We recall various multiple integrals with one parameter, related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility, to the (lattice) form factors, to the two-point correlation functions and to their λ-extensions. The univariate analytic functions defined by these integrals are holonomic and even G-functions: they satisfy Fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties. We recall the explicit forms, found in previous work, of these Fuchsian equations, as well as their russian-doll and direct sum structures. These differential operators are very selected Fuchsian linear differential operators, and their remarkable properties have a deep geometrical origin: they are all globally nilpotent, or, sometimes, even have zero p-curvature. We also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known. Focusing on the factorised parts of all these operators, we find out that the global nilpotence of the factors (resp. p-curvature nullity) corresponds to a set of selected structures of algebraic geometry: elliptic curves, modular curves, curves of genus five, six, . . . , and even a remarkable weight-1 modular form emerging in the three-particle contribution χ (3) of the magnetic susceptibility of the square Ising model. Noticeably, this associated weight-1 modular form is also seen in the factors of the differential operator for another n-fold integral of the Ising class, ΦH , for the staircase polygons counting, and in Apéry's study of ζ(3). G-functions naturally occur as solutions of globally nilpotent operators. In the case where we do not have G-functions, but Hamburger functions (one irregular singularity at 0 or ∞) that correspond to the confluence of singularities in the scaling limit, the p-curvature is also found to verify new structures associated with simple deformations of the nilpotent property. PACS: 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, 14KxxGlobally nilpotent operators 2 Key-words: Globally nilpotent operators, p-curvature, G-functions, arithmetic Gevrey series, Form factors of the square Ising model, susceptibility of the Ising model, Fuchsian linear differential equations, moduli space of curves, two-point correlation functions of the lattice Ising model, complete elliptic integrals, scaling limit of the Ising model, apparent singularities, modular forms, Atkin-Lehmer involutions, Fricke involutions, Dedekind eta functions, Weber modular functions, Calabi-Yau manifolds, three-choice polygons, enumerative combinatorics. IntroductionGenerating large series expansions of physical quantities that are quite often defined as n-fold integrals is the bread and butter of lattice statistical mechanics, enumerative combinatorics, and more generally theoretical physics. The n-fold integrals...

Abstract. We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contributionsχ (n) 's of the susceptibility of the Ising model for n ≤ 6, are linear differential operators "associated with elliptic curves". Beyond the simplest differential operators factors which are homomorphic to symmetric powers of the second order operator associated with the complete elliptic integral E, the second and third order differential operators Z 2 , F 2 , F 3 ,L 3 can actually be interpreted as modular forms of the elliptic curve of the Ising model. A last order-four globally nilpotent linear differential operator is not reducible to this elliptic curve, modular forms scheme. This operator is shown to actually correspond to a natural generalization of this elliptic curve, modular forms scheme, with the emergence of a CalabiYau equation, corresponding to a selected 4 F 3 hypergeometric function. This hypergeometric function can also be seen as a Hadamard product of the complete elliptic integral K, with a remarkably simple algebraic pull-back (square root extension), the corresponding Calabi-Yau fourth-order differential operator having a symplectic differential Galois group SP (4, C). The mirror maps and higher order Schwarzian ODEs, associated with this Calabi-Yau ODE, present all the nice physical and mathematical ingredients we had with elliptic curves and modular forms, in particular an exact (isogenies) representation of the generators of the renormalization group, extending the modular group SL(2, Z) to a GL(2, Z) symmetry group.

We revisit miscellaneous linear differential operators mostly associated with lattice Green functions in arbitrary dimensions, but also Calabi-Yau operators and order-seven operators corresponding to exceptional differential Galois groups. We show that these irreducible operators are not only globally nilpotent, but are such that they are homomorphic to their (formal) adjoints. Considering these operators, or, sometimes, equivalent operators, we show that they are also such that, either their symmetric square or their exterior square, have a rational solution. This is a general result: an irreducible linear differential operator homomorphic to its (formal) adjoint is necessarily such that either its symmetric square, or its exterior square has a rational solution, and this situation corresponds to the occurrence of a special differential Galois group. We thus define the notion of being "Special Geometry" for a linear differential operator if it is irreducible, globally nilpotent, and such that it is homomorphic to its (formal) adjoint. Since many Derived From Geometry n-fold integrals ("Periods") occurring in physics, are seen to be diagonals of rational functions, we address several examples of (minimal order) operators annihilating diagonals of rational functions, and remark that they also seem to be, systematically, associated with irreducible factors homomorphic to their adjoint. -adjoint operators, homomorphism or equivalence of differential operators, Special Geometry, globally bounded series, diagonal of rational functions. † Sorbonne Universités (previously the UPMC was in Paris Universitas). ¶ Their corresponding linear differential operators are necessarily globally nilpotent [8]. ‡ One shows that there are no rational solutions of symmetric powers in degree 2, 3,4,6,8,9,12, using an algorithm in M. van Hoeij et al. [9] ♯ This operator is actually homomorphic to its adjoint (see below) with non-trivial order-two intertwiners. † This operator has an irregular singularity at infinity. At x = ∞ the solutions behave like: t · (1 + 77/72 t 2 + · · · ), and exp(−2/t)/t · (1 + 13/36 t + · · · ) where t = ±1/ √ x. † In maple the exterior power is normalised to be a monic operator (the head polynomial is normalised to 1). § The intertwiner T is given by the command Homomorphisms(L,L) of the DEtools package in Maple [33]. ♯ Note that the constraint on the order rules out the "tautological" intertwining relation, satisfied by any operator, like L · adjoint(T ) = T · adjoint(L) with T = L.¶ It is easy to show, in the case of an homomorphism of an operator L with its adjoint, that the intertwiner on the right-hand-side of (8) is necessarily equal to the adjoint of the intertwiner on the left-hand-side. Actually, from the equivalence L · T = S · adjoint(L), taking adjoint on both sides gives adjoint(T ) · adjoint(L) = L · adjoint(S). For irreducible L, the intertwiner is unique, so S = adjoint(T ). ¶ Corresponding to a change of variable: F (x) → F (x/(1 − 18 x))/(1 − 18 x).

Abstract. In this paper, we examine the non-relativistic stationary Schrödinger equation from a differential Galois-theoretic perspective. The main algorithmic tools are pullbacks of second order ordinary linear differential operators, so as to achieve rational function coefficients ("algebrization"), and Kovacic's algorithm for solving the resulting equations. In particular, we use this Galoisian approach to analyze Darboux transformations, Crum iterations and supersymmetric quantum mechanics. We obtain the ground states, eigenvalues, eigenfunctions, eigenstates and differential Galois groups of a large class of Schrödinger equations, e.g. those with exactly solvable and shape invariant potentials (the terms are defined within). Finally, we introduce a method for determining when exact solvability is possible.