arXiv subjects

J-M. Maillard

Publications and source records attributed to J-M. Maillard.

At least 19 recordsLinked to original sources

arXiv2025

Modular correspondences and replicable functions (unabridged version)

Landen transformation, and more generally modular correspondences, can be seen to be exact symmetries of some integrable lattice models, like the square Ising model, or the Baxter model. They are solutions of remarkable Schwarzian equations and have some compositional properties. Most of the known examples correspond, in an elliptic curves framework, to an automorphy property of pullbacked $_2F_1$ hypergeometric functions, associated with modular forms. It is, however, important to underline that these Schwarzian equations go beyond an elliptic curves, and hypergeometric functions framework. The question of a modular correspondence interpretation of the solutions of these ``Schwarzian'' equations was clearly an open question. This paper tries to shed some light on this open question. We first shed some light on the very nature of a one-parameter series solution of the Schwarzian equation. This one-parameter series is not generically a modular correspondence series, but it actually reduces to an infinite set of modular correspondence series for an infinite set of (N-th root of unity) values of the parameter. We also provide an example of two-parameter series, with a compositional property, solution of a Schwarzian equation. We finally provide simple pedagogical examples that are very similar to modular correspondence series, but are far beyond the elliptic curves framework. These last examples show that the modular correspondence-like series, or the nome-like series, are not necessarily globally bounded. The results of that paper can be seen as an incentive to study differentially algebraic series with integer coefficients, in physics or enumeratice combinatorics.

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arXiv2025

On the diagonals of rational functions: the minimal number of variables (unabridged version)

From some observations on the linear differential operators occurring in the Lattice Green function of the d-dimensional face centred and simple cubic lattices, and on the linear differential operators occurring in the n-particle contributions to the magnetic susceptibility of the square Ising model, we forward some conjectures on the diagonals of rational functions. These conjectures are also in agreement with exact results we obtain for many Calabi-Yau operators, and many other examples related, or not related to physics. Consider a globally bounded power series which is the diagonal of rational functions of a certain number of variables, annihilated by an irreducible minimal order linear differential operator homomorphic to its adjoint. Among the logarithmic formal series solutions, at the origin, of this operator, call n the highest power of the logarithm. We conjecture that this diagonal series can be represented as a diagonal of a rational function with a minimal number of variables N_v related to this highest power n by the relation N_v = n +2. Since the operator is homomorphic to its adjoint, its differential Galois group is symplectic or orthogonal. We also conjecture that the symplectic or orthogonal character of the differential Galois group is related to the parity of the highest power n, namely symplectic for n odd and orthogonal for n even. We also sketch the case where the denominator of the rational function is not irreducible and is the product of, for instance, two polynomials. The analysis of the linear differential operators annihilating the diagonal of rational function where the denominator is the product of two polynomials, sheds some light on the emergence of such mixture of direct sums and products of factors. The conjecture N_v = n +2 still holds for such reducible linear differential operators.

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arXiv2023

Plea for diagonals and telescopers of rational functions

This paper is a plea for diagonals and telescopers of rational, or algebraic, functions using creative telescoping, in a computer algebra experimental mathematics learn-by-examples approach. We show that diagonals of rational functions (and this is also the case with diagonals of algebraic functions) are left invariant when one performs an infinite set of birational transformations on the rational functions. These invariance results generalize to telescopers. We cast light on the almost systematic property of homomorphism to their adjoint of the telescopers of rational, or algebraic, functions. We shed some light on the reason why the telescopers, annihilating the diagonals of rational functions of the form P/Q^k and 1/Q, are homomorphic. For telescopers with solutions (periods) corresponding to integration over non-vanishing cycles, we have a slight generalization of this result. We introduce some challenging examples of generalization of diagonals of rational functions, like diagonals of transcendental functions, yielding simple $_2F_1$ hypergeometric functions associated with elliptic curves, or (differentially algebraic) lambda-extension of correlation of the Ising model.

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arXiv2022

The lambda extensions of the Ising correlation functions C(M, N)

We revisit, with a pedagogical heuristic motivation, the lambda extension of the low-temperature row correlation functions C(M,N) of the two-dimensional Ising model. In particular, using these one-parameter series to understand the deformation theory around selected values of $λ$, namely $λ= \cos(π\, m/n)$ with m and n integers, we show that these series yield perturbation coefficients, generalizing form factors, that are D-finite functions. As a by-product these exact results provide an infinite number of highly non-trivial identities on the complete elliptic integrals of the first and second kind. These results underline the fundamental role of Jacobi theta functions and Jacobi forms, the previous D-finite functions being (relatively simple) rational functions of Jacobi theta functions, when rewritten in terms of the nome of elliptic functions.

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arXiv2022

Diagonals of rational functions: from differential algebra to effective algebraic geometry (unabridged version)

We show that the results we had obtained on diagonals of nine and ten parameters families of rational functions using creative telescoping, yielding modular forms expressed as pullbacked $ _2F_1$ hypergeometric functions, can be obtained, much more efficiently, calculating the $ j$-invariant of an elliptic curve canonically associated with the denominator of the rational functions. In the case where creative telescoping yields pullbacked $ _2F_1$ hypergeometric functions, we generalize this result to other families of rational functions in three, and even more than three, variables. We also generalise this result to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, foliation in elliptic curves. We also extend these results to rational functions in three variables when the denominator is associated with a {\em genus-two curve such that its Jacobian is a split Jacobian} corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked $ _2F_1$ hypergeometric solutions, the denominator corresponding to an algebraic variety being not simply foliated in elliptic curves, but having a selected elliptic curve in the variety explaining the pullbacked $ _2F_1$ hypergeometric solution.

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arXiv2020

The Ising correlation $C(M,N)$ for $ν=-k$

We present Painlev{é} VI sigma form equations for the general Ising low and high temperature two-point correlation functions $ C(M,N)$ with $M \leq N $ in the special case $ν= -k$ where $ν= \, \sinh 2E_h/k_BT/\sinh 2E_v/k_BT$. More specifically four different non-linear ODEs depending explicitly on the two integers $M $ and $N$ emerge: these four non-linear ODEs correspond to distinguish respectively low and high temperature, together with $ M+N$ even or odd. These four different non-linear ODEs are also valid for $M \ge N$ when $ ν= -1/k$. For the low-temperature row correlation functions $ C(0,N)$ with $ N$ odd, we exhibit again for this selected $ν= \, -k$ condition, a remarkable phenomenon of a Painlevé VI sigma function being the sum of four Painlevé VI sigma functions having the same Okamoto parameters. We show in this $ν= \, -k$ case for $ T < T_c $ and also $ T > T_c$, that $ C(M,N)$ with $ M \leq N $ is given as an $ N \times N$ Toeplitz determinant.

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arXiv2020

On Christol's conjecture

We show that the unresolved examples of Christol's conjecture $ \, _3F_{2}\left([2/9,5/9,8/9],[2/3,1],x\right)$ and $_3F_{2}\left([1/9,4/9,7/9],[1/3,1],x\right)$, are indeed diagonals of rational functions. We also show that other $\, _3F_2$ and $\, _4F_3$ unresolved examples of Christol's conjecture are diagonals of rational functions. Finally we give two arguments that show that it is likely that the $\, _3F_2([1/9, 4/9, 5/9], \, [1/3,1], \, 27 \cdot x)$ function is a diagonal of a rational function.

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arXiv2019

Heun functions and diagonals of rational functions (unabridged version)

We provide a set of diagonals of simple rational functions of three and four variables that are squares of Heun functions. These Heun functions obtained through creative telescoping, turn out to be either pullbacked $_2F_1$ hypergeometric functions and in fact classical modular forms. We also obtain Heun functions that are Shimura curves as solutions of telescopers of rational functions.

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arXiv2018

Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)

We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. We find that a seven-parameter rational function of three variables with a numerator equal to one (reciprocal of a polynomial of degree two at most) can be expressed as a pullbacked 2F1 hypergeometric function. This result can be seen as the simplest non-trivial family of diagonals of rational functions. We focus on some subcases such that the diagonals of the corresponding rational functions can be written as a pullbacked 2F1 hypergeometric function with two possible rational functions pullbacks algebraically related by modular equations, thus showing explicitely that the diagonal is a modular form. We then generalise this result to eight, nine and ten parameters families adding some selected cubic terms at the denominator of the rational function defining the diagonal. We finally show that each of these previous rational functions yields an infinite number of rational functions whose diagonals are also pullbacked 2F1 hypergeometric functions and modular forms.

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arXiv2017

Analyticity of the Ising susceptibility: An interpretation

We discuss the implications of studies of partition function zeros and equimodular curves for the analytic properties of the Ising model on a square lattice in a magnetic field. In particular we consider the dense set of singularities in the susceptibility of the Ising model at $H=0$ found by Nickel and its relation to the analyticity of the field theory computations of Fonseca and Zamolodchikov.

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arXiv2016

Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?

We study the class of non-holonomic power series with integer coefficients that reduce, modulo primes, or powers of primes, to algebraic functions. In particular we try to determine whether the susceptibility of the square-lattice Ising model belongs to this class, and more broadly whether the susceptibility is a solution of a differentially algebraic equation. Initial results on Tutte's non-linear ordinary differential equation (ODE) and other simple quadratic non-linear ODEs suggest that a large set of differentially algebraic power series solutions with integer coefficients might reduce to algebraic functions modulo primes, or powers of primes. Here we give several examples of series with integer coefficients and non-zero radius of convergence that reduce to algebraic functions modulo (almost) every prime (or power of a prime). These examples satisfy differentially algebraic equations with the encoding polynomial occasionally possessing quite high degree (and thus difficult to identify even with long series). Additionally, we have extended both the high- and low-temperature Ising square-lattice susceptibility series to 5043 coefficients. We find that even this long series is insufficient to determine whether it reduces to algebraic functions modulo $3$, $5$, etc. This negative result is in contrast to the comparatively easy confirmation that the corresponding series reduce to algebraic functions modulo powers of $2$.

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arXiv2016

The anisotropic Ising correlations as elliptic integrals: duality and differential equations

We present the reduction of the correlation functions of the Ising model on the anisotropic square lattice to complete elliptic integrals of the first, second and third kind, the extension of Kramers-Wannier duality to anisotropic correlation functions, and the linear differential equations for these anisotropic correlations. More precisely, we show that the anisotropic correlation functions are homogeneous polynomials of the complete elliptic integrals of the first, second and third kind. We give the exact dual transformation matching the correlation functions and the dual correlation functions. We show that the linear differential operators annihilating the general two-point correlation functions are factorised in a very simple way, in operators of decreasing orders.

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arXiv2016

Lattice Green Functions: the d-dimensional face-centred cubic lattice, d=8, 9, 10, 11, 12

We previously reported on a recursive method to generate the expansion of the lattice Green function of the $d$-dimensional face-centred cubic lattice (fcc). The method was used to generate many coefficients for d=7 and the corresponding linear differential equation has been obtained. In this paper, we show the strength and the limit of the method by producing the series and the corresponding linear differential equations for d=8, 9, 10, 11, 12. The differential Galois groups of these linear differential equations are shown to be symplectic for d=8, 10, 12 and orthogonal for d= 9, 11. The recursion relation naturally provides a 2-dimensional array $ T_d(n,j)$ where only the coefficients $ t_d(n,0)$ correspond to the coefficients of the lattice Green function of the d-dimensional fcc. The coefficients $ t_d(n,j)$ are associated to D-finite bivariate series annihilated by linear partial differential equations that we analyze.

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arXiv2016

The perimeter generating functions of three-choice, imperfect, and 1-punctured staircase polygons

We consider the isotropic perimeter generating functions of three-choice, imperfect, and 1-punctured staircase polygons, whose 8th order linear Fuchsian ODEs are previously known. We derive simple relationships between the three generating functions, and show that all three generating functions are joint solutions of a common 12th order Fuchsian linear ODE. We find that the 8th order differential operators can each be rewritten as a direct sum of a direct product, with operators no larger than 3rd order. We give closed-form expressions for all the solutions of these operators in terms of $_2F_1$ hypergeometric functions with rational and algebraic arguments. The solutions of these linear differential operators can in fact be expressed in terms of two modular forms, since these $_2F_1$ hypergeometric functions can be expressed with two, rational or algebraic, pullbacks.

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arXiv2015

Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics

We recall that the full susceptibility series of the Ising model, modulo powers of the prime 2, reduce to algebraic functions. We also recall the non-linear polynomial differential equation obtained by Tutte for the generating function of the q-coloured rooted triangulations by vertices, which is known to have algebraic solutions for all the numbers of the form $2 +2 \cos(jπ/n)$, the holonomic status of the q= 4 being unclear. We focus on the analysis of the q= 4 case, showing that the corresponding series is quite certainly non-holonomic. Along the line of a previous work on the susceptibility of the Ising model, we consider this q=4 series modulo the first eight primes 2, 3, ... 19, and show that this (probably non-holonomic) function reduces, modulo these primes, to algebraic functions. We conjecture that this probably non-holonomic function reduces to algebraic functions modulo (almost) every prime, or power of prime numbers. This raises the question to see whether such remarkable non-holonomic functions can be seen as ratio of diagonals of rational functions, or algebraic, functions of diagonals of rational functions.

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arXiv2015

Diagonals of rational functions and selected differential Galois groups

We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. In all the examples emerging from physics, the minimal linear differential operators annihilating these diagonals of rational functions have been shown to actually possess orthogonal or symplectic differential Galois groups. In order to understand the emergence of such orthogonal or symplectic groups, we analyze exhaustively three sets of diagonals of rational functions, corresponding respectively to rational functions of three variables, four variables and six variables. We impose the constraints that the degree of the denominators in each variable is at most one, and the coefficients of the monomials are 0 or $ \pm 1$, so that the analysis can be exhaustive. We find the minimal linear differential operators annihilating the diagonals of these rational functions of three, four, five and six variables. We find that, even for these sets of examples which, at first sight, have no relation with physics, their differential Galois groups are always orthogonal or symplectic groups. We discuss the conditions on the rational functions such that the operators annihilating their diagonals do not correspond to orthogonal or symplectic differential Galois groups, but rather to generic special linear groups.

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arXiv2015

Automata and the susceptibility of the square lattice Ising model modulo powers of primes

We study the full susceptibility of the Ising model modulo powers of primes. We find exact functional equations for the full susceptibility modulo these primes. Revisiting some lesser-known results on discrete finite automata, we show that these results can be seen as a consequence of the fact that, modulo 2^r, one cannot distinguish the full susceptibility from some simple diagonals of rational functions which reduce to algebraic functions modulo 2^r, and, consequently, satisfy exact functional equations modulo 2^r. We sketch a possible physical interpretation of these functional equations modulo 2^r as reductions of a master functional equation corresponding to infinite order symmetries such as the isogenies of elliptic curves. One relevant example is the Landen transformation which can be seen as an exact generator of the Ising model renormalization group. We underline the importance of studying a new class of functions corresponding to ratios of diagonals of rational functions: they reduce to algebraic functions modulo powers of primes and they may have solutions with natural boundaries.

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arXiv2014

Scaling functions in the square Ising model

We show and give the linear differential operators ${\cal L}^{scal}_q$ of order q= n^2/4+n+7/8+(-1)^n/8, for the integrals $I_n(r)$ which appear in the two-point correlation scaling function of Ising model $ F_{\pm}(r)= \lim_{scaling} {\cal M}_{\pm}^{-2} < σ_{0,0} \, σ_{M,N}> = \sum_{n} I_{n}(r)$. The integrals $ I_{n}(r)$ are given in expansion around r= 0 in the basis of the formal solutions of $\, {\cal L}^{scal}_q$ with transcendental combination coefficients. We find that the expression $ r^{1/4}\,\exp(r^2/8)$ is a solution of the Painlevé VI equation in the scaling limit. Combinations of the (analytic at $ r= 0$) solutions of $ {\cal L}^{scal}_q$ sum to $ \exp(r^2/8)$. We show that the expression $ r^{1/4} \exp(r^2/8)$ is the scaling limit of the correlation function $ C(N, N)$ and $ C(N, N+1)$. The differential Galois groups of the factors occurring in the operators $ {\cal L}^{scal}_q$ are given.

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