Abstract. We show that the n-fold integrals χ (n) of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green functions, correspond to a distinguished class of functions generalising algebraic functions: they are actually diagonals of rational functions. As a consequence, the power series expansions of the, analytic at x = 0, solutions of these linear differential equations "Derived From Geometry" are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. We also give several results showing that the unique analytical solution of Calabi-Yau ODEs, and, more generally, Picard-Fuchs linear ODEs with solutions of maximal weights, are always diagonal of rational functions. Besides, in a more enumerative combinatorics context, generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity of ODEs.This paper is the short version of the larger (100 pages) version [19], available on http: // arxiv. org/ abs/ 1211. 6031 , where all the detailed proofs are given and where a larger set of examples is displayed.

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We introduce some multiple integrals that are expected to have the same singularities as the singularities of the n-particle contributions χ(n) to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differential equation satisfied by these multiple integrals for n = 1, 2, 3, 4 and only modulo some primes for n = 5 and 6, thus providing a large set of (possible) new singularities of χ(n). We discuss the singularity structure for these multiple integrals by solving the Landau conditions. We find that the singularities of the associated ODEs identify (up to n = 6) with the leading pinch Landau singularities. The second remarkable obtained feature is that the singularities of the ODEs associated with the multiple integrals reduce to the singularities of the ODEs associated with a finite number of one-dimensional integrals. Among the singularities found, we underline the fact that the quadratic polynomial condition 1 + 3w + 4w2 = 0, that occurs in the linear differential equation of χ(3), actually corresponds to a remarkable property of selected elliptic curves, namely the occurrence of complex multiplication. The interpretation of complex multiplication for elliptic curves as complex fixed points of the selected generators of the renormalization group, namely isogenies of elliptic curves, is sketched. Most of the other singularities occurring in our multiple integrals are not related to complex multiplication situations, suggesting an interpretation in terms of (motivic) mathematical structures beyond the theory of elliptic curves.

Abstract. We calculate very long low-and high-temperature series for the susceptibility χ of the square lattice Ising model as well as very long series for the five-particle contribution χ (5) and six-particle contribution χ (6) . These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150000 CPU hours on computer clusters. The series for χ (low-and high-temperature regime), χ (5) and χ (6) are now extended to 2000 terms. In addition, for χ (5) , 10000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by χ (5) modulo a prime.A diff-Padé analysis of the 2000 terms series for χ (5) and χ (6) confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the n-particle components of the susceptibility, up to a small set of "additional" singularities. The exponents at all the singularities of the Fuchsian linear ODE of χ (5) and the (as yet unknown) ODE of χ (6) are given: they are all rational numbers. We find the presence of singularities at w = 1/2 for the linear ODE of χ (5) , and w 2 = 1/8 for the ODE of χ (6) , which are not singularities of the "physical" χ (5) and χ (6) , that is to say the series-solutions of the ODE's which are analytic at w = 0.Furthermore, analysis of the long series for χ (5) (and χ (6) ) combined with the corresponding long series for the full susceptibility χ yields previously conjectured singularities in some χ (n) , n ≥ 7. The exponents at all these singularities are also seen to be rational numbers.We also present a mechanism of resummation of the logarithmic singularities of the χ (n) leading to the known power-law critical behaviour occurring in the full χ, and perform a power spectrum analysis giving strong arguments in favor of the existence of a natural boundary for the full susceptibility χ.

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...