On globally nilpotent differential equations

Dettweiler, Michael; Reiter, Stefan

doi:10.1016/j.jde.2010.02.009

Cited by 6 publications

(25 citation statements)

References 11 publications

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“…where A = α 2 − 4 and Operator (109) is globally nilpotent [8,40] for any rational value of the parameter A, the Jordan reduction of its p-curvature [8,40] reading…”

Section: The Lattice Green Function Of the Anisotropic Simple Cubic Lmentioning

confidence: 99%

“…where P is a slightly involved ‡ polynomial expression, on exterior square of product of the two (not necessarily self-adjoint) operators These order-seven linear differential operators are globally nilpotent [8,40], the Jordan reduction of their p-curvature [40,8] reading: of characteristic polynomial equal to the minimal polynomial P (λ) = λ 7 . These operators are MUM, so they have the traditional "triangular log structure" [20], the formal series solutions with a log being of the form y (obtained fromÊ 1 by taking the LCLM with D x and rightdividing by D x ), the Yukawa couplings (C.5), as well as these adjoint Yukawa couplings (C.8), as well as the W n 's, as well as the W n 's of the adjoint operators, are still globally bounded (see [18,19]).…”

Section: Appendix B2 Equivalence Of Operators Satisfying the Calabimentioning

confidence: 99%

“…Operator G 4D 4 is a globally nilpotent operator, as can be seen on the Jordan form reduction of its p-curvature [8,40].…”

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confidence: 99%

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Differential algebra on lattice Green functions and Calabi–Yau operators

Boukraa

Hassani

Maillard

et al. 2014

J. Phys. A: Math. Theor.

152

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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).

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“…where A = α 2 − 4 and Operator (109) is globally nilpotent [8,40] for any rational value of the parameter A, the Jordan reduction of its p-curvature [8,40] reading…”

Section: The Lattice Green Function Of the Anisotropic Simple Cubic Lmentioning

confidence: 99%

Section: Appendix B2 Equivalence Of Operators Satisfying the Calabimentioning

confidence: 99%

See 1 more Smart Citation

Differential algebra on lattice Green functions and Calabi–Yau operators

Boukraa

Hassani

Maillard

et al. 2014

J. Phys. A: Math. Theor.

152

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“…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%

Globally nilpotent differential operators and the square Ising model

Bostan¹,

Boukraa²,

Hassani³

et al. 2009

J. Phys. A: Math. Theor.

232

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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...

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“…Hence most of the Lamé equations obtained in the above way are not pull-backs of a Gauss hypergeometric differential equation. In the literature we found only Lamé equations with arithmetic Fuchsian monodromy group (s. [23] and [9]) or geometric Heun equations with nondiscrete monodromy group (s. [12]) being no pull-backs of hypergeometric differential equations, providing counter examples to the above mentioned conjecture of Dwork. (Recently also differential equations with 5 singularities and non-arithmetic Fuchsian monodromy group were computed in [7].)…”

Section: Introductionmentioning

confidence: 99%

Halphen's transform and middle convolution

Reiter

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We show that the Halphen transform of a Lamé equation can be written as the symmetric square of the Lamé equation followed by an Euler transform. We use this to compute a list of Lamé equations with arithmetic Fuchsian monodromy group. It contains all those Lamé equations where the quaternion algebra A over k associated to the arithmetic Fuchsian group is a quaternion algebra A over Q. Further we classify all geometric braid group orbits in SL 2 (Z) 4 with the possible exception of three orbits.

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On globally nilpotent differential equations

Cited by 6 publications

References 11 publications

Differential algebra on lattice Green functions and Calabi–Yau operators

Differential algebra on lattice Green functions and Calabi–Yau operators

Globally nilpotent differential operators and the square Ising model

Halphen's transform and middle convolution

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