From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

Boukraa, S.; Hassani, S.; Maillard, J.‐M.; Zenine, N.

doi:10.3842/sigma.2007.099

Cited by 1 publication

(8 citation statements)

References 113 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us recall briefly Krammer's counter-example † † to Dwork's conjecture [41,42] which comes from the periods of a family of abelian surfaces over a Shimura curve P 1 \ {0, 1, 81, ∞}: [60] one should read ln A i , instead of A i , in the equations defining the A i after equation (H.2) in [60,61].…”

Section: Krammer's Counterexamplementioning

confidence: 99%

“…Global nilpotence is a stronger structure than having regular singularities with rational exponents [53]. It is a very strong arithmetic property with a large number of remarkable consequences: for instance modulo any prime p the Fuchsian linear differential operator factorizes, and for almost all primes, it factorizes into linear differential operators of order one, each operator of order one having rational solutions modulo p. Such a property is quite well illustrated 30 in appendix H of [59,60] on n-fold integrals related to Apéry's analysis of ζ (3). In that case, we even have a factorization into order-one linear differential operators on the rationals Q and not only modulo (almost all) primes.…”

Section: Recalls On Global Nilpotencementioning

confidence: 99%

“…The numerator N 1 is the solution of an order-14 linear operator that can be obtained as the LCLM of L 10 and L 4 . We have calculated 60 the p-curvature for the first two hundred primes, of this order-14 operator (the degree of the polynomial coefficients is 320) and found that its characteristic polynomial is T 14 , its minimal polynomial being T 2 . The global nilpotence of this order-14 operator is a straight consequence of the expression of N 1 in (87).…”

Section: Beyond Holonomic Functions: Ratio Of Holonomic Functionsmentioning

confidence: 99%

“…¶ Note a misprint in[60] one should read ln A i , instead of A i , in the equations defining the A i after equation (H.2) in[60,61]. † † The uniformizing linear differential equation of an arithmetic Fuchsian lattice[41].…”

mentioning

confidence: 99%

“…‡ Note a misprint in[60] one should read ln A i , instead of A i , in the equations defining the A i after equation (H.2) in[61,71].…”

mentioning

confidence: 99%

See 4 more Smart Citations

Globally nilpotent differential operators and the square Ising model

Bostan¹,

Boukraa²,

Hassani³

et al. 2009

J. Phys. A: Math. Theor.

232

View full text Add to dashboard Cite

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...

show abstract