Renormalization, isogenies and rational symmetries of differential equations

We give an example of infinite order rational transformation that leaves a linear differential equation covariant. This example can be seen as a non-trivial but still simple illustration of an exact representation of the renormalization group.


Introduction
There is no need to underline the success of the renormalization group revisited by Wilson [1,2] which is nowadays seen as a fundamental symmetry in lattice statistical mechanics or field theory. It contributed to promote 2-d conformal field theories and/or scaling limits of second order phase transition in lattice statistical mechanics ¶. If one does not take into account most of the subtleties of the renormalization group, the simplest sketch of the renormalization group corresponds to Migdal-Kadanoff decimation calculations where the new coupling constants created at each step of the (real-space) decimation calculations are forced † to stay in some (slightly arbitrary) finite dimensional parameter space. This drastic projection may be justified by the hope that the basin of attraction of the fixed points of the corresponding (renormalization) transformation in the parameter space is "large enough".
One heuristic example is always given because it is one of the very few examples of exact renormalization, the renormalization of the one-dimensional Ising model without a magnetic field. It is a straightforward undergraduate exercise to show that performing various decimations summing over every two, or three or ... N spins, one gets exact generators of the renormalization group reading T N : t → t N where t is (with standard notations) the high temperature variable t = tanh(K). It is easy to see that these transformations T N , depending on the integer N , commute together. Such an exact symmetry is associated with a covariance of the partition function per site Z(t) = C(t) · Z(t 2 ). In this particular case one recovers the (very simple) expression of the partition function per site, 2 cosh(K), as an infinite product of the action of (for instance) T 2 on the cofactor C(t). In this very simple case, this corresponds to using the identity (valid for |x| < 1): For T 3 : t → t 3 one must use the identity and for T N : t → t N a similar identity where the 3 in the exponents is changed into N .
Another simple heuristic example is the one-dimensional Ising model with a magnetic field. Straightforward calculations enable to get an infinite number of exact generators of the corresponding renormalization group, represented as rational transformations ‡: where the first two transformations T 2 and T 3 read in terms of the two (lowtemperature well-suited and fugacity-like) variables x = e 4 K and z = e 2 H : and: x 3 = x · z 2 x + 2 z + 1 z 2 + 2 z + x (z 2 x + z + xz + x) 2 , z 3 = z · z 2 x + 2 z + 1 z 2 + 2 z + x .
One simply verifies that these rational transformations of two (complex) variables commute. This can be checked by formal calculations for T N and T M for any N and M less than 30, and one can easily verify a fundamental property expected for renormalization group generators: where the "dot" denotes the composition of two transformations. The infinite number of these rational transformations of two (complex) variables (1) are thus a rational representation of the positive integers together with their product. Such rational transformations can be studied "per se" as discrete dynamical systems, the iteration of any of these various exact generators corresponding to an orbit of the renormalization group.
Of course these two examples of exact representation of the renormalization group are extremely degenerate since they correspond to one-dimensional models ¶. Migdal-Kadanoff decimation will quite systematically yield rational † † transformations similar to (1) in two, or more, variables . Consequently, they are never (except "academical" self-similar models) exact representations of the renormalization group. The purpose of this paper is to provide simple (but non trivial) examples of exact renormalization transformations that are not degenerate like the previous transformations on onedimensional models §. In several papers [6,7] for Yang-Baxter integrable models with a canonical genus one parametrization [8,9] (elliptic functions of modulus k), we underlined that the exact generators of the renormalization group must necessarily identify with the various isogenies which amounts to multiplying or dividing τ , the ratio of the two periods of the elliptic curves, by an integer. The simplest example is the Landen transformation [7] which corresponds to multiplying (or dividing because of the modular group symmetry τ ↔ 1/τ ), the ratio of the two periods: The other transformations † correspond to τ ↔ N · τ , for various integers N . In the (transcendental) variable τ , it is clear that they satisfy relations like (2). However, in the natural variables of the model (e K , tanh(K), k = sinh(2 K), not transcendental variables like τ ), these transformations are algebraic transformations corresponding in fact to the fundamental modular curves. For instance (3) corresponds to the genus zero fundamental modular curve j 2 · j 2 − (j + j ) · (j 2 + 1487 · j j + j 2 ) + 3 · 15 3 · (16 j 2 − 4027 j j + 16 j 2 ) (4) − 12 · 30 6 · (j + j ) + 8 · 30 9 = 0, ¶ For instance the fixed points of (1) are not isolated fixed points but lie on (an infinite number) of genus zero curves ... † † In well-suited Boltzmann weight variables like x and z in (1), and not in (bad) variables like K, the coupling constants or the temperature. Such representations of the renormalization group are not exact representations (the exact transformation acts in an infinite number of parameters) but some authors tried to define "improved" renormalization transformations imposing the compatibility (commutation) of the renormalization transformations with some known exact symmetries of the model (Kramers-Wannier duality, gauge symmetries ...).
§ For which the partition function or other physical quantities are algebraic functions. † See for instance (2.18) in [10]. or: which relates the two Hauptmoduls u = 12 3 /j(k), v = 12 3 /j(k L ): One verifies easily that (4) is verified with j = j(k) and j = j(k L ).
The selected values of k, the modulus of elliptic functions, k = 0, 1 are actually fixed points of the Landen transformations. The Kramers-Wannier duality k ↔ 1/k maps k = 0 onto k = ∞. For the Ising (resp. Baxter) model these selected values of k correspond to the three selected subcases of the model ( T = ∞, T = 0 and the critical temperature T = T c ), for which the elliptic parametrization of the model degenerates into a rational parametrization [7]. We have the same property for all the other algebraic modular curves corresponding to τ ↔ N · τ . This is certainly the main property most physicists expect for an exact representation of a generator of the renormalization group, namely that it maps a generic point of the parameter space onto the critical manifold (fixed points). Modular transformations are, in fact, the only transformations to be compatible with all the other symmetries of the Ising (resp. Baxter) model like for instance, the gauge transformations, some extended sl(2)×sl(2)×sl(2)×sl(2) symmetry [11], etc. It has also been underlined in [6,7] that seeing (3) as a transformation on complex variables (instead of real variables) provides two other complex fixed points which actually correspond to complex multiplication for the elliptic curve, and are, actually, fundamental new singularities discovered on the χ (3) linear ODE [13,14,15]. In general, this underlines the deep relation between the renormalization group and the theory of elliptic curves in a deep sense, namely isogenies of elliptic curves, Hauptmoduls †, modular curves and modular forms.
Note that an algebraic transformation like (3) or (5) cannot be obtained from any local Migdal-Kadanoff transformation which naturally yields rational transformations: an exact renormalization group transformation like (3) can only be deduced from non-local decimations. The emergence of modular transformations as representations of exact generators of the renormalization group explains, in a quite subtle way, the difficult problem of how renormalization group transformations can be compatible with reversibility ‡ (iteration forward and backwards). An algebraic modular transformation (5) corresponds to τ → 2 τ and τ → τ /2 in the same time, as a consequence of the modular group symmetry τ ↔ 1/τ .
Suggesting an understanding [7,12] of the quite rich structure of infinite number of the singularities of the χ (n) 's in the complex plane from a Hauptmodul approach [7,12]. Furthermore the notion of Heegner numbers is closely linked to the isogenies mentioned here [7]. An exact value of the j-function j(τ ) corresponding one of the first Heegner number is, for instance, j(1 + i) = 12 3 .
† It should be recalled that the mirror symmetry found with Calabi-Yau manifolds [16,17,18,19,20] can be seen as higher order generalizations of Hauptmoduls. We thus have already generalizations of this identification of the renormalization and modular structure when one is not restricted to elliptic curves anymore. ‡ The fact that the renormalization group must be reversible has apparently been totally forgotten by most of the authors who just see a semi-group corresponding to forward iterations converging to the critical points (resp. manifolds).
A simple rational parametrization † † of the genus zero modular curve (5) reads: Note that the previously mentioned reversibility is also associated with the fact that the modular curve (5) is invariant by u ↔ v, and, within the previous rational parametrization (6), with the fact that permuting u and v corresponds § to the Atkin-Lehner involution z ↔ 2 12 /z. For many Yang-Baxter integrable models of lattice statistical mechanics the physical quantities (partition function per site, correlation functions, ...) are solutions of selected ¶ linear differential equations. For instance the partition function per site of the square (resp. triangular, etc.) Ising model is an integral of an elliptic integral of the third kind. It would be too complicated to show the precise covariance of these physical quantities with respect to (algebraic) modular transformations like (5). Instead, let us give, here, an illustration of the non-trivial action of the renormalization group on some elliptic function that actually occurs in the 2-D Ising model: a weightone modular form. This modular form actually, and remarkably, emerged [22] in a second order linear differential operator factor denoted Z 2 occurring [13] for χ (3) , and that the reader can think as a physical quantity solution of a particular linear ODE replacing the too complicated integral of an elliptic integral of the third kind. Let us consider the second order linear differential operator (D z denotes d/dz): which has the (modular form) solution: Do note that the two pull-backs in the arguments of the same hypergeometric function are actually related by the modular curve relation (5) (see (6)). The covariance (7) is thus the very expression of a modular form property with respect to a modular transformation (τ ↔ 2 τ ) corresponding to the modular transformation (5).
Along this line we can also recall that the (modular form) function ‡: verifies: A relation like (7) is a straight generalization of the covariance we had in the one-dimensional model Z(t) = C(t) · Z(t 2 ), which basically amounts to seeing the partition function per site as some "automorphic function" with respect to the renormalization group, the simple renormalization group transformation t → t 2 being replaced by the algebraic modular transformation (5) corresponding to τ ↔ 2 τ (that is the Landen transformation (3)).
We have here all the ingredients for seeing the identification of exact algebraic representations of the renormalization group with the modular curves structures we tried so many times to promote (preaching in the desert) in various papers [6,7]. However, even if there are no difficulties, just subtleties, these Ising-Baxter examples of exact algebraic representations of the renormalization group already require some serious knowledge of the modular curves, modular forms and Hauptmoduls in the theory of elliptic curves, mixed with the subtleties naturally associated with the various branches of such algebraic (multivalued) transformations.
The purpose of this paper is to present another elliptic hypergeometric function and other much simpler (Gauss hypergeometric) second order linear differential operators covariant by infinite order rational transformations.
The replacement of algebraic (modular) transformations by simple rational transformations will enable us to display a complete explicit description of an exact representation of the renormalization group that any graduate student can completely dominate.

Infinite number of rational symmetries on a Gauss hypergeometric ODE
Keeping in mind modular form expressions like (7), let us recall a particular Gauss hypergeometric function introduced by R. Vidunas in [25]: This hypergeometric function corresponds to the integral of a holomorphic form on a genus-one curve P (y, t) = 0: Note that the function ]; z , (13) ‡ Where j is typically the j-function [10,23].
which is exactly an integral of an algebraic function, has an extremely simple covariance property with respect to the infinite order rational transformation z → −4 z/(1 − z) 2 : The occurrence of this specific infinite order transformation is reminiscent of Kummer's quadratic relation but it is crucial to note that, relation (14) does not relate two different functions, but is an "automorphy" relation on the same function.
It is clear from the previous paragraph that we want to see such functions as 'ideal' examples of physical functions covariant by an exact (here, rational) generator of the renormalization group. The function (13) is actually solution of the second order linear differential operator: From the previous expression of ω 1 involving a log-derivative of a rational function it is obvious that this second order linear differential operator has two solutions, the constant function and an integral of an algebraic function. Since these two solutions behave very simply under the infinite order rational transformation z → −4 z/(1−z) 2 , it is totally and utterly natural to see how the linear differential operator Ω transforms under the rational change of variable z → R(z) = −4 z/(1 − z) 2 (which amounts to seeing how the two order-one operators ω 1 and D z transform). It is a straightforward calculation to see that introducing the cofactor C(z) which is the inverse of the derivative of R(z) D z and ω 1 respectively transform under the rational change of variable z → R(z) = −4 z/(1 − z) 2 as: yielding: Since z → −4 z/(1 − z) 2 is of infinite order, the second order linear differential operator (16) has an infinite number of rational symmetries (isogenies): Once we have found a second order linear differential operator (written in a unitary or monic form) Ω, covariant by the infinite order rational transformation z → −4 z/(1 − z) 2 , it is natural to seek for higher order linear differential operators also covariant by z → −4 z/(1−z) 2 . One easily verifies that the successive symmetric powers of Ω are (of course ...) also covariant. The symmetric square of Ω, factorizes in simple order-one operators: and, more generally, the symmetric N -th power † of Ω reads The covariance of such expressions is the straight consequence of the fact that the order-one factors transform very simply under z → −4 z/(1 − z) 2 : More generally, let us consider a rational transformation z → R(z), the corresponding cofactor C(z) = 1/R (z), and the order-one operator ω 1 = D z + A(z). We have the identity: The change of variable z → R(z) on ω 1 reads: We want to impose that this rhs expression can be written (see (18)) as: which, because of (26), occurs if yielding a "Rota-Baxter-like" [26,27] functional equation on A(z) and R(z): Remark: Coming back to the initial Gauss hypergeometric differential operator the covariance of Ω becomes a conjugation.
Let us start with the Gauss hypergeometric differential operator for (11): (28) † Such formula is actually valid for Ω A = (Dz + A(z)) · Dz for any A(z). Denoting S N symmetric N -th power of Ω A one has S N = (Dz + A(z)) · S N −1 .

It is transformed by
where:

A few remarks on the "Rota-Baxter-like" functional equation.
The functional equation † (27) is the (necessary and sufficient) condition for Ω = (D z + A(z)) · D z to be covariant by z → R(z).
Using the chain rule formula of derivatives of composed functions: , one can show that, for A(z) fixed, the "Rota-Baxter-like" functional equation (27) is invariant by the composition of R(z) by itself R(z) −→ R(R(z)), R(R(R(z))), · · · This result can be generalized to any composition of various R(z)'s satisfying (27). This is in agreement with the fact that (27) is the condition for Ω = (D z + A(z))·D z to be covariant by z → R(z): it must be invariant by composition of R(z)'s (for A(z) fixed).
Note that we have not used here the fact that for globally nilpotent [22] operators, A(z) and B(z) are necessarily log-derivatives of N -th roots of rational functions. For The existence of the underlying a(z) in (29) consequence of a global nilpotence of the order-one differential operator, can however be seen in the following remark on the zeros of the lhs and rhs terms in the functional equation (27). When R(z) is a rational function (for instance −4 z/(1 − z) 2 or any of its iterates R (n) (z)), the lhs and rhs of (27) are rational expressions. The zeros are roots of the numerators of these rational expressions. Because of (29) the functional equation (27) can be rewritten (after dividing by R (z)) as: One easily verifies, in our example, that the zeros of the rhs of (30) come from the zeros of A(R(z)) (and not from the zeros of R (z) in the lhs of (30)). The zeros of the log-derivative rhs of (30) correspond to a(z) · R (z) 4 = ρ, where ρ is a constant to be found. Let us consider for R(z) the n-th actually correspond to the general closed formula: More precisely the zeros of 5·R (n) (z) −3 verify (31), or, in other words, the numerator of 5 R (n) (z) − 3 divides the numerator of the lhs of (31). In another case for T (z) given by (43), which also verifies (27) (see below), the relation (31) is replaced by: More generally for a rational function ρ(x), obtained by an arbitrary composition of −4 z/(1 − z) 2 and T (z), we would have: where λ corresponds to: 2.2. Symmetries of Ω, solutions the "Rota-Baxter-like" functional equation.
Let us now analyse all the symmetries of the linear differential operator Ω = (D z + A(z)) · D z by analyzing all the solutions of (27) for a given A(z). For simplicity we will restrict to A(z) = (3 − 5 z)/z/(1 − z)/4 which corresponds to R(z) = −4 z/(z − 1) 2 and all its iterates (20). Let us first seek for other (more general) solutions that are analytic at z = 0: It is a straightforward calculation to get, order by order from (27), the successive coefficients a n in (35) as polynomial expressions (with rational coefficients) of the first coefficient a 1 with · a 1 · (a 1 − 1) · (41 a 2 1 − 232 a 1 + 366), · · · a n = − n 5 · a 1 · (a 1 − 1) · P n (a 1 ) where P n (a 1 ) is a polynomial with integer coefficients of degree n − 2. Since we have here a series depending on one parameter a 1 we will denote it R a1 (z). This is a quite remarkable series depending on one parameter ¶. One can easily verify that this series actually reduces (as it should!) to the successive iterates (20) In other words this one-parameter family of "functions" actually reduces to rational functions for an infinite number of integer values a 1 = (−4) n . Furthermore, one can also verify a quite essential property we expect for a representation of the renormalization group namely that two R a1 (z) for different values of a 1 commute, the result corresponding to the product of these two a 1 : The neutral element must necessarily correspond to a 1 = 1 which is actually the identity transformation R 1 (z) = z. We have an "absorbing" element corresponding to a 1 = 0, namely R 0 (z) = 0. Performing the inverse of R a1 (z) (with respect to the composition of functions) amounts to changing a 1 into its inverse 1/a 1 . Let us explore some "reversibility" property of our exact representation of a renormalization group with the inverse of the rational transformations (20). The inverse of R −4 (z) = −4 z/(1 − z) 2 must correspond to a 1 = −1/4: However, a straight calculation of the inverse of R −4 (z) = −4 z/(1 − z) 2 gives a multivalued function, or if one prefers, two functions: which are the two roots of the simple quadratic relation (R −4 (z ) = z): where it is clear that the product of these two functions is equal to +1. The radius of convergence of S (1) −1/4 (z) is R = 1. Because of our choice to seek for functions analytical at z = 0 our renormalization group representation "chooses" the unique root that is analytical at z = 0, namely S (20) the inverse transformation corresponds to the roots of the polynomial equation of degree four (R 16 (z ) = z): which yields four roots, one of which is analytical at z = 0 and corresponds to a 1 = 1/(−4) 2 in our one-parameter family of (renormalization) transformations: its (multiplicative) inverse S (2) 1/16 (z) = 1/S (1) 1/16 (z): and two (formal) Puiseux series (u = ± √ z): Many of these results are better understood when one keeps in mind that there is a special transformation J : z ↔ 1/z which is also a R-solution of (27) and verifies many compatibility relations with these transformations (Id denotes the identity transformation R 0 (z)): where the dot corresponds, here, to the composition of functions. These symmetries of the linear differential operator Ω correspond to isogenies of the elliptic curve (12). It is clear that we have another one-parameter family corresponding to J · R a1 with an expansion of the form: this family reduces to the (multiplicative) inverse of the successive rational functions displayed in (20) which can also be written as: where we discover some "additive structure" of these successive rational functions.
As far as the reduction of (36) to a rational function is concerned, it is straightforward to see that: where N (n) is a large integer growing with n, and P n is a polynomial with integer coefficients of degree n − 4, or · a 1 · (a 1 − 1) · (a 1 + 7 + 24 i) · (P 6 (a 1 ) + i Q 6 (a 1 )) · z 6 + · · · (45) · a 1 · (a 1 − 1) · (a 1 + 7 + 24 i) · (P n (a 1 ) + i Q n (a 1 )) · z n + · · · where P n and Q n are two polynomials with integer coefficients of degree respectively n − 3 and n − 4. Similar calculations can be performed for T * (z) defined by for which we also have the covariance: It is a simple calculation to check that any iterate of T (z), (resp. T * (z)) is actually a solution of (27) and corresponds to R a1 (z) for the infinite number of values a 1 = (−7 − 24 i) N (resp. (−7 + 24 i) N ). Furthermore, one verifies, as it should (see (37)), that the three rational functions R −4 (z), T (z), and T * (z) commute. It is also a straightforward calculation to see that the rational function built from any composition of R −4 (z), T (z) and T * (z) is actually a solution of (27). We thus have a triple infinity of values of a 1 , namely a 1 = (−4) M · (−7 − 24 i) N · (−7 + 24 i) P for any integer M , N and P , for which R a1 (z) reduces to rational functions. We are in fact describing (some subset of) the isogenies of the elliptic curve (12), and identifying these isogenies with a discrete subset of the renormalization group. Conversely, a functional equation like (27) can be seen as a way to extend the n-fold composition of a rational function R(z) (namely R(R(· · · R(z) · · ·))) to n any complex number. This extension can be revisited as follows. Keeping in mind the well-known example of the parametrization of the standard map z → 4 z · (1 − z) with z = sin 2 (θ), yielding θ → 2 θ, let us seek for a (transcendental) parametrization z = P (u) such that R −4 P (u) = P (−4 u) or: where H a1 denotes the scaling transformation z → a 1 · z (here H −4 : z → −4 · z) and P −1 denotes the inverse transformation of P (for the composition). One can easily find such a (transcendental) parametrization order by order: and similarly for its inverse (for the composition) transformation: This approach is reminiscent of the conjugation introduced in Siegel's theorem [28,29,30]. It is a straightforward matter to see (order by order) that one actually has R a1 P (u) = P (a 1 · u) or: The structure of the (one-parameter) renormalization group and the extension of the composition of n times a rational function R(z) (namely R(R(· · · R(z) · · ·))) to n any complex number, becomes a straight consequence of this relation. Along this line one can define some "infinitesimal composition" ( 0): where one can find, order by order, the "infinitesimal composition" function F (z): It is straightforward to see, from (37), that the function F (z) satisfies the following functional equations involving a rational function R(z) (in the one-parameter family R a1 (z)): where: F (z) cannot be a rational or algebraic function. Let us consider the fixed points of is not equal to 0 or ∞ at any of these fixed points. Therefore one must have F (z) = 0 or F (z) = ∞ for the infinite set of these fixed points: F (z) cannot be a rational or algebraic function, it is a transcendental function, and similarly for the parametrization function P (z). In fact, let us introduce the function z 9 + · · · + g n · z n + · · · One actually finds that the successive g n satisfies the very simple (hypergeometric function) relation: g n+1 g n = 4 n − 9 4 n + 1 .
The function G(z) is actually the hypergeometric function solution of the homogeneous operator (1 − z) 2 · z 2 , or of the inhomogeneous ODE: One deduces the expression of F (z) as a hypergeometric function ]; z = ∂R a1 ∂a 1 | a1 = 1 .
Finally we get the linear differential operator annihilating F (z) which is, in fact, nothing but Ω * the adjoint of linear differential operator Ω (see (16)). One easily checks § that the second order differential equation Ω F (y(z)) = 0 transforms under the change of variable z → −4 z/(1 − z) 2 into the second order differential equation Ω · Ω * · C(z).
derivative ∂R a1 /∂a 1 that can be expressed in terms of hypergeometric functions for for a double infinity of values of a 1 , namely a 1 = (−4) M × (−7 − 24) N . One can, of course, check, order by order, that (53) is actually verified for any function in the one-parameter family R a1 (z): which corresponds to an infinitesimal version of (37).
From (51) one simply deduces that we can check order by order from (48), the series expansion of P (z), and from (52) the series expansion of F (z), but also that we can, check order by order, from (49), the series expansion of Q(z) = P −1 (z) and from (52). We now deduce that the log-derivative of the "well-suited change of variable" Q(z) is nothing but the (multiplicative) inverse of a hypergeometric function F (z): The function Q(z) is solution of the non-linear differential equation where the Q (n) 's denote the n-th derivative of Q(z). At first sight Q(z) would be a non-holonomic function, however, remarkably, it is a holonomic function solution of an order-five operator which factorizes as follows: yielding the exact expression of Q(z) in terms of hypergeometric functions: that is the fourth power of (13), the differential operator (66) being the symmetric fourth power of Ω. From (13) we immediately get the covariance of Q(z): and, more generally, Q R a1 = a 1 · Q(z). Since Q(z) and F (z) are expressed in term of the same hypergeometric function, the relation (63) must be an identity on that hypergeometric function. This is actually the case. This hypergeometric function verifies the ingomogeneous equation: where: ]; z .
Recalling Q(P (z)) = z, one has the following functional relation on P (z) Noting that Q(z 4 ) 1/4 = F(z 4 ) (see (13)) can be expressed in term of an incomplete elliptic integral of the first kind of argument √ −1 one can find that (70) rewrites on P (z) as from which we deduce that the function P (z) is nothing but a Jacobi elliptic function ‡ In Appendix B we display a set of "Painlevé-like" ODEs † verified by P (z). From the simple non-linear ODE on the Jacobi elliptic sinus, namely S" + 2 · S 3 = 0, and the exact expression of P (z) in term of Jacobi elliptic sinus, one can deduce other non-linear ODEs verified by the non-holonomic function P (z) (P (1) = dP (z)/dz, P (2) = d 2 P (z)/dz 2 ): 2.4. Singularities of the Jacobi elliptic function P (z).
Most of the results of this section, and to some extent, of the next one, are straight consequences of the exact closed expression of P (z) in term of an elliptic function. Following the pedagogical approach of this paper we will rather follow a heuristic approach not taking into account the exact result (73), to display simple methods and ideas that can be used beyond exact results on a specific example. From a diff-Padé analyzis of the series expansion of P (z), we got the sixty (closest to z = 0) singularities. In particular we got that P (z) has a radius of convergence R 11.81704500807 · · · corresponding to the following (closest to z = 0) singularity z = z s of P (z): (76) ‡ Denoted JacobiSN in Maple: P (z) = (JacobiSN (z 1/4 , I)) 4 . † As a (non-holonomic) elliptic function P (z) provides elementary examples [24] of non-linear ODEs with the Painlevé property (like the Weierstrass-P function).
This singularity corresponds to a pole of order four: P (z) (z − z s ) −4 . The function P (z) has many other singularities: 3 4 · z s , (161 ± 240 i) · z s , (−7 ± 24 i) · z s , (−119 ± 120 i) · z s , · · · 5 4 · z s , (41 ± 840 i) · z s , (−527 ± 336 i) · z s , (−1519 ± 720 i) · z s , · · · 7 4 · z s , (1241 ± 2520 i) · z s , (−567 ± 1944 i) · z s , (−3479 ± 1320 i) · z s , · · · In fact, introducing x and y the real and imaginary part of these singularities in z s units, one finds out that they correspond to the double infinity of points where m 1 and m 2 are two integers, and they all lie on the intersection of an infinite number of genus zero curves indexed by the fourth power of an integer M = m 4 (m = m 1 or m = m 2 ): The parametrization (77) describes not only the poles of P (z) when m 1 + m 2 is odd, but also the zeros of P (z) when m 1 + m 2 is even. This (infinite) proliferation of singularities confirms the non-holonomic character of P (z). These results are simply inherited from (73). The zeros and poles of the elliptic sinus sn(z, i) correspond to two lattice of periods. Denoting K 1 and K 2 the two periods of the elliptic curve, the location of the poles and zeros reads respectively: making crystal clear the fact that we have complex multiplication for this elliptic curve. The formula (77) just amount to saying that the poles and zeros of sn(z 1/4 , i) are located at P 4 n1,n2 and Z 4 n1,n2 : The correspondence with (77) is m 1 = n 1 + 2n 2 + 1, m 2 = −n 1 for the poles and m 1 = n 1 + 2n 2 , m 2 = −n 1 for the zeros.
Remark: let us consider the a 1 → ∞ limit of the one-parameter series R a1 (see (35), (36)) rewriting R a1 (z) asR b1 (u) In the a 1 → ∞ limit, that is the b 1 → 0 limit, one easily verifies, order by order in u, thatR b1 (u) becomes exactly the transcendental parametrization function (48): For a 1 = (−4) n (n → ∞), one finds that the radius of convergence ¶ of the R a1 (z) series becomes in the n → ∞ limit R n z s /4 n , in agreement with (83). ¶ It is to the absolute value of the inverse of the image of by the n-th iterate of S (1) −1/4 of −1.

P (z) and an infinite number of rational transformations: the sky is the limit.
Note that some non-linear ODEs associated with P (z) and displayed in Appendix B, namely (B.2) and (B.7), and the functional equation (70), are invariant by the change of variable (P (z), z) → (−4 P (z)/(1 − P (z)) 2 , −4 z). In fact (B.2), (70), and (B.7) are invariant by (P (z), z) → (−4 P (z)/(1 − P (z)) 2 , −4 z), but also (P (z), z) → (−(1 − P (z)) 2 /4/P (z), − z/4), and also by (P (z), z) → (1/P (z), z). The function P (z) satisfies the functional equation: but also P ((−7 − 24 i) · z) = T (P (z)), (85) P ((−7 + 24 i) · z) = T * (P (z)), and, more generally, as can be checked order by order on series expansions, For example, considering the "good" branch (39) for the inverse of −4 z/(1 − z) 2 , namely S (1) −1/4 (z), we can even check, order by order, on the series expansions of P (z) and S (1) −1/4 (z) the functional relation: valid for |P (z)| < 1 since the radius of convergence of S (1) −1/4 (z) is R = 1. Recalling the functional equations (85) it is natural to say that if P (z) is singular at z = z s , then, for almost all the rational functions, in particular T (z) (resp. T * (z)) the T (P (z)) is also singular z = z s , and thus, from (85), P (z) is also singular at z = (−7 ± 24 i) · z s . It is thus extremely natural to see the emergence of the infinite number of singularities in (77) of the form z = (N 1 + i · N 2 ) · z s , as a consequence of (86) together with a reduction of the one-parameter series R a1 (z) to a rational function for an infinite number of selected values of a 1 , namely the N 1 + i · N 2 in (77). This is actually the case for all the values displayed in (77). For instance, for a 1 = 3 4 = 81 we get the following simple rational function: for which it is straightforward to verify that this rational transformation commutes with T (z), T * (z), −4 z/(1 − z) 2 , and is a solution of the Rota-Baxter-like functional equation (27). The case a 1 = 5 4 = 625 in (77), is even simpler, since it just requires to compose T (z) and T * (z) which, again verifies (27) and commutes with all the other rational functions, in particular (88). We also obtained the rational function corresponding to a 1 = 7 4 = 2401, namely: The polynomial N 2401 (z) satisfies many functional equations, like, for instance (with R −4 (z) = −4 z/(1 − z 2 )): and also: ( We also obtained the rational function corresponding to a 1 = 11 4 = 14641, namely: and, of course, one can verify that R 14641 (z) actually commutes with R −4 , R 81 , R 625 , R 2401 (z), and is a solution of the Rota-Baxter-like functional equation (27). Similarly to R 2401 (z) (see (93), (94)), we also have the functional equations: and also: ( Next we obtained the rational function corresponding to a 1 = 13 4 = 28561, which verifies (27) namely: We get similar results, mutatis mutandis, than the ones previously obtained (commutation, functional equations like (97), (98)...), namely: The "palindromic" nature of (88) (89), (90), (95) and (99) (see (91), (96)), (99)) corresponds to the fact that these rational transformations commute with J: In fact, more generally, we have R N 4 (1/z) = 1/R N 4 (z) for N any odd integer (N = 9, 21, · · · , · · ·) and R N 4 (1/z) = R N 4 (z) for N any even integer. From (77) one can reasonnably conjecture that the fourth power of any integer will provide a new example of R a1 (z) being a rational function. The simple non-trivial example corresponds to the already found rational function: We already have explicit rational functions for all values of a 1 of the form N 4 for N = 2, 3, · · · , 16 and of course, we can in principle, build explicit rational functions for all the N 's product of the previous integers. Along this line it is worth noticing that the coefficients of the series R a1 (z) are all integers when a 1 is the fourth power of any integer. We are thus starting to build an infinite number of (elementary) commuting rational transformations, any composition of these (infinite number of) rational transformations giving rational transformations satisfying (27) and preserving the linear differential operator Ω. This set of rational transformations is a pretty large set! Actually this set of rational transformations corresponds to the isogenies of the underlying elliptic function.
The proliferation of the singularities of P (z) corresponds to this (pretty large ...) set of rational transformations. Recalling (87), the previous singularity argument is not valid ‡ for the (well-suited) inverse transformations ( S (1) −1/4 (z), ...) of these rational transformations because (87) requires |P (z)| < 1 (corresponding to the radius of convergence of S (1) −1/4 (z)) and the singularity z = z s corresponds precisely to "hit" the value P (z) = 1. For instance, introducing as soon as ‡ c = 1 + a. For instance is an integral of a simple algebraic function and is solution with the constant function of the second order operator: yielding a new A(z): The adjoint of (118) has the simple solution z 1−a · (1 − z) b : Due to the (a, b)-symmetry of (116) we have a similar result for ; z is solution of (118) where a and b have been permuted: yielding another A(z): The adjoint of (121) has the solution (1 − z) a · z 1−b together with the hypergeometric function: where one recovers the previous result (115). We are seeking for (Gauss hypergeometric) second order differential equations § with an infinite number of (hopefully rational, if not algebraic) symmetries: this is another way to say that we are not looking for generic Gauss hypergeometric differential equations, but Gauss hypergeometric differential equations related to elliptic curves, and thus having an infinite set of such isogenies. We are necessarily in the framework where the two parameters a and b of the Gauss hypergeometric are rational numbers in order to have integral of algebraic functions (yielding globally nilpotent [22] second order differential operators). Let us denote by D the common ‡ See for instance (C.2) in Appendix C.
§ More generally in our models of lattice statistical mechanics (or enumerative combinatorics etc.) we are seeking for (high order) globally nilpotent [22] operators that, in fact, factor into globally nilpotent operators of smaller order, which, for Yang-Baxter integrable models with a canonical elliptic parametrization, must necessarily "be associated with elliptic curves". Appendix D provides some calculations showing that the integral for χ (2) , the two-particle contribution of the susceptibility of the Ising model [31,32,33] is clearly, and straightforwardly, associated with an elliptic curve.
denominator of the two rational numbers a = N a /D and b = N b /D, the function (117) is associated to a period of the algebraic curve: We just need to restrict to triplets of integers (N a , N b , D) such that the previous curve is an elliptic curve. Let us give an example (of the c = 1 + b type) that does not correspond to a genus one curve, with , which corresponds to the genus two curve: Again one introduces A(z): and seek for R(z) as series expansions analytical at z = 0. One gets actually, order by order, a one-parameter family: a 1 · (a 1 − 1) · (3674 a 3 1 + 121194 a 2 1 − 552261 a 1 + 2095059) · z 5 + · · · + 1 + n N (n) · a 1 · (a 1 − 1) · P n (a 1 ) · z n + · · · where n = 0 for n odd and n = 1 for n even, and N (n) is a (large) integer depending on n, and P n (a 1 ) is a polynomial with integer coefficients of degree n − 2. One easily verifies, order by order, that one gets a one-parameter family of transformations commuting for different values of the parameter: As far as the "algorithmic complexity" of this series (126) is concerned it is worth noticing that the degree growth [34] of the series coefficients, is actually linear and not exponential as we could expect [35] at first sight. Even if this series is transcendental, it is not a "wild" series.
Seeking for selected values of a 1 such that the previous series (126) reduces to a rational function one can try to reproduce the simple calculations (44), (45), but unfortunately "shooting in the dark" because we have no hint of a well-suited denominator (if any!) like the polynomials in the lhs of (44), (45).
It is also worth noticing that if we slightly change A(z) into: the algebraic curve (125) becomes y N − (1 − t) 2 · t 5 = 0 which has, for instance genus five for N = 11, but genus zero for N = 7. For any of these cases of (127) one can easily get, order by order, a one-parameter series R a1 totally similar to (126) with, again, polynomials P n (a 1 ) of degree n − 2.
The first coefficient a 2 is, in general: and, again we can actually check that this is actually the series expansion of the hypergeometric function z · (1 − z) · 2 F 1 [1, 15 11 ], [ 17 11 ]; z , solution of Ω * the adjoint of the Ω linear differential operator corresponding to this (genus 5) N = 11 case: We have similar results for (117), (118), (119). As far as these one-parameter family of transformations R a1 , are concerned, the only difference between the generic cases corresponding to arbitrary genus and genus one cases like (107), is that, in the generic higher genus case, only a finite number of values of the parameter a 1 can correspond to rational functions. Note that this higher genus result generalizes to the arbitrary genus Gauss hypergeometric functions (117) and associated operators (118) and function (119). In this general case one can also get order by order a one-parameter family of transformations R a1 satisfying a commutation relation (127).
Note that R(z) = 1/z is actually a solution of (27) for this genus-two example (126). Along this line of selected R(z) solutions of (27) many interesting subcases of this general case (117), (118), (119) are given in Appendix C.1.
In our previous genus-one examples, with this close identification between the renormalization group and the isogenies of elliptic curves, we saw that, in order to obtain linear differential operators covariant by an infinite number of transformations (rational or algebraic), we must restrict our second order Gauss hypergeometric differential operator to Gauss hypergeometric associated to elliptic curves (see Appendix C and Appendix D). Beyond this framework we still have one-parameter families (see (127)) but we cannot expect an infinite number of rational (and probably algebraic) transformations to be particular cases of such families of transcendental transformations.

Conclusion
We have shown that several selected Gauss hypergeometric linear differential operators associated to elliptic curves and factorised into order one differential operators actually present an infinite number of rational symmetries that actually identify with the isogenies of the associated elliptic curves that are perfect illustrations of exact representations of the renormalization group. We actually displayed all these calculations, results and structures because they are perfect examples of exact renormalization transformations. For more realistic models (corresponding to Yang-Baxter models with elliptic parametrizations), the previous calculations and structures become more involved and subtle, the previous rational transformations being replaced by algebraic transformations corresponding to modular curves. For instance, in our models of lattice statistical mechanics (or enumerative combinatorics, etc.), we are often getting globally nilpotent linear differential operators [22] of quite high orders [36,37,38,39,40] that, in fact, factor into globally nilpotent operators of smaller orders ¶, which, for Yang-Baxter integrable models with a canonical elliptic parametrization, must necessarily "be associated with elliptic curves". Appendix D provides some calculations showing that the integral for χ (2) , the two-particle contribution of the susceptibility of the Ising model [31,32,33] is clearly and straightforwardly associated with an elliptic curve.
We wanted to highlight the importance of explicit constructions in answering difficult or subtle questions.
All the calculations displayed in this paper are elementary calculations given explicitly for heuristic reasons. The simple calculations (in particular with the introduction of a simple Rota-Baxter like functional equation) should be seen as some undergraduate training to more realistic renormalization calculations that will require a serious knowledge of fundamental modular curves, modular forms, Hauptmoduls, Gauss-Manin or Picard-Fuchs structures [42,43] and, beyond, some knowledge of mirror symmetries [16,17,18,19,20] of Calabi-Yau manifolds, these mirror symmetries generalizing § the Hauptmodul structure for elliptic curves.
Recall that the corresponding second order differential operator is invariant under the permutation of a and b which is not obvious ‡ on the Euler integral representations of the hypergeometric functions (this amounts to permuting 0 and ∞). The permutation of a and b is always floating around in this paper.
When the three parameters a, b and c of the Gauss hypergeometric functions are rational numbers we have integrals of algebraic functions and, therefore, we know [22,46,47,48,49] that the corresponding second order differential operator is necessarily globally nilpotent [22,46,47,48,49]. Let us restrict to a, b and c being rational numbers a = N a /D, b = N b /D and c = N c /D where D is the common denominator of these three rational numbers. The Gauss hypergeometric functions are naturally associated to the pencil of algebraic curves: where we associated (see (12)) to 2 F 1 [1/2, 1/4], [5/4]; z the elliptic curve In the more general (117), (118), (119), (resp. (121), (122)) framework, one can find many interesting subcases.
• The previous R(z) = 1/z involution is solution of the functional relation (27) when a = 2 b if c = 1 + b, or b = 2 a if c = 1 + a.
• The involution R(z) = 1 − z is solution of the functional relation (27) when a + b = 1 if c = 1 + b, or c = 1 + a.