypergeometric functions and modular forms (unabridged version)[END_REF][START_REF] Abdelaziz | Diagonals of rational functions, pullbacked 2 F 1 hypergeometric functions and modular forms[END_REF], are always obtained using the creative telescoping approach, being globally bounded series, solutions of the telescopers of these rational functions. Finally we have also seen a case where the rational function has a telescoper with Heun function solutions, that can be expressed as pullbacked 2 F 1 hypergeometric functions that are not globally bounded, and happen to be associated with one of the 77 cases of Shimura curves [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF]. Such remarkable 2 F 1 hypergeometric functions solutions of a telescoper of a rational function are not diagonals of that rational function (the series are not globally bounded). They can be interpreted as "Periods" [START_REF] Kontsevich | Periods[END_REF][START_REF] Zagier | Arithmetic and Topology of Differential Equations[END_REF] of an algebraic variety over some non-evanescent † cycles. With these 2 F 1 Shimura examples one sees clearly that solutions, analytic at x = 0, of telescopers of rational functions are not diagonals of these rational functions.

All these examples seem to suggest an algebraic geometrical link between the diagonals/solutions of the telescopers, and the original rational functions, and this link should be investigated. This study should help shed light on the geometrical nature of the algebraic varieties associated with the denominators of the rational functions (K3, Calabi-Yau threefolds, extremal rational elliptic surfaces, Shimura varieties). In a forthcoming paper which is a work in progress at the current stage, we intend to introduce an algebraic geometry approach that proves to be efficient in explaining this link, in the cases where the order-two linear differential telescopers of the rational functions or the diagonals of rational functions are related to classical modular forms.

Table 1 page 181 in [START_REF] Ronveaux | Factorization of Heun's differential operator[END_REF] gives a set of six cases for which the order-two linear differential operator factors into the product of two order-one linear differential operators. Let us recall the conditions of Table 1 in [START_REF] Ronveaux | Factorization of Heun's differential operator[END_REF] to get a factorization of the order-two Heun linear differential operator. The order-two Heun linear differential operator reads:

H 2 = D 2 x + γ x + δ x -1 + x -a • D x + α β x -q x • (x -1) • (x -a) . (A.3)
where one has the Fuchsian constraint = α + β -γ -δ + 1. One can easily verify that the order-two Heun linear differential operator (A.3) factorizes into two order-one linear differential operators when q = a δ • (γ -1) + + γ -2 and:

(γ + -2)

• (δ + 1) = α β (A.4)
i.e. after using the Fuchsian constraint:

q = a δ • (γ -1) + α + β -δ -1,
and:

(α + β -δ -1) • (δ + 1) = α β. (A.5)
One has the following factorization H 2 = L 1 • M 1 where:

L 1 = (x -1) • D x + δ, M 1 = x • (x -a) • D x + (γ + -2) • x + a • (1 -γ) (A.6) = x • (x -a) • D x + (α + β -δ -1) • x + a • (γ -1).
The case VI in Table 1 of Ronveaux [START_REF] Ronveaux | Factorization of Heun's differential operator[END_REF], α = δ + 1, and thus β = γ + -2 (since one has the Fuchsian constraint = α + β -γ -δ + 1), corresponds to this case. Along this line, let us consider the Heun function †: Heun a, a δ • (γ -1) + β, δ + 1, β, γ, δ, x .

(A.7)

The previous factorisation (A.6) yields the RHS of (A.8) being an algebraic function when the parameter δ is a rational number. The LHS of (A.8) can thus be written as the diagonal of a rational function [START_REF] Furstenberg | Algebraic functions over finite fields[END_REF][START_REF] Denef | Algebraic power series and diagonals[END_REF]. The Heun function (A.7) is not necessarily the diagonal of a rational function but the order-one operator M 1 acting on that Heun function is a diagonal ¶ of a rational function [START_REF] Furstenberg | Algebraic functions over finite fields[END_REF][START_REF] Denef | Algebraic power series and diagonals[END_REF]. This Heun function (A.7) is locally bounded. For instance Heun(9, 92, 6, 2, 3, 5, 9 x) is not globally bounded, but the series expansion of (θ + 2)[Heun(9, 92, 6, 2, 3, 5, 9 x)] is actullay globally bounded (it is a series with integer coefficients corresponding to the expansion of an algebraic function):

M 1 Heun a, a δ • (γ -1) + β, δ + 1, β, γ, δ, x = λ • (1 -x) -δ , (A.
x • dHeun(9, 92, 6, 2, 3, 5, 9 x) dx + 2 • Heun(9, 92, 6, 2, 3, 5, 9 x) = 2 + 92 x + 2522 x 2

+ 53552 x 3 + 972092 x 4 + 15852440 x 5 + 239057660

x 6 + • • • = 2 (1 -x) • (1 -9 x) 5 .
(A.9)

• For the parameters a = 9, q = 3, α = 1/2, β = 1, γ and δ being deduced from (A. [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF]), (A.21), one finds that

Heun 9, 3, 1/2, 1, 40, - 73 2 , x = 1 + 1 120 • x + 1 6560 • x 2 - 1 1353984 • x 4 - 3 19858432 • x 5 - 1 36106240 • x 6 - 13 2491330560 • x 7 + • • • (A.10)
is an algebraic function. Let us consider the Laurent series expansion of the algebraic function A(x)

A(x) = 2 73 729027183996402643275 • (1 -x) 75/2 • (x -12) x 39 = - 2 75 243009061332134214425 x 39 + • • • - 292448 5 x 3 + 9880 3 x 2 - 104 x + 1 + 1 120 • x + 1 6560 • x 2 - 1 1353984 • x 4 + • • • = P P (A(x)) + Heun 9, 3, 1/2, 1, 40, - 73 2 , x , (A.11)
where P P (A(x)) denotes the principal part (negative powers) of the Laurent expansion of the algebraic function A(x).

Note that the Heun function Heun 9, 3, 1/2, 1, 40, -73/2, x is solution of an order-two linear differential operator L 2 , which is the direct-sum (LCLM) L 2 = L 1 ⊕ M 1 of two order-one linear differential operators. One order-one linear differential operator

L 1 = D x + 3 2 x -26 (x -1) • x - 1 x -12 , (A.12)
has the algebraic function A(x) as solution, and the other one M 1 is a quite large order-one linear differential operator having the rational function P P (A(x)) = P (x)/x 39 given in (A.11) as solution.

• For other values of the parameters a = 9, q = 3, α = 1/2, β = 1, γ and δ being deduced from (A. [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF]), (A.21), one finds that the Heun function is solution of an order-two linear differential operator L 2 which factorises in the product of two order-one operators L 2 = N 1 • P 1 where:

Heun(9, 3, 1, 1, 37, -33, x) = 1 + x 111 - x 2 2109 - x 3 9139 - x 4 54834 - 7 x 5 2248194 - 3 x 6 5245786 - 11 x 7 96672342 - x 8 40899837 + • • • (A.
N 1 = (x -9) • D x -1, P 1 = x • (x -1) • (x -15) • D x + x 2 -65 x + 540. (A.14)
where the order-one linear differential operator P 1 has the rational function solution:

R(x) = (x -15) • (x -1) 34 x 36 . (A.15)
One deduces:

x • (x -1) • (x -15) • d dx Heun(9, 3, 1, 1, 37, -33, x) + (x 2 -65 x + 540) • Heun(9, 3, 1, 1, 37, -33, x) + 60 • (x -9) = P 1 Heun(9, 3, 1, 1, 37, -33, x) + 60 • (x -9) = 0. (A.16)
Note that the series (A.13) is not globally bounded but P 1 (Heun(9, 3, 1, 1, 37, -33, x)) is globally bounded † †.

• For other values of the parameters a = 10, q = 3, α = 1/2, β = 3/2, γ and δ being deduced from (A. [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF]), (A.21), one finds that the corresponding Heun function Heun(10, 3, 1/2, 3/2, 81/2, -73/2, x) is solution of an order-two linear differential operator

L 2 = 4 x • (x -1) • (x -10) • D 2 x + 6 • (2 x 2 -53 x + 270) • D x + 3 • (x -4), (A.17)
which factorises † in the product of two order-one linear differential operators L 2 = N 1 • P 1 , where the order-one linear differential operator P 1 has an algebraic solution of the form x -79/2 • P 38 (x), where P 38 (x) is a polynomial of degree 38, when the order-one linear differential operator N 1 has an algebraic solution of the form (1 -x) 73/2 • (x -10)/x/P 38 (x), with the same polynomial P 38 (x). This polynomial of degree 38 is solution of the non-linear ODE:

2 • (152 x 2 -1579 x + 770) • P (x) P (x) -4 • x • (x -1) • (x -10) • P (x) P (x) = 24 • (247 x -2410). (A.18)
Besides this selected polynomial solution P 38 (x), the solutions of (A.18) can be expressed in terms of Heun functions like Heun(10, 84, 5/2, 7/2, 81/2, -73/2, x) or Heun(10, 14383, -37, -36, -77/2, -73/2, x), which are not globally bounded series ¶.

Appendix A.2. Heun functions where the fourth singularity is an apparent singularity Let us recall some results of [START_REF] Maier | identities[END_REF].

Let us consider a Heun function Heun(a, q, α, β, γ, δ, x) where ‡:

δ = α + β -γ + 2.
(A.19) † † A situation we already encountered [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF]. † The fact that the second order linear differential operator associated with the Heun function factorises is noticed in the Remark of page 15 of [START_REF] Maier | identities[END_REF].

¶ These Heun functions are not diagonals of rational functions, they are just solutions of the nonlinear ODE (A.18). ‡ i.e. = -1 see the definition of in ( 1) or (A.3).

The fourth singularity a will be an apparent singularity when [START_REF] Joyce | On the Simple Cubic Lattice Green Function[END_REF] or:

q 2 + (γ -1) -(2 α β + α + β) • a • q + α β a • (α + 1) • (β + 1) • a -γ = 0, (A.
γ = a 2 α β • (α + 1) • (β + 1) -a q • (2 α β + α + β) + q • (q -1) α β a -q . (A.21)
This condition (A.20) can also be rationally parametrized as:

a = e • (e -γ + 1) (e -α) (e -β) , q = α β • (e + 1) • (e -γ + 1) (e -α) (e -β) . (A.22)
Introducing the order-three linear differential operator

L 3 = x 3 • (x -1) • L 1 • L 2
where

L 1 = D x + e + 1 x + 1 x -1 + 1 x -a , (A.23)
and where L 2 is the previous Heun operator (A.3) for = -1

L 2 = D 2 x + γ x + δ x -1 - 1 x -a • D x + α β x -q x • (x -1) • (x -a) , (A.24)
one finds that the order-three linear differential operator L 3 reads:

L 3 = (x -1) • x 3 • D 3 x + (α + β + e + 4) • x -e -γ -1 • x 2 • D 2 x (A.25) + α β + (α + β + 1) • (e + 2) • x -e γ • x • D x + α β (e + 1) • x,
which, in term of the homogeneous derivative θ = x D x , can be written in a more compact way:

L 3 = (θ + e + 1) • (θ + β) • (θ + α) • x -θ • (θ + γ -1) • (θ + e -1). (A.26)
This last expression (A.26) shows, very clearly, that L 3 corresponds to a 3 F 2 hypergeometric function (see page 16 of [START_REF] Maier | identities[END_REF]):

3 F 2 [α, β, e + 1], [γ, e], x . (A.27)
The singularity x = a in the Heun linear differential operator (A.24) is an apparent singularity. The head polynomial of (A.25) does not have this apparent singularity x = a. This apparent singularity can thus be removed introducing a (non minimal) higher order linear differential operator L 3 : this is called [START_REF] Chen | Desingularization of Ore Operators[END_REF][START_REF] Barkatou | Apparent Singularities of Differential Systems with Rational Function Coefficients[END_REF][START_REF] Barkatou | Removing Apparent Singularities of Linear Differential Systems with Rational Function Coefficients[END_REF] the desingularization † of the linear differential operator L 2 . Introducing the order-two linear differential operator M 2 having the 2 F 1 hypergeometric solution

2 F 1 ([α, β], [γ],
x), one finds that the Heun linear differential operator (A.24), and M 2 are actually homomorphic:

L 2 • (θ + e) = θ + e + 2 + 1 x -1 - a x -a • M 2 .
(A.28) † Often the desingularization removes the apparent singularities but creates unpleasant irregular singularities (for instance at x = ∞). Such desingularization destroys the Fuchsian character of the original linear differential operators. In physics one is interested in desingularization preserving the Fuchsian character of the linear differential operators. This is precisely the case here.

One also has the identity:

Heun 9, 3 4 , 1 4 , 3 4 , 1, 1 2
, 36 • u • (u -1) 2 • (u -4) • (u 2 -4) (u 2 -2 u -2) 2 • (u 2 -2 u + 4) 2 (B.5) = (2 + 2 u -u 2 ) • (4 -2 u + u 2 ) 8 • (1 -u) 3 1/2 • 2 F 1 [ 1 2 , 1 2 ], [1], - u • (4 -u) 3 16 • (1 -u) 3 .
Appendix B.2. Simple cubic lattice Green function: focus on Heun(4, 1 2 , 1 2 , 1 2 , 1, 1 2 , x) Also note that the square of this Heun function can be written, in a different way, as a product of pullbacked 2 F 1 hypergeometric functions. Let us recall the identity [START_REF] Joyce | On the Simple Cubic Lattice Green Function[END_REF] Heun 4, 1

= 2 F 1 [ 1 2 , 1 2 ], [1], H + • 2 F 1 [ 1 2 , 1 2 ], [1], H -, 2 , 1 2 , 1 2 , 1, 1 2 , x 2
where:

H ± = 1 2 ± x 2 • 1 - x 4 1/2 - 1 2 • 1 - x 2 • 1 -x 1/2 . (B.7)
Their series expansion reads: The relation between x and these two Hauptmoduls gives the genus-zero quartic relation †:

H + = x - 1 8 x 2 -
256 • H 2 ± • (1 -H ± ) 2 (B.9) -32 • x • (2 x 2 -9 x + 8) • H ± • (1 -H ± ) + x 4 = 0,
the relation between these two Hauptmoduls reading the genus-zero modular equation:

-256 • A 3 B 3 + 384 A 2 B 2 • (A + B) + A 4 + B 4 -132 • A B • (A 2 + B 2 ) -762 • A 2 B 2 + 384 • A B • (A + B) -256 • A B = 0, (B.10)
corresponding to q ↔ q 3 in the nome q.

One can rewrite the two 2 F 1 hypergeometric functions in the RHS of (B.6)

2 F 1 [ 1 2 , 1 2 ], [1], H ± = 2 F 1 [ 1 4 , 1 4
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 4

H ± • (1 -H ± ) = 2 F 1 [ 1 4 , 1 4 ], [1], H ± ,
where the two pullbacks

H ± H ± = 1 4 • (2 x 2 -9 x + 8) • x ± (x -2) • (1 -x) 1/2 • 1 - x 4 1/2 • x, (B.11)
are solutions of the genus-zero quadratic relation

16 • H 2 ± -8 • x • (2 x 2 -9 x + 8) • H ± + x 4 = 0, (B.12)
their expansions reading:

H -= 4 x - 9 2 x 2 + 63 64 x 3 - 9 512 x 4 - 261 16384 x 5 - 1791 131072 x 6 + • • • , H + = 1 64 x 3 + 9 512 x 4 + 261 16384 x 5 + 1791 131072 x 6 + • • • (B.13)
Thus (B.6) can also be rewritten as:

Heun 4, 1 2 , 1 2 , 1 2 , 1, 1 2 , x 2 = 2 F 1 [ 1 4 , 1 4
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF],

H + • 2 F 1 [ 1 4 , 1 4
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF]

H (+) ± = 1 2 ± x 2 • 1 - x 4 1/2 + 1 2 • 1 - x 2 • 1 -x 1/2
H 4 = 256 • (x -1) 2 • (x -4) 2 • x 2 • D 4 x + 256 • (x -1) (x -4) • (8 x 2 -33 x + 16) • x • D 3 x (B.16) + 32 • (115 x 4 -991 x 3 + 2370 x 2 -1696 x + 256) • D 2 x + 16 • (76 x 3 -693 x 2 + 1128 x -448) • D x + 9 • x • (x -16).
Note that this order-four linear differential operator is homomorphic to its adjoint with an order-one linear differential intertwiner

H 4 • L 1 = adjoint(L 1 ) • adjoint(H 4 )
where:

L 1 = 2 • (x -1) • (x -4) • x • D x + 5 x 2 -4 x -4, (B.17)
and with an order-three intertwiner

adjoint(H 4 ) • L 3 = adjoint(L 3 ) • H 4 , (B.18)
suggesting that the differential Galois group of this order-four linear differential operator could be SO(3, C). This is not the case. In fact this order-four linear differential operator is not irreducible †: it has an absolute factorization (see (B.24) below). Introducing the parametrization

x = - u • (4 -u) • (4 -u 2 ) 4 • (1 -u) 2 , (B.19)
the product identity (B.6) becomes:

Heun 4, 1 2 , 1 2 , 1 2 , 1, 1 2 , - u • (4 -u) • (4 -u 2 ) 4 • (1 -u) 2 2 (B.20) = 2 F 1 [ 1 2 , 1 2 ], [1], - u • (4 -u) 3 16 • (1 -u) 3 • 2 F 1 [ 1 2 , 1 2 ], [1], - u 3 • (4 -u) 3 16 • (1 -u) 3 .
The relation between the two Hauptmoduls H ± corresponds to the genus-zero modular equation † 262144000000000

• A 3 B 3 • (A + B) + 4096000000 • A 2 B 2 • 27 • (A 2 + B 2 ) -45946 • A B + 15552000 • A B • (A + B) • (A 2 + B 2 + 241433 • A B) + 729 • A 4 + B 4 -1069956 • A B (A 2 + B 2 ) + 2587918086 • A 2 B 2 + 2811677184 • A B • (A + B) -2176782336 • A B = 0, (B.29)
The Anecdotal remark: The order-three intertwiner L 3 in (B.18) is, in fact ‡, the symmetric square of an order-two linear differential operator

L 2 = D 2 x + 3 x 2 -10 x + 4 2 x • (x -4) • (x -1) • D x + (x + 4) • (x -2) 32 • (x -1) • (x -4) • x 2 , (B.32)
which has the two Heun solutions

S ± = x (1 ± √ 2)/4
• h ± where:

h ± = Heun 4, 11 8 ± 7 • 2 1/2 8 , 1 ± 3 • 2 1/2 8 , 1 ± 2 1/2 8 , 1 ± 2 1/2 2 , x . (B.33)
It is clear, from the x (1 ±2 1/2 )/4 factors, that the order-two operator L 2 is not even ¶ globally nilpotent [START_REF] Bostan | Globally nilpotent differential operators and the square Ising model[END_REF]. Furthermore, the two series h + + h -and 2 -1/2 • (h + -h -) are series with rational number coefficients that are not globally bounded series. Such Heun functions (B.33) cannot be written as pullbacked 2 F 1 hypergeometric functions.

Appendix C. Polynomials of the degree six equation for the Hauptmodul in [START_REF] Elkies | Shimura curve computations[END_REF].

The polynomials of the degree six equation for the Hauptmodul in [START_REF] Elkies | Shimura curve computations[END_REF] + 15237120 x 13 + 324000 x 14 , (C.2) † Corresponding to q ↔ q 3 in the nome q. ‡ When written in a unitary way D 3

x + • • • ¶ The intertwiners of Fuchsian (resp. globally nilpotent) linear differential operators have no reason to be Fuchsian (resp. globally nilpotent) linear differential operators as well. Remark: Taking the resultant (eliminating x) between equation (32), in the example 2, with the following genus-zero curve (E.4)

8 z 3 x 3 -12 z 2 x 2 • (x + z) + x z • (6 x 2 -83 x z + 6 z 2 ) -(x + z) • (x 2 + 17 x z + z 2 ) + x z = 0, (C.
× 2 F 1 [ 1 6 , 2 3 ], [1], - 108 z • (1 + 3 z) • (1 -4 z -12 z 2 ) 2 (1 -36 z -108 z 2 ) 2 2 = (1 + 6 z) 1/3 (1 -6 z -18 z 2 ) 2/3 (E.5) × 2 F 1 [ 1 6 , 2 3
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], -108

z 2 • (1 + 3 z) 2 • (1 -6 z) • (1 + 2 z) (1 + 6 z) 2 • (1 -6 z -18 z 2 ) 2 = (1 + 6 z) 1/2 (1 + 234 z + 972 z 2 + 1080 z 3 ) 1/2 × 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],

1728 z • (1 + 3 z) 3 • (1 -6 z) 6 • (1 + 2 z) 2 (1 + 6 z) 3 • (1 + 234 z + 972 z 2 + 1080 z 3 ) 3 2 = (1 + 6 z) 1/2 (1 -6 z + 12 z 2 + 120 z 3 ) 1/2 × 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], 1728 Consider the identity:

z 3 • (1 + 3 z) • (1 -6 z) 2 • (1 + 2 z) 6 (1 + 6 z) 3 • (1 -6 z + 12 z 2 + 120 z 3
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], x (F.1)

= (1 + 8 x) -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], 64

• x • (1 -x) 3 (1 + 8 x) 3 .
The nome associated to the linear differential operator of order two having 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], x) as a solution is given by:

Q(x) = x
τ τ [ 1 3 , 2 3 ], [1], x = µ • τ [ 1 12 , 5 12
], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 64

• x • (1 -x) 3 (1 + 8 x) 3 , (F.4)
where µ is a constant, which gives after exponentiation:

64 • Q(x) = q 64 • x • (1 -x) 3 (1 + 8 x) 3 . (F.5)
Now, the RHS of (F.5) is necessarily globally bounded, which agrees with the globally bounded character of the nome (F.2). In contrast, let us consider 2 F 1 ([1/5, 1/5], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], x). The corresponding series is globally bounded ¶, however the corresponding nome which reads is not globally bounded. Therefore, it is not possible to find any algebraic (or rational) pullback p(x) such that

Q [1/5,
µ • Q [1/5,1/5] (x) = q p(x) , (F.7)
since the RHS of (F.7) is necessarily globally bounded when µ • Q [1/5,1/5] (x) cannot be globally bounded regarless of the constant µ.

Appendix F.1. The globally bounded nome condition to be a classical modular form for Heun functions.

Let us recall the order-two linear differential operator (1), which has Heun(a, q, α, β, γ, δ, x) as a solution. Since we are interested in diagonals of rational functions, we focus on series expansions at x = 0. To have a nome requires the other solution to have a formal series expansion with a logarithm which corresponds to impose that the parameter † γ is an integer. To have a logarithm is a necessary but not sufficient condition to have a nome which is analytic at x = 0 (mirror map).

When ‡ γ = 1 we have a nome Q, analytic at x = 0, which series expansion reads:

Q = x + δ -2 + α + β -δ a • x 2 + • • • (F.8)
The next terms become more and more involved rational expressions of the parameters of the Heun function. For a given Heun function with γ = 1 one can easily use the globally bounded nome condition. In practice this is an efficient way to discard the γ = 1 Heun functions which are not classical modular forms. However, finding exhaustively the Heun functions with γ = 1 corresponding to classical modular forms, remains a quite involved task (see [START_REF] Maier | On reducing the Heun equation to the hypergeometric equation[END_REF] and Appendix K below).

¶ Any 2 F 1 (a, b], [c],
x) with c = 1 is globally bounded since it is of weight zero: it is of the form nFn-1, and has c given by an integer and not a fractional number.

† The condition to have a logarithm for the formal series of (1) at x = 1 is that the parameter δ is an integer. The condition to have a logarithm at x = a is = α + β -γ -δ + 1 being an integer. ‡ This actually corresponds to what we had called "premodular condition" in [START_REF] Abdelaziz | Modular forms, Schwarzian conditions, and symmetries of differential equations in physics[END_REF]. The premodular condition W (x) = -1/2/x 2 + • • • yields exactly γ = 1. In contrast with 2 F 1 hypergeometric functions, this premodular condition is far from imposing that the corresponding Heun series is globally bounded: Heun(3, 5, 7, 11, 1, 13, x), for instance, is not globally bounded.

1 6 , 1 6 ], [1], x = (1 -x) 2/3 • (1 -6 • θ) 2 F 1 [ -1 6 , 5 6 ], [1], x , (H.21)
and thus, by elimination, one finds that the order-two operators annihilating respectively

2 F 1 ([ -1 6 , 5 6 ], [1], x) and 2 F 1 ([ 1 6 , 1 6 ], [1], x) read: L 2 = (1 -x) 1/3 • (1 + 6 • θ) • (1 -x) 2/3 • (1 -6 • θ) -1 (H.22) = x • (6 θ + 5) • (6 θ -1) -36 θ 2 , M 2 = (1 -x) 2/3 • (1 -6 • θ) • (1 -x) 1/3 • (1 + 6 • θ) -1 (H.23) = x • (6 θ + 1) • (6 θ + 1) -36 θ 2 .
These two order-two linear differential operators L 2 and M 2 are equivalent † with order-one intertwiners (with algebraic coefficients):

M 2 • (1 -x) 2/3 • (6 θ -1) = (1 -x) 2/3 • (6 θ -1) • L 2 , (H.24)
and:

L 2 • (1 -x) 1/3 • (6 θ + 1) = (1 -x) 1/3 • (6 θ + 1) • M 2 , (H.25)
Furthermore we also have the identities

2 F 1 [ 5 6 , 7 6 ], [1], x = (1 -x) -5/6 • 2 F 1 [- 1 6 , 5 6 ], [1], - x 1 -x , (H.26) or: 2 F 1 [- 1 6 , 5 6 ], [1], x = (1 -x) -5/6 • 2 F 1 [ 5 6 , 7 6 ], [1], - x 1 -x . (H.27)
Like all the Belyi coverings [START_REF] Musty | A database of Belyi maps[END_REF], the pullback 27 4

• x 2 (1 -x) 3 in (145) is "special". It is such that: 27 4 • x 2 (1 -x) 3 • 480 • (1 -x) • x (17 x -32) 2 = 27 4 • x 2 (1 -x) 3 • 15 • (17 x -32) • x 1024 • (1 -x) 2 . (J.1)
It has already been seen to occur in another framework [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF], namely † the walk in a Weyl chamber of the Lie algebra sl 3 (C). It actually occurs in the well-known "Kernel equation" for that particular walk described in [START_REF] Raschel | Random walks in the quarter plane, discrete harmonic functions and conformal mappings[END_REF] G(x, y) + G(0, 0) = G(x, 0) + G(0, y), (J.2)

where:

G(x, y) = L(x, y) • H(x, y), (J.3)
and where the generating function H(x, y) of the walk, and the Kernel of the walk L(x, y), read respectively:

H(x, y) = 1 -x y (1 -x) 3 • (1 -y) 3 , L(x, y) = 27 4 • (y + x y 2 + x 2 -3 x y).
Noticeably, G(x, y) is the sum of the particular rational function pullback w(x) = The genus-zero curve L(x, y) = 0 has the rational parametrisation

y = 225 t 2 32 • (1 -t) • (17 t -32)
, and:

x = 480 • (1 -t) • t (17 t -32) 2 or: x = 15 • (17 t -32) • t 1024 • (1 -t) 2 , (J.6)
where one recovers, in the last two rational parametrisations (J.6) for x, the two rational functions in the identity (J.1). The two rational functions are the rational parametrisation of the (symmetric) genus-zero curve x 2 x 2 -3 x x + x + x = 0 (corresponding to the elimination of y in L(x, y) = 0 and L(x , y) = 0). These two rational functions (J.6) are simply related by an involution: The order-two linear differential operator

x (t) = x 32
L 2 = θ • θ - 1 6 -x • θ + 1 24 • θ + 7 24 , (J.8)
has the two 2 F 1 hypergeometric solutions:

2 F 1 [ 1 24 , 7 24 ], [ 5 6 ], x , x 1/6 • 2 F 1 [ 5 24 , 11 24
], [

], x , (J.9)

A modular equation of level five has been given in [START_REF] Voight | Shimura curves of genus at most two[END_REF] on this example of 2 F 1 hypergeometric function corresponding to automorphic forms associated with Shimura curves. This modular equation of level five corresponds to the elimination of u between

A(u) = - 1350000 • u 6 225 u 2 + 18 u + 1 , B(u) = A 11 u + 2 252 u -11 = - 2160 • (11 u + 2) 6 (225 u 2 + 18 u + 1) (252 u -11) 4 , (J.10)
Changing x into 1/x, the order-two linear differential operator (J.8) becomes the order-two linear differential operator

L 2 x → 1 x = θ - 1 24 • θ - 7 24 -x • θ • θ + 1 6 , (J.11)
which has the two 2 F 1 hypergeometric solutions:

x 1/24 • 2 F 1 [ 1 24 , 5 24 ], [ 3 4 ]
, x , x 7/24 • 2 F 1 [ 7 24 , 11 24 ], [ 5 4 ], x .
(J.12)

Instead of (J.11) one can also introduce the conjugate of (J.11) by x 1/24 One can also introduce the order-two linear differential operator

L 2 = D 2 x + 135 x 2 -167 x + 140 576 • x 2 • (x -1) 2 , (J.15)
which has the two hypergeometric solutions:

x 5/12 • (1 -x) 1/4 • 2 F 1 [ 1 24 , 7 24 ], [ 5 6
], x ,

x 7/12 • (1 -x) 1/4 • 2 F 1 [ 5 24 , 11 24
], [

], x , (J.16)

and

M 2 = D 2 x + 140 x 2 -167 x + 135 576 • x 2 • (x -1) 2 , (J.17)
which has the two hypergeometric solutions:

S 1 = x 3/8 • (1 -x) 1/4 • 2 F 1 ([ 1 24 , 5 24 ], [ 3 4
], x), (J.18) corresponding to the x ↔ y symmetry. In other words, the modular equation of level five is also parametrised by the automorphic Schwarz function (J.33). We thus have two parametrisations of the modular equation of level five: a rational parametrisation (J.24), and a parametrisation (uniformisation) by automorphic functions †.

S 2 = x 5/8 • (1 -x)
Recalling the two 2 F 1 hypergeometric identities (J.21) or (J.20), and taking their ratio, one finds the following simple covariance of the ratio τ (x) by the algebraic series (J. (J.44) † This is the well-known Poincaré result [START_REF] Whittaker | On the Connexion of Algebraic Functions with Automorphic Functions[END_REF] that, whatever the genus of an algebraic curve, this algebraic curve is uniformised by automorphic functions of a new variable (here τ ). ‡ See th 2.5 in J. Voight and J. Willis paper [START_REF] Voight | Computations with Modular Forms, chapter Computing Power Series Expansions of Modular Forms[END_REF] for the simplicity of the complex numbers like E.

If one of the modular correspondence x → y(x) is such that (see (J.38)) τ (y(x)) = E • τ (x), the other modular correspondences x → y(x) of the form (J.44) commuting with that modular correspondence will also be such that τ (y(x)) = α • τ (x).

The action of the modular correspondence on the Schwarz map τ , given by this simple relation (J.38), is an infinite order transformation. Such a simple relation (J. [START_REF] Ronveaux | Factorization of Heun's differential operator[END_REF] for Shimura automorphic forms has to be compared with the action of the modular correspondence on the Schwarz map τ for classical modular forms (cusp forms ‡ with a nome q) where we have transformations like q → q N where N is an integer.

Appendix J.5. Derivative of Shimura automorphic functions.

Let us introduce the order-two linear differential operator Along this line, it is tempting to revisit the arguments of section (2.5) for classical modular forms, to (Shimura) automorphic forms. Let us consider the rational function (138) of section (3), which yields the Heun functions (141) and therefore the Shimura it is tempting to calculate the telescoper of the homogeneous partial derivative with respect to u of the rational function (138). One remarks that the homogeneous partial derivative with respect to u of this rational function (138), is very simple: it is equal to u times the square of that rational function. The telescoper of u • ∂R(x, y, z, u) ∂u (J.49) = u • x 2 y 2 z 2 (1 -x y z u + x y z • (x + y + z) + x y + y z + x z) 2 , is an order-three linear differential operator M 3 which is actually homomorphic to the order-three linear differential operator L 3 given by (139) and yielding the Heun functions (141) or equivalently squares of Shimura 2 F 1 functions (145). This homomorphism reads:

M 3 • θ = x 2 • θ • L 3 .
(J.50) ‡ This correspond to the fact that one of solutions has logarithmic terms in its formal series solution expansion at x = 0. From (J.50) one finds that the solutions of M 3 (and thus the solutions of the telescoper of (J.49)) are simply obtained from the action of the homogeneous derivative θ on the Shimura solutions of the order-three linear differential operator S 3 , i.e. on the Heun functions (141) or the (square of) Shimura 2 F 1 hypergeometric functions (145). This example of rational function (J.49) thus yields a telescoper which solutions are not Shimura 2 F 1 automorphic forms, but derivatives of Shimura 2 F 1 automorphic forms.

Anecdotal remark: recalling (A.29) where the parameters in (A. [START_REF] Tu | Algebraic Transformations of hypergeometric functions arising from theory of Shimura curves[END_REF]

],

x 2 • (1 -x) • (49 x -81) 7 4 • (6561 -13851 x -9261 x 2 + 16807 x 3 ) 3 , where the 2 F 1 hypergeometric function on the RHS of the identity (J.53) corresponds to a Shimura curve with elliptic points [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF][START_REF] Bostan | Diagonal of rational functions and selected differential Galois groups[END_REF][START_REF] Takemura | The Heun equation and the Calogero-Moser-Sutherland system I: the Bethe Ansatz method[END_REF]. To see that the hypergeometric function on the LHS also corresponds to a Shimura curve is not obvious on the difference of exponants † of the hypergeometric function. However considering the order-two linear differential operator N 2 annihilating 2 F 1 ([ 5 42 , 19 42 ], [ 5 7 ], x), one finds that the pullback of N 2 by one of the Euler's hypergeometric transformations, namely x → 1/x, has the solution x 5/42 • 2 F 1 ([11/42, 23/42], [2/3], x) which is a Shimura hypergeometric function of the [START_REF] Bostan | Diagonal of rational functions and selected differential Galois groups[END_REF][START_REF] Bostan | Diagonal of rational functions and selected differential Galois groups[END_REF][START_REF] Takemura | The Heun equation and the Calogero-Moser-Sutherland system I: the Bethe Ansatz method[END_REF] type.

Note that the set of Gauss hypergeometric functions that are associated with Shimura curves is a finite set [START_REF] Voight | Shimura curves of genus at most two[END_REF][START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF].

Similarly in [START_REF] Tu | Algebraic Transformations of Hypergeometric Functions and automorphic Forms on Shimura Curves[END_REF] we find on page 2, equation (3), the identity:

2 F 1 [ 1 20 , 1 4 ], [ 4 5
], 64

• x • (1 -x -x 2 ) 5 (1 -x 2 ) (1 + 4 x -x 2 ) 5 (J.54) = (1 -x 2 ) 1/20 • (1 + 4 x -x 2 ) 1/4 • 2 F 1 [ 3 10 , 2 5 ], [ 9 10
], x 2 , (classical modular forms) solutions are actually expandable at x = 0 and correspond to globally bounded series expansions. The solution of L (p)

2 , analytic at x = 0, can be written as a Heun function

x • Heun 1 2 - i √ 3 2 , 1 2 - i √ 3 6
, 1, 1, 1, 1, -3 + i √ 3 18

• x = x -1 9

x 2 + 1 81 x 3 -7 6561

x 4 + 1 59049

x 5 + 11 531441

x 6 + • • • (L.14) to be compared with (117). Again these nice classical modular forms globally bounded series expansions are different from the diagonal of (L.13) which is equal to zero.

8 )

 8 † For a = 9, β = γ = δ = 1, the Heun function (A.7) is the simple rational function Heun(9, 1, 2, 1, 1, 1, 27 x) = 1/(1 -3 x).

  13) ¶ We have, of course, a similar result for the previous Heun function (A.1).

7 )( 1 -

 71 one obtains immediately equation[START_REF] Kontsevich | Periods[END_REF] and polynomials (C.1)-(C.6) of this appendix. a rational parametrisation is given in Appendix E. The previous genus-zero modular equation (D.11) has the following rational parametrisation (see (E.4) and (E.36 z -108 z 2 ) 2/3

) 3 2 .= 1 + 6 z

 216 + 12 z 2 + 96 z 3 + 360 z 4 + 2160 z 5 + 10488 z 6 + 58464 z 7 + • • • Appendix F. A nome necessary condition to be a classical modular form: why 2 F 1 ([1/5, 1/5], [1], x) is not a classical modular form.

27 4 • x 2 ( 1 4 )

 27214 -x)3 and of another rational function of y:Note that this additional rational function of y corresponds to the duality x ↔ 1/x: G(x, y) = L(x, y) • H(x, y) = w(x) -

2 = 140 y(x) 2 -

 22 [START_REF] Maier | On rationally parametrized modular equations[END_REF]:τ (y(x)) = w 1/4 • τ (x),where:Do note that w 1/4 takes a simpler form ‡:τ (y(x)) = E • τ (x), where: rule relation for the Schwarzian derivative of composition of functions {τ (y(x)), x} = {τ (y), y} y=y(x) • y (x) 2 + {y(x), x}, (J.39) one deduces from the Schwarzian equation (J.31) that: {τ (y(x)), x} = {E • τ (x), x} = {τ (x), x} = 140 x 2 -167 x + 135 288 • x 2 • (x -1) 167 y(x) + 135 288 • y(x) 2 • (y(x) -1) 2 • y (x) 2 + {y(x), x}. (J.40) One thus recovers the Schwarzian equation (J.28) on the algebraic correspondence x → y(x). Conversely, and more generally, if the Schwarz map τ (x) verifies the Schwarzian equation {τ (x), x} -W (x) = 0, (J.41) and the algebraic correspondence x -→ y(x) verifies the Schwarzian equation {y(x), x} + W (y(x)) • y (x) 2 -W (x) = 0, (J.42) with the same rational function W (x), one deduces using the the chain rule relation (J.40) that: {τ (y(x)), x} = {τ (x), x}. (J.43) Consequently this means that τ (y(x) can only be a linear fractional transformation of τ (x): τ (y(x)) = a • τ (x) + b c • τ (x) + d .

2 F 1

 21 functions (145). Recalling the identity (132) of section (2.5) that the diagonal of the partial homogeneous derivative of a rational function is the homogeneous derivative of that diagonalx • d dx Diag(R(x, y, z, u)) = Diag u • ∂R(x, y, z, u) ∂u , (J.48)

  , H -.

	(B.14)
	Besides (B.7), let us introduce the two other Hauptmoduls:

  series expansion of the two Hauptmoduls H ± reads respectively:

	27 4	• x 2 +	81 16	• x 3 -	8181 512	• x 4 -	27351 1024	• x 5 + • • • ,	and:	(B.30)
	27 262144	• x 6 +	243 1048576	• x 7 +	11421 33554432	• x 8 +	14013 33554432	• x 9 + • • •	(B.31)

  -126 x + 6657 x 2 -191100 x 3 + 3224004 x 4 -32165952 x 5 + 179161346 x 6 -459836304 x 7 + 116094384 x 8 + 1082203136 x 9 -247538592 x 10 -690095616 x 11 -102971392 x 12

		read
	respectively:
	p 6 (x) = 1 -53324 x + 3340572 x 2 -47158880 x 3 + 453452848 x 4
	+ 867240000 x 5 + 729000000 x 6 ,	(C.1)
	p 14 (x) = 1

  p 16 (x) = 1 -144 x + 12624 x 2 + 42210112 x 3 + 35493701376 x 4 + 4373215830144 x 5 + 146527536091776 x 6 + 1709973141608448 x 7 + 8301405990184512 x 8 + 19700334651209215 x 9 + 25456135068016191 x 10 + 18571208108112576 x 11 + 7732095471912574 x 12 + 1556868770685984 x 13 + 183059891926656 x 14 -189 x + 15939 x 2 -790713 x 3 + 25604460 x 4 -567479130 x 5 + 8729096106 x 6 -129136524678 x 7 -3128791781472 x 8 -301592422936140 x 9 + 9302223231205632 x 10 -898829709897155904 x 11 -3001729628561501184 x 12 -14056123657998705984 x 13-75146837553583537200 x 14 -220865053128551921712 x15

	+ 2050894080000 x 15 + 43740000000 x 16 ,	(C.3)
	p 20 (x) = 496 + 14229477 x + 10755437160 x 2 + 607313973993 x 3
	+ 21837165846834 x 4 -8741350651741356 x 5 + 602696000526139688 x 6
	-18362650954659075270 x 7 + 237729206666512798092 x 8
	-755131861209486545984 x 9 -4078730236814710350912 x 10
	+ 4455455555487369556416 x 11 + 8298505398959353031040 x 12
	-501211331403375909060096 x 13 + 32930923081507234916352 x 14
	-7365760252808436401159680 x 15 -14299198145937514719360000 x 16
	-5550618706232520960000000 x 17 -2323303457201280000000000 x 18
	-2534505901920000000000000 x 19 + 114791256000000000000000 x 20 ,	(C.4)
	p 21 (x) = 1 -233016707230759517184 x 16 -25485724878879707232 x 17
	+ 57908320494660830720 x 18 + 11705232438547200000 x 19
	-65745768960000000 x 20 -3149280000000000 x 21 ,	(C.5)
	p 23 (x) = 1 -207 x + 21552 x 2 + 41491618 x 3 + 32829303696 x 4
	+ 2194878922992 x 5 -81778493396032 x 6 -2027922617204064 x 7
	+ 50756763414324000 x 8 + 304451170309086240 x 9 -5117266473854922240 x 10
	-23872757678772284352 x 11 + 92761784722387529728 x 12
	-1131857205540040786944 x 13 + 4168576271341671432192 x 14
	-35184687910528656122881 x 15 + 2169420555967834017888 x 16
	-270473856235208160230976 x 17 -471011087555724046299136 x 18
	-105170018593490449009152 x 19 -71294201738328407040000 x 20
	-141314879220788736000000 x 21 + 1238649615360000000000 x 22
	-127545840000000000000 x 23 ,	(C.6)

  Hypergeometric functions like (J.46) are not Shimura 2 F 1 hypergeometric functions: they do not correspond to automorphic forms, but derivatives of automorphic forms.

		θ • θ +	3 4		-x • θ +	29 24	• θ +	25 24	,	(J.45)
	which has the two 2 F 1 hypergeometric solutions:
	2 F 1 [	25 24	,	29 24	], [	7 4	], x ,			x -3/4 • 2 F 1 [	7 24	,	11 24	], [	1 4	], x .	(J.46)
	The linear differential operator (J.45) is homomorphic to (J.13). One deduces from
	that homomorphism the following identity
	(4 θ + 1) 2 F 1 [	7 24	,	11 24	], [	5 4	], x	= 2 F 1 [	7 24	,	11 24	], [	1 4	], x .	(J.47)

  verify (A.[START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF]) and (A.[START_REF] Peters | A pencil of K3-surfaces related to Apéry's recurrence for ζ(3) and Fermi surfaces for potential zero[END_REF], one can also write (J.47) as 3 F 2 ([7/24, 11/24, 5/4], [5/4, 1/4], x) but also, since here e = 1/4, as the Heun function This simple form of the Heun function is a consequence of the fact that one has γ = e + 1 in this case. With e = 1/2 the identities (A.29) become: Appendix J.6. Identities linking hypergeometric functions that are related with Shimura curves.In fact, several identities linking 2 F 1 hypergeometric functions related with Shimura curves, appear in the litterature. For example we see in[START_REF] Vidunas | Parametric transformations between the Heun and Gauss hypergeometric functions[END_REF] (equation (4.8) page 14):

										Heun 0, 0,	7 24	,	11 24	,	5 4	,	3 2	, x .	(J.51)
	Heun	72 5	,	231 40	,	7 24	,	11 24	,	5 4	,	3 2	, x	= (2 θ + 1) 2 F 1 ([	7 24	,	11 24	], [	5 4	], x
							= 3 F 2 ([	7 24	,	11 24	,	3 2	], [	1 2	,	5 4	], x	(J.52)
	2 F 1 [	5 42	,	19 42	], [	5 7	], x			=				6561 -13851 x -9261 x 2 + 16807 x 3 6561	-1/28	(J.53)
			× 2 F 1 [	1 84	,	29 84	], [

† These calculations were performed using the creative telescoping program of C. Koutschan[6].

† All these 2 F 1 hypergeometric functions are Shimura hypergeometric functions[START_REF] Voight | Shimura curves of genus at most two[END_REF][START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF] corresponding to[START_REF] Abdelaziz | Diagonals of rational functions, pullbacked 2 F 1 hypergeometric functions and modular forms (unabridged version)[END_REF][START_REF] Abdelaziz | Diagonals of rational functions, pullbacked 2 F 1 hypergeometric functions and modular forms (unabridged version)[END_REF][START_REF] Bostan | Diagonal of rational functions and selected differential Galois groups[END_REF] and (6, 6, 2) difference of exponents in Table[START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF] of[START_REF] Voight | Shimura curves of genus at most two[END_REF][START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF].

† Diagonals are periods over evanescent cycles.

† Note that this quartic relation (B.9) is invariant by H ± → 1 -H ± .

† At first sight using the command DFactor of Maple, one could imagine that this linear differential operator is irreducible.

† Its parametrisation is in agreement with the rational parametrisation given below in Appendix E.

F 1 [

§ See[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: integrality versus modularity Preprint[END_REF], and the hypergeometric functions in the previous sections in this paper. † By related to classical modular forms, we mean, from now, that any of the hypergeometric functions below can be rewritten as a pullbacked 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], x) function, and hence is necessarily a classical modular form.

† Use the command "equiv" of Mark van Hoeij and not the command "Homomorphisms" of DEtools in Maple. This command is the algebraic extension of the command "Homomorphisms".

† The difference of exponants of a 2 F 1 ([a, b], [c], x)hypergeometric function is the triplet (c -ab, b -a, 1 -c).These rational numbers (c -a -b, b -a, 1 -c) have to be (up to a sign) reciprocals of integer, and furthermore in some table given by Takeuchi[START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF].

† Note that the normalised accessory parameter q/α/β is a equal to the constant η.

† This series can be recast into a series with integer coefficients after the x → 2304 • x rescaling. ‡ We cannot totally exclude the fact that a nome series like (K.9) could correspond to an order-one linear differential operator acting on a classical modular form (see the second 2 F 1 hypergeometric function of (H.28) and equation (H.29) in Appendix H.2).

Acknowledgments. Y. A. and J-M. M. would like to thank the Stat. and Math. Department of Melbourne University for hospitality, where part of this paper was written. Y. A. and J-M. M. would like to thank A. J. Guttmann for numerous discussions on lattice Green functions. Y. A. would like to thank A. Malmendier for several enlightening discussions on the periods of rational extremal surfaces. Y. A. would like to thank Dahlia Abdelaziz, Rashida Abdelaziz and Siwa Abdelaziz for the great time spent together while working on this paper, and for being the joyful and loving cousins they are. Y. A. would like to thank Mark van Hoeij for the explanations he provided him on Heun to Hypergeometric functions pullbacks. Y. A. would like to thank John Voight for enlightening explanations on Shimura curves and for pointing out the reference [19]. Y. A. would like to thank Yifang Yang for several very explanatory emails on automorphic functions and Shimura curves. Y. A. would like to thank Wadim Zudilin for an enlightening exchange on the modular parametrization of hypergeometric functions. J-M. M. would like to thank Pierre Charollois for many enlightening discussions on modularity and automorphic forms. J-M. M. would like to thank Raschel Kilian for enlightening discussions on walks on the quarter of a plane. S. B. would like to thank the LPTMC and the CNRS for kind support. We thank the Research Institute for Symbolic Computation, for access to the RISC software packages. We thank M. Quaggetto for technical support.

Appendix A. Trivialization cases of Heun functions

We already encountered in [START_REF] Abdelaziz | Modular forms, Schwarzian conditions, and symmetries of differential equations in physics[END_REF] (see section 3 equations ( 47), [START_REF] Baba | Quaternionic Modular Forms and Exceptional Sets of Hypergeomtric Functions[END_REF] and (C.12) in [START_REF] Abdelaziz | Modular forms, Schwarzian conditions, and symmetries of differential equations in physics[END_REF]) an interesting example of Heun function solution of an order-two linear differential operator which factorises into two order-one operators. The Heun function

satisfies the identity:

For M = 2, 1/2, -1, the Heun function (A.1) can be written as pullbacked 2 F 1 hypergeometric functions. Let us recall, in the next subsections, the more general results of Ronveaux [START_REF] Ronveaux | Factorization of Heun's differential operator[END_REF][START_REF] Hounkonnou | Generalized Heun and Lamé equations: factorization[END_REF].

Consequently, as far as series expansions at x = 0 are concerned, the Heun function ‡ such that its parameters verify (A. [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF]) and (A. [START_REF] Peters | A pencil of K3-surfaces related to Apéry's recurrence for ζ(3) and Fermi surfaces for potential zero[END_REF], can be written in several ways:

Heun(a, q, α, β, γ, δ, x) = 3 Note that such a Heun function, like [START_REF] Glasser | Lattice Green function (at 0) for the 4D hypercubic lattice[END_REF], can also be written as a Heun(9, 3, 1, 1, 1, 1, A(x)) function, where A(x)) is an algebraic function, using the identity [START_REF] Maier | identities[END_REF] Heun 9, 3, 1, 1, 1, 1, x = (B.1)

1 -

or

which is a special case of:

Heun a, q, 1, 1, 1, 1, x = (B.2)

1 -

Appendix B.1. Other Heun functions for the simple cubic lattice Green function.

One has the identity

which can be written using the parametrization (B. [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF]):

, y . ‡ When γ is not an integer, the series expansions of these Heun functions (A.29) are not generically globally bounded.

Note that these two pullbacked 2 F 1 hypergeometric functions are simply related:

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], -u • (4 -u) 3 16

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], -u 3 • (4 -u) 3 16

Therefore the identity (B.20), in fact, relates the square of a Heun function with the square of a pullbacked 2 F 1 hypergeometric function, or more simply, gives this Heun function as a function of 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], -u • (4 -u) 3 16 • (1 -u) 3 , (B. [START_REF] Peters | A pencil of K3-surfaces related to Apéry's recurrence for ζ(3) and Fermi surfaces for potential zero[END_REF] or:

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], -t • (3 t + 4) 3 16

Let us note that the order-four linear differential operator H 4 , given by (B. [START_REF] Hallouin | Computation of a cover of Shimura curves using a Hurwitz space[END_REF]), pullbacked by the x → x(u) pullback (B. [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF], gives an order-four linear differential operator which, not only factorizes, but has a direct-sum factorization into two ordertwo linear differential operators that are simply conjugated:

These two linear differential operators L 2 and M 2 have respectively the pullbacked 2 F 1 hypergeometric solutions:

Recalling the identity 3 , one finds that the 2 F 1 hypergeometric functions in (B.6) can be rewritten as:

where H ± is no longer solution of a quartic equation, but of a quadratic genus-zero equation:

Let us consider example 6 of section (2.1), which diagonal is the square of the Heun function (96), which 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

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

, where the two pullbacks H

± , H

± are square root algebraic functions

where

± ) 12 are simple algebraic functions respectively solutions of 64 + p 3 (x)

where

and where

± ) 8 are simple algebraic functions, respectively solutions of:

and

and are related by the genus-zero † modular equation:

which is the same modular equation as (54). Note that one can get rid of these square root algebraic expressions for the Hauptmoduls introducing some rational parametrisation such that all the arguments of the miscellaneous 2 F 1 hypergeometric functions (94) introduced to rewrite the Heun solution, are rational functions. Such

The two Hauptmoduls H

(2)

± have the following series expansions:

and are related by the genus-zero modular equation:

Remark: The modular equation (D.15) is actually the same as the modular equation (59) of example 3. This can be seen as a consequence of identity (95), showing a relation between example 3 and example 6.

Appendix E. A rational parametrisation for Heun( The diagonal of the following rational function of the four variables x, y, z and w: R(x, y, z) = 1 1 + x y + y z + z w + w x + y w + x z . (E.1) reads:

(E.2)

This Heun function can be rewritten as miscellaneous 2 F 1 with rational pullbacks. Let us introduce the following rational parametrisation:

With this rational parametrisation (E.3) the diagonal of the rational function (E.1) can be rewritten in the following ways:

Appendix G. Periods of an extremal rational surface: a rational parametrisation.

Let us introduce a rational parametrisation for the linear differential operator M 2 and its solution (122):

With that rational change of variable (G.1) the solution (122) of M 2 becomes:

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

where

With the same rational change of variable (G.1), the solution (117) of the order-two linear differential operator L 2 can be written in term of a very simple Heun function associated with the following remarkable Heun identity:

The solution S 1 can also be written:

(G.9) † Corresponding to q ↔ q 2 in the nome, see equation ( 4) in [START_REF] Abdelaziz | Modular forms, Schwarzian conditions, and symmetries of differential equations in physics[END_REF].

Appendix H. Special 2 F 1 hypergeometric functions associated with classical modular forms

The Heun functions of this paper can all be rewritten in terms of pullbacked 2 F 1 hypergeometric functions which turn out to correspond to classical modular curves (with the exception of the "Shimura" Heun functions of section [START_REF] Bostan | Diagonal of rational functions and selected differential Galois groups[END_REF], see also Appendix J below). These 2 F 1 hypergeometric functions are not arbitrary, they are "special"

2 F 1 's corresponding to selected parameters, namely 2 F 1 's related to classical modular curves. These various 2 F 1 are often simply related

where 1 + ω + ω 2 = 0 ( ω is a third root of unity), or:

], [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], 144 x

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 256 x , (H.5)

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 108 x , (H.6)

], [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],

(H.9)

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 64

, (H.12)

In fact, all these "special" 2 F 1 's hypergeometric functions correspond to classical modular forms because they can be rewritten [START_REF] Abdelaziz | Modular forms, Schwarzian conditions, and symmetries of differential equations in physics[END_REF] as

where the pullback p(x) is in general more involved than simple rational pullbacks like (H.12) or (H.13), being often algebraic functions. For instance one has the following identities

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], H(x) , (H. [START_REF] Shaska | Genus 2 fields with degree 3 elliptic subfields[END_REF] where H(x) reads

and where A(x) reads:

, (H.17)

which is solution of

where Z = A(x) 12 . Generalizing the globally bounded nome condition approach of the previous Appendix F, we looked for all possible 2 F 1 hypergeometric functions related § to pullbacked 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], x), looking at the globally bounded condition of their nome (see (F.7)). We give here a finite list of only 28 hypergeometric functions that have integer coefficient series, that are related † to modular forms.

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 432 x ,

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 64 x ,

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 256 x ,

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 27 x ,

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 1728 x ,

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 1728 x , (H. [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF])

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 1728 x .

Using this globally bounded condition of the nome criterion, we wrote a program that went through all the values of a and b in [-1, 1] (with small increments like 1/200), with c = 1, singling out the 2 F 1 hypergeometric functions that have integer coefficients (or more generally globally bounded) series expansions, both for the 2 F 1 hypergeometric functions, and for the nome. Running this program returned to us exactly the 2 F 1 hypergeometric functions in the above list (H. [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF]), and only this list of twenty eight hypergeometric functions.

Appendix H.1. Derivatives of classical modular forms

Recalling identities (A.29), where the parameters in (A.29) verify (A. [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF]) and (A. [START_REF] Peters | A pencil of K3-surfaces related to Apéry's recurrence for ζ(3) and Fermi surfaces for potential zero[END_REF], one can also deduce, for instance for e = 1/2, some identities on some (homogeneous) derivatives of the classical modular form 2 F 1 ([1/6, 1/6], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], x) of the previous list (H.19):

Such homogeneous derivatives of classical modular forms are not classical modular forms. We have however seen, in section (2.5), that when a diagonal of a rational function can be expressed as a classical modular form, the homogeneous derivatives of that classical modular form are also diagonals of other rational functions simply deduced from the first rational function.

Appendix H.

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], x , Therefore, we see that hypergeometric functions like

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], x , (H.28)

are not classical modular forms, but are related to classical modular forms, being an order-one operator acting on a classical classical modular form.

Remark 1: The order-two linear differential operators for 2 F 1 ([ 5 6 , 7 6 ], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], x), and for 2 F 1 ([ 1 6 , 1 6 ], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], x) which corresponds to a classical modular form, are equivalent, up to a x → -x/(1 x) pullback. The of the order-two linear differential operators for 2 F 1 ([ ], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], x), one has the homomorphism

yielding the relation:

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

Using Koutschan's creative telescoping program we have obtained the telescoper of the rational function 1/(1 + x y + z x + 3 (x 2 + y 2 )) . It is an order-four linear differential operator W 4 which factors ¶ as W 4 = U 2 • V 2 where V 2 is a linear differential operator of order two with two algebraic solutions. A well-suited linear combination of these two algebraic solutions gives the integer coefficient series expansion

which is actually the diagonal of the rational function 1/(1 + x y + z x + 3 (x 2 + y 2 )). Therefore the order-two linear differential operator V 2 is the minimal order linear differential operator for the diagonal (H.32). The creative telescoping method, however, provides a larger order ‡ telescoper W 4 and, consequently, a "companion" ‡ Note that this is not the case for Heun functions. Recalling [START_REF] Musty | A database of Belyi maps[END_REF] and changing γ → γ + 1, one sees easily that the linear differential operators for Heun(-1/27, 2/27, 1/3, 2/3, 1, 1/2, -x) and Heun(-1/27, 2/27, 1/3, 2/3, 2, 1/2, -x) are not equivalent.

¶ But no direct-sum factorisation. ‡ Non minimal order linear differential operator as far as the diagonal of the rational function is concerned.

to this minimal order operator V 2 . The linear differential operator "companion" of order two, U 2 , admits the solution:

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], -315 16

The series expansion of (H.33) is a globally bounded series †. From (H.26) this last hypergeometric function is related to

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 315 x 2 16 + 315 x 2 , (H.34)

which is related, using (H.21), to:

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 315

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

Hence, if a 2 F 1 hypergeometric function appears as the diagonal of a rational function, or a (globally bounded) solution of a factor of a (non minimal order) telescoper, it seems often to be still related to a classical modular form: in this case here, it is the derivative of a classical modular form. The diagonal of (I.1) has the following series expansion with integer coefficients:

-30 x 3 + 840 x 4 -20790x 5 + 504504 x 6 -12252240 x 7 + 299304720 x 8

-7362064710 x 9 + 182298745200

The telescoper of this rational function of three variables (I.1) gives an order-three linear differential operator L 3 = L 1 ⊕ L 2 which is the direct sum (LCLM) of an order-one linear differential operator L 1 and an order-two linear differential operator L 2 , where:

† The series (H.33) becomes a series with integer coefficients with x → 96 x.

The order-one linear differential operator L 1 has the simple solution y(x) = x, and the order-two linear differential operator has the following Heun solution:

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], -27 • x (I.4)

where:

It can also be written alternatively as

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF], 108

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 1728

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF], 1728

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], 1728

], [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF], -1728 x 3 (1 + 27 x) (1 + 24 x) 3 , (I.8)

where:

(I.9)

Again we see that the derivative of a classical modular form, or more generally an order-one linear differential operator like (I.4) acting on a classical modular form, is no longer a classical modular form. With this example we see, one more time (recall section (2.5)), that a Heun function which has a series expansion with integer coefficients, is not necessarily a classical modular form but can be an order-one linear differential operator acting on a classical modular form.

Appendix J. Automorphic forms associated with a Shimura curve

Appendix J.1. The pullback in 2 F 1 ([ 1 24 , 7 24 ], [ 5 6 ], x) and 2 F 1 ([ 5 24 , 11 24 ], [ 7 6 ], x) is special.

Appendix J.3. Identities on Shimura 2 F 1 hypergeometric functions and modular equations.

There exists an algebraic series y(x) such that the two hypergeometric (J.18), (J. [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF]) actually verify the two following identities:

], y(x) , (J. [START_REF] Joyce | On the Simple Cubic Lattice Green Function[END_REF] and (with the same ρ and w )

], x

], y(x) , (J.21)

where the two complex constants ρ and w are given by:

The level of the (quaternionic) modular equation is the reduced norm of α such that Φ L (j(τ ), j(α τ )) = 0, see section 2.1 in [START_REF] Baba | Differential equations and expansions for quaternionic modular forms in the discriminant 6 case[END_REF]. For the definition of modular polynomials for these quaternionic cases see section 3.1 page 8 of [START_REF] Baba | Quaternionic Modular Forms and Exceptional Sets of Hypergeomtric Functions[END_REF]. It depends only on the integer index

which is parametrised by:

The algebraic series y(x) in (J. It is straightforward to check that the Taylor expansion (J. [START_REF] Maier | On rationally parametrized modular equations[END_REF]) is such that the two identities (J.21) or (J.20) are actually verified. This algebraic series (J.25) is actually a solution of the Schwarzian equation:

This is a consequence of the fact that the algebraic transformation ‡ x → y(x), given by the modular equation of level five (J.23), is a symmetry of the order-two linear ‡ Such a transformation is called a modular correspondence.

differential operator (J.17) (see for instance [START_REF] Abdelaziz | Modular forms, Schwarzian conditions, and symmetries of differential equations in physics[END_REF]):

The (Shimura) modular correspondences x → y(x) are algebraic solutions of that Schwarzian equation (J.28) which thus "encapsulates" all the modular correspondences.

Let us introduce τ the ratio † of the two hypergeometric (J.18), (J. [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF]) solutions of (J.17 ). One has the following identity:

This identity corresponds to the (automorphy theory) relation that one of the two solutions of (J.17) can be written (up to an overall factor) as f (τ ) 1/2 (the square root of the τ -derivative of an automorphic function x = f (τ )):

when the other solution S 2 of (J.17) can be written (up to an overall factor) as

The Schwarz map τ , given by (J.29), seen as a function of x, is a differentially algebraic function. It verifies the Schwarzian equation:

Relation (J.29) which can be rewritten

yields the following expansion of the compositional inverse of the Schwarz map called the Schwarz function (here z denotes z = τ 4 )

which can be seen as an automorphic function of the variable τ . This automorphic function verifies the Schwarzian equation:

where the 2 F 1 hypergeometric function on the RHS of the identity corresponds to a Shimura curve with elliptic points [START_REF] Abdelaziz | Diagonals of rational functions, pullbacked 2 F 1 hypergeometric functions and modular forms[END_REF][START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity Preprint[END_REF][START_REF] Abdelaziz | Diagonals of rational functions, pullbacked 2 F 1 hypergeometric functions and modular forms[END_REF]. The hypergeometric function on the LHS also corresponds to a Shimura curve. We also see the identity: 6 , for a 2 + 3 = 0 and b 2 + 2 = 0, where the 2 F 1 hypergeometric function on the left of the identity, corresponds to an automorphic form associated with a Shimura curve with elliptic points (4, 4, 4).

Appendix J.7. A level three modular equation for Shimura Heun functions.

Similar calculations can be performed on the Heun example (164). This Heun example has a level three modular equation [START_REF] Voight | Shimura curves of genus at most two[END_REF]. Introducing the following ratio (Schwarz map):

.

the elimination of the variable t yields the (genus-zero) level three modular equation [START_REF] Voight | Shimura curves of genus at most two[END_REF] which reads:

Appendix K. Heun functions that are pullbacked 2 F 1 hypergeometric functions but are not related to classical modular forms or Shimura automorphic forms.

Let us recall Maiers's identity (see Theorem 3.8 in [START_REF] Maier | On reducing the Heun equation to the hypergeometric equation[END_REF])

where ω is a sixth root of unity and η read †:

Such a Heun function is thus a pullbacked 2 F 1 hypergeometric function for all the values of the parameters α, β. This two parameters ( α, β) space is large enough to encapsulate a set of interesting subcases.

• For selected values of the two parameters α, β, the Heun function (K.1) actually reduces to Shimura 2 F 1 hypergeometric functions. For instance for α = 1 and β = 1/4, we recover the Shimura 2 F 1 hypergeometric function (153) related to the telescoper of the rational function (138). The relation (K.1) becomes:

Many other values of (α, β) yield Shimura 2 F 1 hypergeometric functions:

• In Appendix F.1 we have seen that one can introduce a nome, for that linear differential Heun operator (1), when γ = 1. This condition is, of course, necessary but not sufficient to reduce to a classical modular form. In that γ = 1 subcase where one can introduce a nome, the previous Heun-to-2 F 1 reduction (K.1) reads (since we have γ = (α + β + 1)/3 = 1):

For selected values of α we actually get 2 F 1 hypergeometric functions corresponding to classical modular curves (see (H. [START_REF] Takeuchi | Commensurability classes of arithmetic triangle groups[END_REF])). The selected values are α = 1, 1/2, 1/4, corresponding respectively to:

Let us now, consider a rational value of α, different from 1, 1/2, 1/4, for instance α = 5/8. The identity (K.5) becomes

Heun ω, 55 64

The series expansion of (K.7) gives a 2 F 1 globally bounded series †. The Heun function (K.7) is annihilated by the order-two linear differential operator Appendix L. The x ↔ 1/x and x ↔ A/x dualities.

Using the (multi-Taylor) definition of the diagonal of a rational function, it is straightforward to show, for any positive integer n, that the diagonal of a rational function R(x, y, z, w) and the diagonal of a rational function R(x n , y n , z n , w n ) are simply related. Denoting these two diagonals respectively D 1 (x) = Diag(R(x, y, z, w)) and D n (x) = Diag(R(x n , y n , z n , w n )), one has the simple relation: D n (x) = D 1 (x n ). Of course this demonstration cannot be extended to negative integers n, in particular n = -1.

Let us revisit example 5 where the diagonal of the rational function (80) is the Heun series expansion (83). Let us consider, instead of the rational function (80), the rational function where the four variables (x, y, z, w) have been changed into their reciprocal (1/x, 1/y, 1/z, 1/w) in the rational function R(x, y, z, w) given by (80):

The diagonal of this "reciprocal" rational function (L.1) reads:

The telescoper of the diagonal of this rational function (L.1) of four variables reads:

This order-three linear differential operator M 3 given by (L.3) actually corresponds to the telescoper L 3 given by (82) pullbacked by x → 1/x. Recalling the solution (83) of the telescoper L 3 given by (82), one finds easily that the diagonal series expansion (L.2), solution of the order-three linear differential operator M 3 , can be written as

In other words there is a simple relation between the diagonal of (80) and the diagonal of its "reciprocal" (L.1):

This is confirmed by the homomorphism between the two telescopers L 3 and M 3 :

Recalling the fact that the order-three linear differential operator M 3 corresponds to the telescoper L 3 pullbacked by x → 1/x, this homomorphism means a slightly puzzling homomorphism between the telescoper L 3 and itself pullbacked by x → 1/x.

Appendix L.1. The x ←→ A/x dualities: dualities on the telescopers.

Let us revisit example 1, considering, similarly, the diagonal of the rational function of four variables [START_REF] Bostan | Globally nilpotent differential operators and the square Ising model[END_REF], its telescoper L 3 given by ( 13), and the corresponding Heun solutions [START_REF] Shaska | Genus 2 fields with degree 3 elliptic subfields[END_REF]. We introduce, as previously, a new rational function (for the rational function [START_REF] Bostan | Globally nilpotent differential operators and the square Ising model[END_REF] of four variables R(x, y, z, w)) corresponding to simple involutions of the form t → A/t on the four variables: The telescoper of this rational function (L.7) reads:

This order-three linear differential operator (L.8) is exactly the telescoper L 3 given by ( 13), pullbacked by x → -1/64/x. The telescoper M 3 has the following Heun solutions (to be compared with (15)) These (non globally bounded) solutions (L.9), (L.10), and (L.11) are different from the diagonal of the rational function (L.7) which is trivial in that case. We are in the situation where the solutions of the telescoper are different from the diagonal of the rational function: they are "Periods" [START_REF] Kontsevich | Periods[END_REF] over non-evanescent cycles of the algebraic variety corresponding to the rational function. Example 1 was seen to correspond to a classical modular form with an integer series and a formal series with a logarithm at x = 0. The point at infinity x = ∞ is an elliptic point with no logarithm, and series that are not globally bounded. Let us note that this order-three linear differential operator (L.8) has the same singularities as the order-three operator L 3 given by [START_REF] Voight | Shimura curves of genus at most two[END_REF]. Consequently the first parameter of these Heun functions solutions and of the Heun function ( 15 One can easily perform the same calculations for examples 2, 3, 4, 6 of section (2.1), and find exactly similar results †, namely the fact that the telescoper T R of the new rational function (L.12) is exactly the telescoper T R of the rational function, pullbacked by x → A/x. Note that, for these examples, there exists a choice of the constant A such that the singularities of T R and T R remain the same. For examples 2, 3, 4, 6 one must take respectively A = -1/27, -1/128, -1/16, 1/144. For the first lattice Green example of section (1.1) take A = 1/4.

Remark: Do note that the other Shimura example of section (3.1) has been build that way: the rational function (156) is actually the rational function of (138) where (x, y, z, w) has been changed into (1/x, 1/y, 1/z, 1/w) . Again we have that the telescoper M 3 of the "recipocal" rational function is exactly the telescoper L 3 of the rational function, pullbacked by x → 1/x. Appendix L.2. Periods of extremal rational surfaces: the x ↔ 1/x duality on the telescoper.

Let us recall the rational function (110) which diagonal was the sum of two classical modular forms. Let us consider the rational function where the three variables (x, y, z) have been changed into their reciprocal (1/x, 1/y, 1/z) in the rational function R(x, y, z) given by (110):

The telescoper of this rational function (L.13) is an order-four linear differential operator which is the direct sum of two order-two operators L