Renormalization, Isogenies, and Rational Symmetries of Differential Equations

Abstract

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

1. 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 2d conformal field theories and/or scaling limits of second-order phase transition in lattice statistical mechanics.¹ 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 forced² 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, three, or𝑁 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(𝑡²). In this particular case one recovers the (very simple) expression of the partition function per site, 2cosh(𝐾), as an infinite product of the action of (for instance) 𝑇₂ on the cofactor 𝐶(𝑡). In this very simple case, this corresponds to the using of the identity (valid for |𝑥|<1):^∞_𝑛=0(1+𝑥^2𝑛1)=.1−𝑥 (1.1) For 𝑇₃∶𝑡→𝑡³ one must use the identity^∞_𝑛=0(1+𝑥^3𝑛+𝑥^2⋅3𝑛)=^∞_𝑛=0(1−𝑥^3𝑛+11−𝑥^3𝑛1)=,1−𝑥 (1.2) 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 transformations³𝑇_𝑁∶(𝑥,𝑧)→𝑇_𝑁(𝑥,𝑧)=(𝑥_𝑁,𝑧_𝑁), (1.3) where the first two transformations 𝑇₂ and 𝑇₃ read in terms of the two (low-temperature well-suited and fugacity-like) variables 𝑥=𝑒^4𝐾 and 𝑧=𝑒^2𝐻: 𝑥₂=(𝑥+𝑧)(1+𝑥𝑧)𝑥⋅(1+𝑧)²,𝑧₂=𝑧⋅(1+𝑥𝑧),𝑥𝑥+𝑧₃=𝑥⋅(𝑧²𝑥+2𝑧+1)(𝑧²+2𝑧+𝑥)(𝑧²𝑥+𝑧+𝑥𝑧+𝑥)²,𝑧₃𝑧=𝑧⋅²𝑥+2𝑧+1𝑧².+2𝑧+𝑥 (1.4) 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:𝑇_𝑁⋅𝑇_𝑀=𝑇_𝑀⋅𝑇_𝑁=𝑇_𝑁𝑀, (1.5) where the “dot” denotes the composition of two transformations. The infinite number of these rational transformations of two (complex) variables (1.3) 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.⁴ Migdal-Kadanoff decimation will quite systematically yield rational⁵ transformations similar to (1.3) in two, or more, variables.⁶ 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 nontrivial) examples of exact renormalization transformations that are not degenerate like the previous transformations on one-dimensional models.⁷ In several papers [3, 4] for Yang-Baxter integrable models with a canonical genus one parametrization [5, 6] (elliptic functions of modulus 𝑘), we underlined that the exact generators of the renormalization group must necessarily identify with the various isogenies which amount 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 [4] which corresponds to multiplying (or dividing because of the modular group symmetry 𝜏↔1/𝜏), the ratio of the two periods is𝑘→𝑘_𝐿=2√𝑘1+𝑘,𝜏⟷2𝜏. (1.6) The other transformations⁸ correspond to 𝜏↔𝑁⋅𝜏, for various integers 𝑁. In the (transcendental) variable 𝜏, it is clear that they satisfy relations like (1.5). However, in the natural variables of the model (𝑒^𝐾,tanh(𝐾),𝑘=sinh(2𝐾), not transcendental variables like 𝜏), these transformations are algebraic transformations corresponding in fact to the fundamental modular curves. For instance, (1.6) corresponds to the genus zero fundamental modular curve 𝑗²⋅𝑗^′2−(𝑗+𝑗^′)⋅(𝑗²+1487⋅𝑗𝑗^′+𝑗^′2)+3⋅15³⋅(16𝑗²−4027𝑗𝑗^′+16𝑗^′2)−12⋅30⁶⋅(𝑗+𝑗^′)+8⋅30⁹=0, (1.7) or5⁹𝑣³𝑢³−12⋅5⁶𝑢²𝑣²⋅(𝑢+𝑣)+375𝑢𝑣⋅(16𝑢²+16𝑣²−4027𝑣𝑢)−64(𝑣+𝑢)⋅(𝑣²+1487𝑣𝑢+𝑢²)+2¹²⋅3³⋅𝑢𝑣=0, (1.8) which relates the two Hauptmoduls 𝑢=12³/𝑗(𝑘), 𝑣=12³/𝑗(𝑘_𝐿):𝑗(𝑘)=256⋅(1−𝑘²+𝑘⁴)³𝑘⁴⋅(1−𝑘²)²,𝑗(𝑘_𝐿)=16⋅(1+14𝑘²+𝑘⁴)³(1−𝑘²)⁴⋅𝑘². (1.9) One verifies easily that (1.7) 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 [4]. 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 [7], and so forth. It has also been underlined in [3, 4] that seeing (1.6) 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 singularities⁹ discovered on the 𝜒⁽³⁾ linear ODE [8–10]. 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,¹⁰ modular curves and modular forms.

Note that an algebraic transformation like (1.6) or (1.8) cannot be obtained from any local Migdal-Kadanoff transformation which naturally yields rational transformations; an exact renormalization group transformation like (1.6) can only be deduced from nonlocal 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 reversibility¹¹ (iteration forward and backwards). An algebraic modular transformation (1.8) corresponds to 𝜏→2𝜏 and 𝜏→𝜏/2 in the same time, as a consequence of the modular group symmetry 𝜏↔1/𝜏.

A simple rational parametrization¹² of the genus zero modular curve (1.8) reads:𝑧𝑢=1728(𝑧+16)³𝑧,𝑣=1728²(𝑧+256)³2=𝑢(¹²𝑧). (1.10) Note that the previously mentioned reversibility is also associated with the fact that the modular curve (1.8) is invariant by 𝑢↔𝑣, and, within the previous rational parametrization (1.10), with the fact that permuting 𝑢 and 𝑣 corresponds¹³ to the Atkin-Lehner involution 𝑧↔2¹²/𝑧.

For many Yang-Baxter integrable models of lattice statistical mechanics the physical quantities (partition function per site, correlation functions, etc.) are solutions of selected¹⁴ 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 (1.8). Instead, let us give, here, an illustration of the nontrivial action of the renormalization group on some elliptic function that actually occurs in the 2D Ising model: a weight-one modular form. This modular form actually, and remarkably, emerged [11] in a second-order linear differential operator factor denoted 𝑍₂ occurring [8] 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):𝛼=𝐷²_𝑧+(𝑧²+56𝑧+1024)𝑧⋅(𝑧+16)(𝑧+64)⋅𝐷_𝑧−240z⋅(𝑧+16)²,(𝑧+64) (1.11) which has the (modular form) solution₂𝐹₁1([,512𝑧12],[1];1728(𝑧+16)³)=2⋅(𝑧+256)𝑧+16^−1/4⋅₂𝐹₁1([,512𝑧12],[1];1728²(𝑧+256)³). (1.12) Do note that the two pull-backs in the arguments of the same hypergeometric function are actually related by the modular curve relation (1.8) (see (1.10)). The covariance (1.12) is thus the very expression of a modular form property with respect to a modular transformation (𝜏↔2𝜏) corresponding to the modular transformation (1.8).

The hypergeometric function at the rhs of (1.12) is solution of the second-order linear differential operator𝛽=𝐷²_𝑧+𝑧²+416𝑧+16384(𝑧+256)(𝑧+64)𝑧⋅𝐷_𝑧−60(𝑧+64)(𝑧+256)², (1.13) which is the transformed of operator 𝛼 by the Atkin-Lehner duality 𝑧↔2¹²/𝑧, and, also, a conjugation of 𝛼:𝛽=(𝑧+16)𝑧+256^−1/4⋅𝛼⋅(𝑧+16)𝑧+256^1/4. (1.14)

Along this line we can also recall that the (modular form) function¹⁵𝐹(𝑗)=𝑗^−1/12⋅₂𝐹₁1([,51212],[1];12³𝑗), (1.15) verifies:𝐹((𝑧+16)³𝑧)=2⋅𝑧^−1/12⋅𝐹((𝑧+256)³𝑧²). (1.16)

A relation like (1.12) is a straight generalization of the covariance we had in the one-dimensional model 𝑍(𝑡)=𝐶(𝑡)⋅𝑍(𝑡²), which basically amounts to seeing the partition function per site as some “automorphic function” with respect to the renormalization group, with the simple renormalization group transformation 𝑡→𝑡² being replaced by the algebraic modular transformation (1.8) corresponding to 𝜏↔2𝜏 (i.e., the Landen transformation (1.6)).

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 [3, 4]. 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.

2. Infinite Number of Rational Symmetries on a Gauss Hypergeometric ODE

Keeping in mind modular form expressions like (1.12), let us recall a particular Gauss hypergeometric function introduced by Vidunas in [12]₂𝐹₁1([2,145],[41];𝑧)=4⋅𝑧^−1/4⋅^𝑧₀𝑡^−3/4(1−𝑡)^−1/2𝑑𝑡=(1−𝑧)^−1/2⋅₂𝐹₁1([2,145],[4];−4𝑧(1−𝑧)²). (2.1) This hypergeometric function corresponds to the integral of a holomorphic form on a genus-one curve 𝑃(𝑦,𝑡)=0:𝑑𝑡𝑦,with∶𝑦⁴−𝑡³⋅(1−𝑡)²=0. (2.2) Note that the functionℱ(𝑧)=𝑧^1/4⋅₂𝐹₁1([2,145],[4];𝑧), (2.3) which is exactly an integral of an algebraic function, has an extremely simple covariance property with respect to the infinite-order rational transformation 𝑧→−4𝑧/(1−𝑧)²:ℱ(−4𝑧(1−𝑧)²)=(−4)^1/4⋅ℱ(𝑧). (2.4) The occurrence of this specific infinite-order transformation is reminiscent of Kummer's quadratic relation ₂𝐹₁([𝑎,𝑏],[1+𝑎−𝑏];𝑧)=(1−𝑧)^−𝑎⋅₂𝐹₁𝑎([2,1+𝑎2−𝑏],[1+𝑎−𝑏];−4𝑧(1−𝑧)²), (2.5) but it is crucial to note that, relation (2.4) 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 (2.3) is actually solution of the second-order linear differential operator:Ω=𝐷²_𝑧+143−5𝑧𝑧⋅(1−𝑧)⋅𝐷_𝑧=𝜔₁⋅𝐷_𝑧𝜔,with₁=𝐷_𝑧+143−5𝑧𝑧⋅(1−𝑧)=𝐷_𝑧+14⋅𝑑ln(𝑧³(1−𝑧)²).𝑑𝑧 (2.6) From the previous expression of 𝜔₁ 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→−4𝑧/(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)=−4𝑧/(1−𝑧)² (which amounts to seeing how the two-order-one operators 𝜔₁ 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)1𝐶(𝑧)=−4⋅(1−𝑧)³,11+𝑧=𝐶(𝑧)𝑑𝑅(𝑧),𝑑𝑧 (2.7)𝐷_𝑧 and 𝜔₁, respectively, transform under the rational change of variable 𝑧→𝑅(𝑧)=−4𝑧/(1−𝑧)² as𝐷_𝑧→𝐶(𝑧)⋅𝐷_𝑧,𝜔₁→(𝜔₁)^(𝑅)=𝐶(𝑧)²⋅𝜔₁⋅1𝐶(𝑧),yielding∶Ω→𝐶(𝑧)².⋅Ω (2.8) Since 𝑧→−4𝑧/(1−𝑧)² is of infinite-order, the second-order linear differential operator (2.6) has an infinite number of rational symmetries (isogenies):𝑧→−4𝑧(1−𝑧)²→16⋅(1−𝑧)²⋅𝑧(1+𝑧)⁴→−64⋅(1−𝑧)²(1+𝑧)⁴𝑧(1−6𝑧+𝑧²)⁴→⋯. (2.9)

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→−4𝑧/(1−𝑧)², it is natural to seek for higher-order linear differential operators also covariant by 𝑧→−4𝑧/(1−𝑧)². One easily verifies that the successive symmetric powers of Ω are (of course) also covariant. The symmetric square of Ω,𝐷³_𝑧+343−5𝑧(1−𝑧)𝑧⋅𝐷²_𝑧+381−5𝑧(1−𝑧)𝑧²⋅𝐷_𝑧, (2.10) factorizes in simple order-one operators(𝐷_𝑧+243−5𝑧(1−𝑧)𝑧)⋅(𝐷_𝑧+143−5𝑧(1−𝑧)𝑧)⋅𝐷_𝑧, (2.11) and, more generally, the symmetric 𝑁th power¹⁶ of Ω reads(𝐷_𝑧+𝑁43−5𝑧𝑧(1−𝑧))⋅(𝐷_𝑧+𝑁−143−5𝑧𝑧(1−𝑧))⋯(𝐷_𝑧+143−5𝑧𝑧(1−𝑧))⋅𝐷_𝑧. (2.12) The covariance of such expressions is the straight consequence of the fact that the order-one factors𝜔_𝑘=𝐷_𝑧+𝑘43−5𝑧𝑧⋅(1−𝑧),𝑘=0,1,…,𝑁, (2.13) transform very simply under 𝑧→−4𝑧/(1−𝑧)²:𝜔_𝑘→(𝜔_𝑘)^(𝑅)=(𝐶(𝑧))^𝑘+1⋅𝜔_𝑘⋅(𝐶(𝑧))^−𝑘. (2.14) More generally, let us consider a rational transformation 𝑧→𝑅(𝑧), the corresponding cofactor 𝐶(𝑧)=1/𝑅^′(𝑧), and the order-one operator 𝜔₁=𝐷_𝑧+𝐴(𝑧). We have the identity𝐶(𝑧)⋅𝐷_𝑧1⋅(𝐶(𝑧))=𝐷_𝑧−𝑑ln(𝐶(𝑧)).𝑑𝑧 (2.15) The change of variable 𝑧→𝑅(𝑧) on 𝜔₁ reads𝐷_𝑧+𝐴(𝑧)→𝐶(𝑧)⋅𝐷_𝑧+𝐴(𝑅(𝑧))=𝐶(𝑧)⋅(𝐷_𝑧+𝐵(𝑧)). (2.16) We want to impose that this rhs expression can be written (see (2.8)) as𝐶(𝑧)²⋅(𝐷_𝑧1+𝐴(𝑧))⋅,𝐶(𝑧) (2.17) which, because of (2.15), occurs if𝐵(𝑧)=𝐴(𝑧)−𝑑ln(𝐶(𝑧)),𝑑𝑧 (2.18) yielding a “Rota-Baxter-like” [13, 14] functional equation on 𝐴(𝑧) and 𝑅(𝑧)(𝑑𝑅(𝑧))𝑑𝑧²⋅𝐴(𝑅(𝑧))=𝑑𝑅(𝑧)𝑑𝑑𝑧⋅𝐴(𝑧)+²𝑅(𝑧)𝑑𝑧². (2.19)

Remark 2.1. 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 (2.1): 𝐻₀=8𝑧⋅(1−𝑧)⋅𝐷²_𝑧+2⋅(5−7𝑧)⋅𝐷_𝑧−1. (2.20) It is transformed by 𝑧→𝑅(𝑧)=−4𝑧/(1−𝑧)² into 𝐻₁=8𝑧⋅(1−𝑧)⋅𝐷²_𝑧−2(3𝑧−5)⋅𝐷_𝑧+41−𝑧=(1−𝑧)^1/2⋅𝐻₀⋅(1−𝑧)^−1/2, (2.21) then by 𝑧→𝑅(𝑅(𝑧))=𝑅₂(𝑧)=16𝑧(1−𝑧)²/(1+𝑧)⁴ into 𝐻₂=8𝑧⋅(1−𝑧)⋅𝐷²_𝑧−2(3𝑧−1)(𝑧+5)𝑧+1⋅𝐷_𝑧+16𝑧−1(𝑧+1)²=(𝑧+1√𝑧−1)⋅𝐻₀⋅(𝑧+1√)𝑧−1⁻¹, (2.22) and more generally for 𝑧→𝑅_𝑁=𝑅(𝑅(𝑅⋯(𝑅(𝑧)⋯)𝐻_𝑁=𝐶_𝑁⋅𝐻₀⋅𝐶_𝑁⁻¹,where∶𝐶_𝑁=𝑧^1/4⋅𝑅_𝑁^−1/4. (2.23)

2.1. A Few Remarks on the “Rota-Baxter-Like” Functional Equation

The functional equation¹⁷(2.19) 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:𝑑𝑅(𝑅(𝑧))=𝑑𝑧𝑑𝑅(𝑧)𝑑𝑧⋅[𝑑𝑅(𝑧)𝑑𝑑𝑧(𝑅(𝑧))],₂𝑅(𝑅(𝑧))𝑑𝑧²=𝑑²𝑅(𝑧)𝑑𝑧²⋅[𝑑𝑅(𝑧)𝑑𝑧(𝑅(𝑧))]+(𝑑𝑅(𝑧))𝑑𝑧²𝑑⋅[²𝑅(𝑧)𝑑𝑧²(𝑅(𝑧))], (2.24) one can show that, for 𝐴(𝑧) fixed, the “Rota-Baxter-like” functional equation (2.19) 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 (2.19). This is in agreement with the fact that (2.19) 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 [11] operators, 𝐴(𝑧) and 𝐵(𝑧) are necessarily log derivatives of 𝑁th roots of rational functions. For 𝑅(𝑧)=−4𝑧/(1−𝑧)²:1𝐴(𝑧)=4⋅𝑑ln(𝑎(𝑧))1𝑑𝑧,𝐵(𝑧)=4⋅𝑑ln(𝑏(𝑧)),𝑑𝑧𝑎(𝑧)=(1−𝑧)²⋅𝑧³,𝑏(𝑧)=𝑧³⋅(1+𝑧)⁴(1−𝑧)¹⁰. (2.25) The existence of the underlying 𝑎(𝑧) in (2.25) 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 (2.19). When 𝑅(𝑧) is a rational function (e.g., −4𝑧/(1−𝑧)² or any of its iterates 𝑅^(𝑛)(𝑧)), the lhs and rhs of (2.19) are rational expressions. The zeros are roots of the numerators of these rational expressions. Because of (2.25) the functional equation (2.19) can be rewritten (after dividing by 𝑅^′(𝑧)) as(𝑑𝑅(𝑧)𝑑𝑑𝑧)⋅𝐴(𝑅(𝑧))=𝐴(𝑧)+𝑑𝑧(ln(𝑑𝑅(𝑧)1𝑑𝑧))=4⋅𝑑𝑑𝑧(ln(𝑎(𝑧)⋅(𝑑𝑅(𝑧))𝑑𝑧⁴)). (2.26) One easily verifies, in our example, that the zeros of the rhs of (2.26) come from the zeros of 𝐴(𝑅(𝑧)) (and not from the zeros of 𝑅^′(𝑧) in the lhs of (2.26)). The zeros of the log-derivative rhs of (2.26) correspond to 𝑎(𝑧)⋅𝑅^′(𝑧)⁴=𝜌, where 𝜌 is a constant to be found. Let us consider for 𝑅(𝑧) the 𝑛th iterates of −4𝑧/(1−𝑧)² that we denote 𝑅^(𝑛)(𝑧). A straightforward calculation shows that the zeros of 𝐴(𝑅^(𝑛)(𝑧)) or 𝑎^′(𝑅^(𝑛)(𝑧)) (where 𝑎^′(𝑧) denotes the derivative of 𝑎(𝑧) namely, (𝑧−1)(5𝑧−3)⋅𝑧²) actually correspond to the general closed formula:5⁵⋅𝑎(𝑧)⋅(𝑑𝑅^(𝑛)(𝑧))𝑑𝑧⁴−4⋅3³⋅(−4)^𝑛=0. (2.27) More precisely the zeros of 5⋅𝑅^(𝑛)(𝑧)−3 verify (2.27), or, in other words, the numerator of 5𝑅^(𝑛)(𝑧)−3 divides the numerator of the lhs of (2.27).

In another case for 𝑇(𝑧) given by (2.45), which also verifies (2.19) (see below), the relation (2.27) is replaced by5⁵⋅𝑎(𝑧)⋅(𝑑𝑇^(𝑛)(𝑧))𝑑𝑧⁴−4⋅3³⋅(−7−24𝑖)^𝑛=0. (2.28) More generally for a rational function 𝜌(𝑥), obtained by an arbitrary composition of −4𝑧/(1−𝑧)² and 𝑇(𝑧), we would have5⁵⋅𝑎(𝑧)⋅(𝑑𝜌(𝑧))𝑑𝑧⁴−4⋅3³⋅𝜆^𝑛=0. (2.29) where 𝜆 corresponds to𝜌(𝑥)=𝜆⋅𝑧+⋯,𝜆=[𝑑𝜌(𝑧)]𝑑𝑧_𝑧=0. (2.30)

2.2. Symmetries of Ω, Solutions to 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 (2.19) for a given 𝐴(𝑧). For simplicity we will restrict to 𝐴(𝑧)=(3−5𝑧)/𝑧/(1−𝑧)/4 which corresponds to 𝑅(𝑧)=−4𝑧/(𝑧−1)² and all its iterates (2.9). Let us first seek for other (more general) solutions that are analytic at 𝑧=0:𝑅(𝑧)=𝑎₁⋅𝑧+𝑎₂⋅𝑧²+𝑎₃⋅𝑧³+⋯. (2.31) It is a straightforward calculation to get, order by order from (2.19), the successive coefficients 𝑎_𝑛 in (2.31) as polynomial expressions (with rational coefficients) of the first coefficient 𝑎₁ with𝑎₂2=−5⋅𝑎₁⋅(𝑎₁−1),𝑎₃=175⋅𝑎₁⋅(𝑎₁−1)⋅(7𝑎₁𝑎−17),₄2=−4875⋅𝑎₁⋅(𝑎₁−1)⋅(41𝑎²₁−232𝑎₁𝑎+366),…,_𝑛𝑛=−5⋅𝑎₁⋅(𝑎₁𝑃−1)⋅^𝑛(𝑎₁)𝑃_𝑛,(−4) (2.32) where 𝑃_𝑛(𝑎₁) is a polynomial with integer coefficients of degree 𝑛−2. Since we have here a series depending on one parameter 𝑎₁ we will denote it 𝑅_𝑎1(𝑧). This is a quite remarkable series depending on one parameter.¹⁸ One can easily verify that this series actually reduces (as it should!) to the successive iterates (2.9) of −4𝑧/(1−𝑧)² for 𝑎₁=(−4)^𝑛. In other words this one-parameter family of “functions” actually reduces to rational functions for an infinite number of integer values 𝑎₁=(−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 𝑎₁ commute, the result corresponding to the product of these two 𝑎₁:𝑅_𝑎1(𝑅_𝑏1(𝑧))=𝑅_𝑏1(𝑅_𝑎1(𝑧))=𝑅_{𝑎1⋅𝑏1}(𝑧). (2.33) The neutral element must necessarily correspond to 𝑎₁=1 which is actually the identity transformation 𝑅₁(𝑧)=𝑧. We have an “absorbing” element corresponding to 𝑎₁=0, namely, 𝑅₀(𝑧)=0. Performing the inverse of 𝑅_𝑎1(𝑧) (with respect to the composition of functions) amounts to changing 𝑎₁ into its inverse 1/𝑎₁. Let us explore some “reversibility” property of our exact representation of a renormalization group with the inverse of the rational transformations (2.9). The inverse of 𝑅₋₄(𝑧)=−4𝑧/(1−𝑧)² must correspond to 𝑎₁=−1/4:𝑅_−1/41(𝑧)=−41⋅𝑧−8𝑧²−5𝑧64³−7𝑧128⁴−21𝑧512⁵+⋯. (2.34) However, a straight calculation of the inverse of 𝑅₋₄(𝑧)=−4𝑧/(1−𝑧)² gives a multivalued function, or if one prefers, two functions𝑆⁽¹⁾_−1/4√(𝑧)=𝑧−2+21−𝑧𝑧1=−41⋅𝑧−8𝑧²𝑆+⋯,_(2)−1/4√(𝑧)=𝑧−2−21−𝑧𝑧4=−𝑧1+2+41𝑧+8𝑧²+⋯, (2.35) which are the two roots of the simple quadratic relation (𝑅₋₄(𝑧^′)=𝑧):𝑧^′22−2⋅(1−𝑧)⋅𝑧^′+1=0, (2.36) 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 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⁽¹⁾_−1/4(𝑧). For the next iterate of 𝑅₋₄(𝑧)=−4𝑧/(1−𝑧)² in (2.9) the inverse transformation corresponds to the roots of the polynomial equation of degree four (𝑅₁₆(𝑧^′)=𝑧):𝑧^′4+(4−16𝑧)⋅𝑧^′3+(6+32𝑧)⋅𝑧^′2+(4−16𝑧)⋅𝑧^′+1=0, (2.37) which yields four roots, one of which is analytical at 𝑧=0 and corresponds to 𝑎₁=1/(−4)² in our one-parameter family of (renormalization) transformations:𝑆⁽¹⁾_1/161(𝑧)=316𝑧+𝑧128²+53𝑧4096³+277𝑧32768⁴+3181𝑧524288⁵+⋯, (2.38) its (multiplicative) inverse 𝑆⁽²⁾_1/16(𝑧)=1/𝑆⁽¹⁾_1/16(𝑧):𝑆⁽²⁾_1/16(𝑧)=16𝑧−6−1716𝑧−67𝑧128²−1333𝑧4096³−7445𝑧32768⁴+⋯, (2.39) and two (formal) Puiseux series (𝑢=±√𝑧):𝑆⁽³⁾_1/161(𝑧)=1+𝑢+2𝑢²+38𝑢³+14𝑢⁴+27𝑢128⁵+5𝑢32⁶+⋯. (2.40) 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 (2.19) and verifies many compatibility relations with these transformations (𝐼𝑑 denotes the identity transformation 𝑅₀(𝑧)):𝑅₋₄⋅𝐽=𝑅₋₄,𝑆⁽²⁾_−1/4⋅𝑅₋₄=𝐽,𝑅₋₄⋅𝑆⁽¹⁾_−1/4=𝑆⁽¹⁾_−1/4⋅𝑅₋₄𝑆=𝐼𝑑,_(1)1/16(𝑧)=𝑆⁽¹⁾_−1/4⋅𝑆⁽¹⁾_−1/4,𝑆⁽²⁾_1/16(𝑧)=𝑆⁽¹⁾_−1/4⋅𝑆⁽²⁾_−1/4,𝐽⋅𝑆⁽¹⁾_−1/4=𝑆⁽²⁾_−1/4,𝐽⋅𝑆⁽²⁾_−1/4=𝑆⁽¹⁾_−1/4,…, (2.41) 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 (2.2).

It is clear that we have another one-parameter family corresponding to 𝐽⋅𝑅_𝑎1 with an expansion of the form𝐽⋅𝑅_𝑎1=𝑏₁𝑧−25⋅(𝑏₁1−1)−⋅𝑏15²¹−1𝑏₁2⋅𝑧−⋅975(𝑏₁−1)(4𝑏₁+1)(4𝑏₁+3)𝑏²₁⋅𝑧²−1⋅248625(𝑏₁−1)(4𝑏₁+1)(1268𝑏²₁+951𝑏₁+91)𝑏³₁⋅𝑧³−2⋅2071875(𝑏₁−1)(4𝑏₁+1)(3688𝑏³₁+2766𝑏²₁+404𝑏₁+17)𝑏⁴₁⋅𝑧⁴+⋯. (2.42) For 𝑏₁=−1/4, 𝑏₁=(−1/4)², 𝑏₁=(−1/4)³, this family reduces to the (multiplicative) inverse of the successive rational functions displayed in (2.9)−14⋅(1−𝑧)²𝑧→1⋅16(1+𝑧)⁴(1−𝑧)²1⋅𝑧→−⋅64(1−6𝑧+𝑧²)⁴(1−𝑧)²(1+𝑧)⁴⋅𝑧→⋯, (2.43) which can also be written as: −141⋅(𝑧+𝑧1)+2,1116⋅(𝑧+𝑧3)+8+𝑧(1−𝑧)²,−1164⋅(𝑧+𝑧)+13−𝑧324⋅17−60𝑧+102𝑧²−60𝑧³+17𝑧⁴(1−𝑧)²(1+𝑧)⁴,11256⋅(𝑧+𝑧)+51+𝑧128⋅1617−60𝑧+102𝑧²−60𝑧³+17𝑧⁴(1−𝑧)²(1+𝑧)⁴+𝑧(−4)^𝑛−6⋅(1−𝑧)²(1+𝑧)⁴(𝑧²−6𝑧+1)⁴,−111024⋅(𝑧+𝑧)+205−𝑧512⋅16417−60𝑧+102𝑧²−60𝑧³+17𝑧⁴(1−𝑧)²(1+𝑧)⁴−4𝑧⋅(1−𝑧)²(1+𝑧)⁴(𝑧²−6𝑧+1)⁴−64𝑧⋅(1−𝑧)²(1+𝑧)⁴(𝑧²−6𝑧+1)⁴(1+20𝑧−26𝑧²+20𝑧³+𝑧⁴)⁴1,…,(−4)^𝑛1⋅(𝑧+𝑧2)+54^𝑛(4^𝑛−(−1)^𝑛)+𝑧(−4)^𝑛−2⋅17−60𝑧+102𝑧²−60𝑧³+17𝑧⁴(1−𝑧)²(1+𝑧)⁴+𝑧(−4)^𝑛−6⋅(1−𝑧)²(1+𝑧)⁴(𝑧²−6𝑧+1)⁴+𝑧(−4)^𝑛−8⋅(1−𝑧)²(1+𝑧)⁴(𝑧²−6𝑧+1)⁴(1+20𝑧−26𝑧²+20𝑧³+𝑧⁴)⁴+⋯, (2.44) where we discover some “additive structure” of these successive rational functions.

In fact, due to the specificity of this elliptic curve (occurrence of complex multiplication), we have another remarkable rational transformation solution of (2.19), preserving covariantly Ω. Let us introduce the rational transformation (𝑖 denotes √−1):𝑇(𝑧)=𝑧⋅(𝑧−(1+2𝑖))1−(1+2𝑖)⋅𝑧⁴, (2.45) we also have the remarkable covariance [12]:₂𝐹₁1([2,145],[4];𝑧)=1−𝑧/(1+2𝑖)⋅1−(1+2𝑖)𝑧²𝐹₁1([2,145],[4];𝑇(𝑧)), (2.46) which can be rewritten in a simpler way on (2.3) (see (2.4)).

It is a straightforward matter to see that 𝑇(𝑧) actually belongs to the 𝑅_𝑎1(𝑧) one-parameter family:𝑇(𝑧)=𝑅_𝑎1(𝑧)=−(7+24𝑖)⋅𝑧+⋯,𝑎₁=−25⋅𝜌,𝜌=(7+24𝑖)25,|𝜌|=1. (2.47)

As far as the reduction of (2.32) to a rational function is concerned, it is straightforward to see that:(1−𝑧)²⋅(1+𝑧)⁴⋅𝑅_𝑎1(𝑧)=𝑎₁2⋅𝑧+⋯−175746796875⋅𝑎₁⋅(𝑎₁−1)⋅(𝑎₁+4)⋅(𝑎₁−16)⋅𝑃₈(𝑎₁)⋅𝑧⁸−1+⋯𝑁(𝑛)⋅𝑎₁⋅(𝑎₁−1)⋅(𝑎₁+4)⋅(𝑎₁−16)⋅𝑃_𝑛(𝑎₁)⋅𝑧^𝑛+⋯, (2.48) where 𝑁(𝑛) is a large integer growing with 𝑛, and 𝑃_𝑛 is a polynomial with integer coefficients of degree 𝑛−4, or(1−(1+2𝑖)⋅𝑧)⁴⋅𝑅_𝑎1(𝑧)=𝑎₁4⋅𝑧+⋯−1243125⋅𝑎₁⋅(𝑎₁−1)⋅(𝑎₁+7+24𝑖)⋅(𝑃₆(𝑎₁)+𝑖𝑄₆(𝑎₁))⋅𝑧⁶+1+⋯𝑁(𝑛)⋅𝑎₁⋅(𝑎₁−1)⋅(𝑎₁+7+24𝑖)⋅(𝑃_𝑛(𝑎₁)+𝑖𝑄_𝑛(𝑎₁))⋅𝑧^𝑛+⋯, (2.49) 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 by𝑇^∗(𝑧)=𝑧⋅(𝑧−(1−2𝑖))(1−2𝑖)𝑧−1⁴, (2.50) for which we also have the covariance₂𝐹₁1([2,145],[4];𝑧)=1−𝑧/(1−2𝑖)⋅1−(1−2𝑖)𝑧²𝐹₁1([2,145],[4];𝑇^∗(𝑧)). (2.51)

It is a simple calculation to check that any iterate of 𝑇(𝑧) (resp. 𝑇^∗(𝑧)) is actually a solution of (2.19) and corresponds to 𝑅_𝑎1(𝑧) for the infinite number of values 𝑎₁=(−7−24𝑖)^𝑁 (resp. (−7+24𝑖)^𝑁). Furthermore, one verifies, as it should (see (2.33)), 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 (2.19). We thus have a triple infinity of values of 𝑎₁, namely 𝑎₁=(−4)^𝑀⋅(−7−24𝑖)^𝑁⋅(−7+24𝑖)^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 (2.2), and identifying these isogenies with a discrete subset of the renormalization group. Conversely, a functional equation like (2.19) 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.

2.3. Revisiting the One-Parameter Family of Solutions of the “Rota-Baxter-Like” Functional Equation

This extension can be revisited as follows. Keeping in mind the well-known example of the parametrization of the standard map 𝑧→4𝑧⋅(1−𝑧) with 𝑧=sin²(𝜃), yielding 𝜃→2𝜃, let us seek for a (transcendental) parametrization 𝑧=𝑃(𝑢) such that𝑅₋₄(𝑃(𝑢))=𝑃(−4𝑢)or∶𝑅₋₄=𝑃⋅𝐻₋₄⋅𝑃⁻¹, (2.52) where 𝐻_𝑎1 denotes the scaling transformation 𝑧→𝑎₁⋅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 order2𝑃(𝑧)=𝑧−5𝑧²+7𝑧75³−82𝑧4875⁴+1078𝑧414375⁵−452𝑧1243125⁶+57311𝑧1212046875⁷−1023946𝑧175746796875⁸+⋯, (2.53) and similarly for its inverse (for the composition) transformation𝑄(𝑧)=𝑃⁻¹2(𝑧)=𝑧+5𝑧²+17𝑧75³+244𝑧1625⁴+45043𝑧414375⁵+2302𝑧27625⁶+128941𝑧1939275⁷[phone]𝑧281194875⁸+⋯. (2.54) This approach is reminiscent of the conjugation introduced in Siegel's theorem [15–17]. It is a straightforward matter to see (order by order) that one actually has𝑅_𝑎1(𝑃(𝑢))=𝑃(𝑎₁⋅𝑢)or∶𝑅_𝑎1=𝑃⋅𝐻_𝑎1⋅𝑃⁻¹. (2.55) 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 become a straight consequence of this relation. Along this line one can define some “infinitesimal composition” (𝜖≃0):𝑅_1+𝜖(𝑧)=𝑃⋅𝐻_1+𝜖⋅𝑃⁻¹(𝑧)=𝑧+𝜖⋅𝐹(𝑧)+⋯, (2.56) where one can find, order by order, the “infinitesimal composition” function 𝐹(𝑧):2𝐹(𝑧)=𝑧−5𝑧²−2𝑧15³−14𝑧195⁴−154𝑧3315⁵−22𝑧663⁶−418𝑧16575⁷−9614𝑧480675⁸−2622𝑧160225⁹+⋯. (2.57) It is straightforward to see, from (2.33), that the function 𝐹(𝑧) satisfies the following functional equations involving a rational function 𝑅(𝑧) (in the one-parameter family 𝑅_𝑎1(𝑧)):𝑑𝑅(𝑧)𝑑𝑧⋅𝐹(𝑧)=𝐹(𝑅(𝑧)),𝑑𝑅^(𝑛)(𝑧)𝑑𝑧⋅𝐹(𝑧)=𝐹(𝑅^(𝑛)𝑅(𝑧)),where∶_(𝑛)(𝑧)=𝑅(𝑅(⋯𝑅(𝑧))⋯). (2.58)F (𝑧) cannot be a rational or algebraic function. Let us consider the fixed points of 𝑅^(𝑛)(𝑧). Generically 𝑑𝑅^(𝑛)(𝑧)/𝑑𝑧 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 function7𝐺(𝑧)=(1−𝑧)⋅𝐹(𝑧),𝐺(𝑧)=𝑧−5𝑧²+4𝑧15³+4𝑧65⁴+28𝑧1105⁵+44𝑧3315⁶+44𝑧5525⁷+836𝑧160225⁸+1748𝑧480675⁹+⋯+𝑔_𝑛⋅𝑧^𝑛+⋯. (2.59) One actually finds that the successive 𝑔_𝑛 satisfies the very simple (hypergeometric function) relation:𝑔_𝑛+1𝑔_𝑛=4𝑛−9.4𝑛+1 (2.60) The function 𝐺(𝑧) is actually the hypergeometric function solution of the homogeneous operator𝐷²_𝑧+1413𝑧−3𝑧⋅(1−𝑧)⋅𝐷_𝑧+346𝑧²−3𝑧+1(1−𝑧)²⋅𝑧², (2.61) or of the inhomogeneous ODE4𝑧⋅(1−𝑧)⋅𝑑𝐺(𝑧)𝑑𝑧+(9𝑧−3)⋅𝐺(𝑧)−𝑧⋅(1−𝑧)²=0. (2.62) One deduces the expression of 𝐹(𝑧) as a hypergeometric function𝐹(𝑧)=𝑧⋅(1−𝑧)^1/2⋅₂𝐹₁1([4,125],[4];𝑧)=𝜕𝑅_𝑎1𝜕𝑎₁|_𝑎1=1. (2.63)

Finally we get the linear differential operator annihilating 𝐹(𝑧)Ω_𝐹=𝐷²_𝑧+14⋅5𝑧−3𝑧(1−𝑧)⋅𝐷_𝑧+14⋅3−6𝑧+5𝑧²(1−𝑧)²𝑧²=𝐷_𝑧⋅(𝐷_𝑧−14⋅3−5𝑧𝑧⋅(1−𝑧)), (2.64) which is, in fact, nothing but Ω^∗ the adjoint of linear differential operator Ω (see (2.6)). One easily checks¹⁹ that the second-order differential equation Ω_𝐹(𝑦(𝑧))=0 transforms under the change of variable 𝑧→−4𝑧/(1−𝑧)² into the second-order differential equation Ω_𝐹^(𝑅)(𝑦(𝑧))=0 with Ω_𝐹^(𝑅)=𝐶(𝑧)²⋅𝜔_𝐹^(𝑅) where the unitary (monic) operator 𝜔_𝐹^(𝑅) is the conjugate of Ω_𝐹:𝜔_𝐹^(𝑅)=𝐷²_𝑧−14⋅11𝑧²+30𝑧+3𝑧⋅(1−𝑧)(1+𝑧)⋅𝐷_𝑧+14⋅3+12𝑧+50𝑧²+12𝑧³+3𝑧⁴𝑧²⋅(1−𝑧)²(1+𝑧)²1=(𝐶(𝑧))⋅𝐷_𝑧⋅(𝐷_𝑧−14⋅3−5𝑧1𝑧⋅(1−𝑧))⋅𝐶(𝑧)=(𝐶(𝑧))⋅Ω_𝐹1⋅𝐶(𝑧)=(𝐶(𝑧))⋅Ω^∗⋅𝐶(𝑧) (2.65) with 𝐶(𝑧)=1/𝑅^′(𝑧) and the “dot” denotes the composition of operators. Actually, the factors in the adjoint Ω^∗ transform under the change of variable 𝑧→−4𝑧/(1−𝑧)² as follows²⁰:𝐷_𝑧→𝐶(𝑧)⋅𝐷_𝑧,𝜔^∗₁=→(𝜔^∗₁)^(𝑅)=𝜔^∗₁⋅𝐶(𝑧),Ω^∗→Ω_𝐹^(𝑅)=𝐶(𝑧)⋅Ω^∗⋅𝐶(𝑧) (2.66) which is precisely the transformation we need to match with (2.58) and see the ODE Ω^∗(𝐹(𝑧))=0 compatible with the change of variable 𝑧→−4𝑧/(1−𝑧)²:Ω^∗(𝐹(𝑧))=0→(𝐶(𝑧)⋅Ω^∗⋅𝐶(𝑧))(𝐹(𝑅(𝑧)))=(𝐶(𝑧)⋅Ω^∗⋅𝐶(𝑧))(𝑅^′(𝑧)⋅𝐹(𝑧))=𝐶(𝑧)⋅Ω^∗(𝐹(𝑧))=0. (2.67) This is, in fact, a quite general result that will be seen to be valid in a more general (higher genus) framework (see (2.148), (2.150) in what follows).

Not surprisingly one can deduce from (2.33) and the previous results, in particular (2.63), the following results for 𝑅_𝑎1(𝑧):−4⋅𝜕𝑅_𝑎1𝜕𝑎₁|_𝑎1=−4=𝐹(𝑅(𝑧)),(−4)^𝑛⋅𝜕𝑅_𝑎1𝜕𝑎₁|_{𝑎1=(−4)𝑛}=𝐹(𝑅^(𝑛)(𝑧)), (2.68) where 𝑅(𝑧)=−4𝑧/(1−𝑧)² and 𝑅^(𝑛)(𝑧) denotes 𝑅(𝑅(⋯𝑅(𝑅(𝑧)))). Of course we have similar relation for 𝑇(𝑧), −4 being replaced by −7−24𝑖. Therefore the partial derivative 𝜕𝑅_𝑎1/𝜕𝑎₁ that can be expressed in terms of hypergeometric functions for for a double infinity of values of 𝑎₁, namely, 𝑎₁=(−4)^𝑀×(−7−24𝑖)^𝑁.

One can, of course, check, order by order, that (2.58) is actually verified for any function in the one-parameter family 𝑅_𝑎1(𝑧):𝑑𝑅_𝑎1(𝑧)𝑑𝑧⋅𝐹(𝑧)=𝐹(𝑅_𝑎1(𝑧)), (2.69) which corresponds to an infinitesimal version of (2.33).

From (2.56) one simply deduces𝑧⋅𝑑𝑃(𝑧)𝑑𝑧=𝐹(𝑃(𝑧)), (2.70) that we can check, order by order from (2.53), the series expansion of 𝑃(𝑧), and from (2.57) the series expansion of 𝐹(𝑧), but also𝑑𝑄(𝑧)𝑑𝑧⋅𝐹(𝑧)=𝑄(𝑧) (2.71) that we can, check order by order, from (2.54), the series expansion of 𝑄(𝑧)=𝑃⁻¹(𝑧) and from (2.57). 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):𝑑ln(𝑄(𝑧))=1𝑑𝑧𝐹(𝑧),𝑄(𝑧)=𝜆⋅exp(^𝑧𝑑𝑧𝐹(𝑧)). (2.72) The function 𝑄(𝑧) is solution of the nonlinear differential equation−4𝑧²⋅(1−𝑧)²⋅(𝑄⋅𝑄⁽¹⁾⋅𝑄⁽³⁾+(𝑄⁽¹⁾)²⋅𝑄⁽²⁾−2𝑄⋅(𝑄⁽²⁾)²)+𝑧⋅(3−5𝑧)(1−𝑧)⋅𝑄⁽¹⁾⋅(𝑄⋅𝑄⁽²⁾−(𝑄⁽¹⁾)²)+(5𝑧²−6𝑧+3)⋅𝑄⋅(𝑄⁽¹⁾)²=0, (2.73) where the 𝑄^(𝑛)'s denote the 𝑛th derivative of 𝑄(𝑧). At first sight 𝑄(𝑧) would be a nonholonomic function, however, remarkably, it is a holonomic function solution of an order-five operator which factorizes as follows:Ω_𝑄=(𝐷_𝑧+3−5𝑧(1−𝑧)⋅𝑧)⋅(𝐷_𝑧+34⋅3−5𝑧(1−𝑧)⋅𝑧)⋅×(𝐷_𝑧+24⋅3−5𝑧(1−𝑧)⋅𝑧)⋅(𝐷_𝑧+14⋅3−5𝑧(1−𝑧)⋅𝑧)⋅𝐷_𝑧, (2.74) yielding the exact expression of 𝑄(𝑧) in terms of hypergeometric functions:𝑄(𝑧)=𝑧⋅(₂𝐹₁1([2,145],[4];𝑧))⁴=𝑧1−𝑧⋅(₂𝐹₁1([4,345],[4𝑧];−1−𝑧))⁴, (2.75) that is, the fourth power of (2.3), with the differential operator (2.74) being the symmetric fourth power of Ω. From (2.3) we immediately get the covariance of 𝑄(𝑧):𝑄(−4𝑧(1−𝑧)²)=−4⋅𝑄(𝑧), (2.76) and, more generally, 𝑄(𝑅_𝑎1)=𝑎₁⋅𝑄(𝑧). Since 𝑄(𝑧) and 𝐹(𝑧) are expressed in terms of the same hypergeometric function, the relation (2.71) must be an identity on that hypergeometric function. This is actually the case. This hypergeometric function verifies the inhomogeneous equation:4⋅𝑧⋅𝑑ℋ(𝑧)𝑑𝑧+ℋ(𝑧)−(1−𝑧)^−1/2=0, (2.77) whereℋ(𝑧)=₂𝐹₁1([2,145],[4];𝑧). (2.78)

Recalling 𝑄(𝑃(𝑧))=𝑧, one has the following functional relation on 𝑃(𝑧):𝑃(𝑧)⋅₂𝐹₁1([4,125],[4];𝑃(𝑧))⁴=𝑧. (2.79)

Noting that 𝑄(𝑧⁴)^1/4=ℱ(𝑧⁴) (see (2.3)) can be expressed in term of an incomplete elliptic integral of the first kind of argument √−1𝑧⋅₂𝐹₁1([4,125],[4];𝑧⁴)=Elliptic𝐹(𝑧,√−1), (2.80) one can find that (2.79) rewrites on 𝑃(𝑧) asElliptic𝐹(𝑃(𝑧)^1/4,√−1)=𝑧^1/4, (2.81) from which we deduce that the function 𝑃(𝑧) is nothing but a Jacobi elliptic function ²¹𝑃(𝑧)=(sn(𝑧^1/4,√−1))⁴. (2.82) In Appendix B we display a set of “Painlevé-like” ODEs²² verified by 𝑃(𝑧). From the simple nonlinear ODE on the Jacobi elliptic sinus, namely, 𝑆^′′+2⋅𝑆³=0, and the exact expression of 𝑃(𝑧) in term of Jacobi elliptic sinus, one can deduce other nonlinear ODEs verified by the nonholonomic function 𝑃(𝑧) (𝑃⁽¹⁾=𝑑𝑃(𝑧)/𝑑𝑧, 𝑃⁽²⁾=𝑑²𝑃(𝑧)/𝑑𝑧²):𝑧^3/2⋅(𝑃⁽¹⁾)²−(1−𝑃)⋅𝑃^3/2𝑃=0,₍₂₎−34⋅(𝑃⁽¹⁾)²𝑃+34⋅𝑃⁽¹⁾𝑧+12⋅𝑃^3/2𝑧^3/2=0. (2.83)(2.84)

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 terms 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 (2.82), to display simple methods and ideas that can be used beyond exact results on a specific example.

From a diff-Padé analysis 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):𝑧_𝑠=−11.817045008077115768316337283432582087420697⋯=(−4)⋅₂𝐹₁1([4,125],[4];1)⁴1=−⋅𝜋16Γ(3/⁶4)⁸. (2.85) This singularity corresponds to a pole of order four: 𝑃(𝑧)≃(𝑧−𝑧_𝑠)⁻⁴. The function 𝑃(𝑧) has many other singularities:3⁴⋅𝑧_𝑠,(161±240𝑖)⋅𝑧_𝑠,(−7±24𝑖)⋅𝑧_𝑠,(−119±120𝑖)⋅𝑧_𝑠5,⋯₄⋅𝑧_𝑠,(41±840𝑖)⋅𝑧_𝑠,(−527±336𝑖)⋅𝑧_𝑠,(−1519±720𝑖)⋅𝑧_𝑠7,⋯₄⋅𝑧_𝑠,(1241±2520𝑖)⋅𝑧_𝑠,(−567±1944𝑖)⋅𝑧_𝑠,(−3479±1320𝑖)⋅𝑧_𝑠,⋯ (2.86)

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 points𝑥=(𝑚²₁−2𝑚₁𝑚₂−𝑚²₂)⋅(𝑚²₁+2𝑚₁𝑚₂−𝑚²₂),𝑦=4⋅𝑚₁𝑚₂⋅(𝑚₂−𝑚₁)⋅(𝑚₂+𝑚₁), (2.87) where 𝑚₁ and 𝑚₂ 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=𝑚⁴ (𝑚=𝑚₁ or 𝑚=𝑚₂):2¹²𝑀⁴−2¹¹⋅𝑥⋅𝑀³−2⁷⋅(17𝑦²+14𝑥²)⋅𝑀²−2⁵⋅𝑥⋅(8𝑥²+7𝑦²)⋅𝑀+𝑦⁴=0. (2.88) The parametrization (2.87) describes not only the poles of 𝑃(𝑧) when 𝑚₁+𝑚₂ is odd, but also the zeros of 𝑃(𝑧) when 𝑚₁+𝑚₂ is even. This (infinite) proliferation of singularities confirms the nonholonomic character of 𝑃(𝑧).

These results are simply inherited from (2.82). The zeros and poles of the elliptic sinus 𝑠𝑛(𝑧,𝑖) correspond to two lattice of periods. Denoting 𝒦₁ and 𝒦₂ the two periods of the elliptic curve, the location of the poles and zeros reads, respectively,𝑃_𝑛1,𝑛2=2𝑛₁⋅𝒦₁+(2𝑛₂+1)⋅𝒦₂,𝑍_𝑛1,𝑛2=2𝑛₁⋅𝒦₁+2𝑛₂⋅𝒦₂,𝒦₁=𝜋^3/22^3/2⋅1Γ(3/4)²,𝒦₂=(1−√−1)⋅𝐾₁ (2.89) making crystal clear the fact that we have complex multiplication for this elliptic curve. The formula (2.87) just amount to saying that the poles and zeros of 𝑠𝑛(𝑧^1/4,𝑖) are located at 𝑃⁴_𝑛1,𝑛2 and 𝑍⁴_𝑛1,𝑛2:𝑃⁴_𝑛1,𝑛2𝑧=−^𝑠4⋅((2𝑛₁+2𝑛₂+1)+𝑖⋅(2𝑛₂+1))⁴,𝑍⁴_𝑛1,𝑛2𝑧=−^𝑠4⋅((2𝑛₁+2𝑛₂+1)+𝑖⋅2𝑛₂)⁴. (2.90) The correspondence with (2.87) is 𝑚₁=𝑛₁+2𝑛₂+1, 𝑚₂=−𝑛₁ for the poles and 𝑚₁=𝑛₁+2𝑛₂, 𝑚₂=−𝑛₁ for the zeros.

Remark 2.2. let us consider the 𝑎₁→∞ limit of the one-parameter series 𝑅_𝑎1 (see (2.31), (2.32)) rewriting 𝑅_𝑎1(𝑧) as 𝑅_𝑏1(𝑢)𝑅_𝑏1(𝑢)=𝑅_𝑎1(𝑧),with∶𝑏₁=1𝑎₁𝑧,𝑢=𝑏₁. (2.91) In the 𝑎₁→∞ limit, that is the 𝑏₁→0 limit, one easily verifies, order by order in 𝑢, that 𝑅_𝑏1(𝑢) becomes exactly the transcendental parametrization function (2.53): 𝑅_𝑏1(𝑢)→𝑃(𝑢)when𝑏₁→0. (2.92) For 𝑎₁=(−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 agreement²³ with (2.91).

2.5. 𝑃(𝑧) and an Infinite Number of Rational Transformations: The Sky Is the Limit

Note that some nonlinear ODEs associated with 𝑃(𝑧) and displayed in Appendix B, namely (B.3) and (B.10), and the functional equation (2.79), are invariant by the change of variable (𝑃(𝑧),𝑧)→(−4𝑃(𝑧)/(1−𝑃(𝑧))²,−4𝑧). In fact (B.3), (2.79), and (B.10) are invariant by (𝑃(𝑧),𝑧)→(−4𝑃(𝑧)/(1−𝑃(𝑧))²,−4𝑧), 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:𝑃(−4⋅𝑧)=−4𝑃(𝑧)1−𝑃(𝑧)², (2.93) but also𝑃((−7−24𝑖)⋅𝑧)=𝑇(𝑃(𝑧)),𝑃((−7+24𝑖)⋅𝑧)=𝑇^∗(𝑃(𝑧)), (2.94) and, more generally, as can be checked order by order on series expansions𝑃(𝑎₁⋅𝑧)=𝑅_𝑎1(𝑃(𝑧)). (2.95) For example, considering the “good” branch (2.35) for the inverse of −4𝑧/(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 relation𝑆⁽¹⁾_−1/4𝑧(𝑃(𝑧))=𝑃(−4), (2.96)valid for |𝑃(𝑧)|<1 since the radius of convergence of 𝑆⁽¹⁾_−1/4(𝑧) is 1.

Recalling the functional equations (2.94) 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 (2.94), 𝑃(𝑧) is also singular at 𝑧=(−7±24𝑖)⋅𝑧_𝑠. It is thus extremely natural to see the emergence of the infinite number of singularities in (2.87) of the form 𝑧=(𝑁₁+𝑖⋅𝑁₂)⋅𝑧_𝑠, as a consequence of (2.95) together with a reduction of the one-parameter series 𝑅_𝑎1(𝑧) to a rational function for an infinite number of selected values of 𝑎₁, namely the 𝑁₁+𝑖⋅𝑁₂ in (2.87). This is actually the case for all the values displayed in (2.87). For instance, for 𝑎₁=3⁴=81 we get the following simple rational function:𝑅₈₁𝑧(𝑧)=𝑧⋅(²+6𝑧−33𝑧²)−6𝑧−1⁴, (2.97) for which it is straightforward to verify that this rational transformation commutes with 𝑇(𝑧), 𝑇^∗(𝑧), −4𝑧/(1−𝑧)², and is a solution of the Rota-Baxter-like functional equation (2.19). The case 𝑎₁=5⁴=625 in (2.87), is even simpler, since it just requires to compose 𝑇(𝑧) and 𝑇^∗(𝑧)𝑅₆₂₅(𝑧)=𝑇(𝑇^∗(𝑧))=𝑇^∗𝑧(𝑇(𝑧))=𝑧⋅(²−2𝑧+55𝑧²)−2𝑧+1⁴⋅(1−12𝑧−26𝑧²+52𝑧³+𝑧⁴1+52𝑧−26𝑧²−12𝑧³+𝑧⁴)⁴, (2.98) which, again verifies (2.19) and commutes with all the other rational functions, in particular (2.97). We also obtained the rational function corresponding to 𝑎₁=7⁴=2401, namely: 𝑅₂₄₀₁𝑁(𝑧)=𝑧⋅(²⁴⁰¹(𝑧)𝐷₂₄₀₁)(𝑧)⁴, (2.99) with:𝑁₂₄₀₁(𝑧)=𝑧¹²⋅𝐷₂₄₀₁(1𝑧𝐷),₂₄₀₁(𝑧)=1+196𝑧−1302𝑧²+14756𝑧³−15673𝑧⁴−42168𝑧⁵+111916𝑧⁶−82264𝑧⁷+35231𝑧⁸−19852𝑧⁹+2954𝑧¹⁰+308𝑧¹¹−7𝑧¹². (2.100)(2.101)

The polynomial 𝑁₂₄₀₁(𝑧) satisfies many functional equations, like, for instance (with𝑅₋₄(𝑧)=−4𝑧/(1−𝑧²)):4¹²⋅𝐷₂₄₀₁(1𝑅₋₄(𝑧))=𝐷₂₄₀₁(𝑧)⋅𝐷₂₄₀₁(1𝑧) (2.102) and also(1−𝑧)⁴⁹⋅𝐷₂₄₀₁(𝑅₋₄(𝑧))²=𝐷₂₄₀₁(𝑧)⁴−𝑧⁴⁹⋅𝐷₂₄₀₁(1𝑧)⁴. (2.103)

We also obtained the rational function corresponding to 𝑎₁=11⁴=14641, namely, 𝑅₁₄₆₄₁𝑁(𝑧)=𝑧⋅(¹⁴⁶⁴¹(𝑧)𝐷₁₄₆₄₁)(𝑧)⁴, (2.104) with:𝑁₁₄₆₄₁(𝑧)=𝑧³⁰⋅𝐷₁₄₆₄₁(1𝑧), (2.105)  𝐷₁₄₆₄₁(𝑧)=1+1210𝑧−33033𝑧²+2923492𝑧³+5093605𝑧⁴−385382514𝑧⁵[phone]𝑧⁶−14323974808𝑧⁷[phone]𝑧⁸−291359180310𝑧⁹[phone]𝑧¹⁰−1642552094436𝑧¹¹[phone]𝑧¹²[phone]𝑧¹³−6610669151537𝑧¹⁴[phone]𝑧¹⁵−8608181312269𝑧¹⁶[phone]𝑧¹⁷−3223489742187𝑧¹⁸[phone]𝑧¹⁹−1197743580033𝑧²⁰[phone]𝑧²¹−50897017743𝑧²²−7864445336𝑧²³[phone]𝑧²⁴−815789634𝑧²⁵[phone]𝑧²⁶−5092956𝑧²⁷+207691𝑧²⁸+2794𝑧²⁹−11𝑧³⁰, (2.106) and, of course, one can verify that 𝑅₁₄₆₄₁(𝑧) actually commutes with 𝑅₋₄, 𝑅₈₁, 𝑅₆₂₅, 𝑅₂₄₀₁(𝑧), and is a solution of the Rota-Baxter-like functional equation (2.19). Similarly to 𝑅₂₄₀₁(𝑧) (see (2.102), (2.103)), we also have the functional equations:4³⁰⋅𝐷₁₄₆₄₁(1𝑅₋₄(𝑧))=𝐷₁₄₆₄₁(𝑧)⋅𝐷₁₄₆₄₁(1𝑧), (2.107) and also(1−𝑧)^(4⋅30+1)⋅𝐷₁₄₆₄₁(𝑅₋₄(𝑧))²=𝐷₁₄₆₄₁(𝑧)⁴−𝑧^(4⋅30+1)⋅𝐷₁₄₆₄₁(1𝑧)⁴. (2.108)

Next we obtained the rational function corresponding to 𝑎₁=13⁴=28561, which verifies (2.19) namely,𝑅₂₈₅₆₁𝑁(𝑧)=𝑧⋅(²⁸⁵⁶¹(𝑧)𝐷₂₈₅₆₁)(𝑧)⁴,with𝑁₂₈₅₆₁(𝑧)=𝑧⁴²⋅𝐷₂₈₅₆₁(1𝑧𝑁),₂₈₅₆₁(𝑧)=𝑧⁴²⋅𝐷₂₈₅₆₁(1𝑧𝐷),₂₈₅₆₁(𝑧)=(1−22𝑧+235𝑧²−228𝑧³+39𝑧⁴+26𝑧⁵+13𝑧⁶)⋅𝐷⁽³⁶⁾₂₈₅₆₁𝐷(𝑧)_(36)28561(𝑧)=1+2388𝑧−61098𝑧²[phone]𝑧³[phone]𝑧⁴−1543922656𝑧⁵[phone]𝑧⁶−221753896032𝑧⁷[phone]𝑧⁸−4542779886736𝑧⁹[phone]𝑧¹⁰[phone]𝑧¹¹−200879613202428𝑧¹²[phone]𝑧¹³−934179604482832𝑧¹⁴+1235038888776160𝑧¹⁵−1788854212778642𝑧¹⁶+3018407750933816𝑧¹⁷−4349780716415868𝑧¹⁸+4419228090228152𝑧¹⁹−2899766501472914𝑧²⁰[phone]𝑧²¹[phone]𝑧²²−857795672629664𝑧²³[phone]𝑧²⁴−304241349909008𝑧²⁵[phone]𝑧²⁶−14593362219920𝑧²⁷[phone]𝑧²⁸[phone]𝑧²⁹−23660433008𝑧³⁰[phone]𝑧³¹−29540327𝑧³²+3238420𝑧³³−73386𝑧³⁴−492𝑧³⁵+𝑧³⁶. (2.109)(2.110) We get similar results, mutatis mutandis, than the ones previously obtained (commutation, functional equations like (2.107), (2.108), etc.), namely,4⁴²⋅𝐷₂₈₅₆₁(1𝑅₋₄(𝑧))=𝐷₂₈₅₆₁(𝑧)⋅𝐷₂₈₅₆₁(1𝑧),…. (2.111)

The “palindromic” nature of (2.97) (2.98), (2.99), (2.104) and (2.109) (see (2.100), (2.105)), (2.109)) corresponds to the fact that these rational transformations commute with 𝐽:1𝑅₈₁(𝑧)=𝑅₈₁(1𝑧1),𝑅₆₂₅(𝑧)=𝑅₆₂₅(1𝑧),…. (2.112) In fact, more generally, we have 𝑅_𝑁⁴(1/𝑧)=1/𝑅_𝑁⁴(𝑧) for 𝑁 any odd integer (𝑁=9,21,…) and 𝑅_𝑁⁴(1/𝑧)=𝑅_𝑁⁴(𝑧) for any even integer 𝑁.

From (2.87) 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 nontrivial example corresponds to the already found rational function𝑅₁₆(𝑧)=16⋅𝑧⋅(1−𝑧)²(𝑧+1)⁴. (2.113) We already have explicit rational functions for all values of 𝑎₁ of the form 𝑁⁴ 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 𝑎₁ 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 (2.19) 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 (2.96), the previous singularity argument is not valid ²⁴ for the (well-suited) inverse transformations (𝑆⁽¹⁾_−1/4(𝑧), … ) of these rational transformations because (2.96) 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.

2.6. Other Examples of Selected Gauss Hypergeometric ODEs

For heuristic reasons we have focused on 𝐴(𝑧)=(3−5𝑧)/𝑧/(1−𝑧)/4, but of course, one can find many other examples and try to generalize these examples.

For instance, introducing1𝐴(𝑧)=6⋅𝑑ln((1−𝑧)³𝑧⁵)=1𝑑𝑧6⋅5−8𝑧(1−𝑧)𝑧 (2.114) the rational transformation𝑧𝑅(𝑧)=−27⋅(1−4𝑧)³, (2.115) verifies the “Rota-Baxter-like” functional relation (2.19). This example corresponds to the following covariance [12] on a Gauss hypergeometric integral (of the 𝑐=1+𝑏 type, see below):₂𝐹₁1([2,167],[6𝑧];𝑧)=^−1/66⋅^𝑧₀𝑡^−5/6(1−𝑡)^−1/2⋅𝑑𝑡=(1−4𝑧)^−1/2⋅₂𝐹₁1([2,167],[6𝑧];−27(1−4𝑧)³) (2.116) which is associated with the elliptic curve𝑦⁶−(1−𝑡)³⋅𝑡⁵=0. (2.117) Another example (of the 𝑐=1+𝑎 type, see below) is1𝐴(𝑧)=3⋅𝑑ln((1−𝑧)²𝑧²)=2𝑑𝑧3⋅1−2𝑧,(1−𝑧)𝑧 (2.118) where the rational transformation𝑅(𝑧)=𝑧⋅(𝑧−2)³(1−2𝑧)³=−8𝑧−36𝑧²−126𝑧³−387𝑧⁴+⋯ (2.119) verifies the “Rota-Baxter-like” functional relation (2.19). This example corresponds to the following covariance [12] on a Gauss hypergeometric integral:₂𝐹₁1([3,234],[3𝑧];𝑧)=^−1/33⋅^𝑧₀𝑡^−2/3(1−𝑡)^−2/3=1⋅𝑑𝑡2⋅2−𝑧⋅1−2𝑧²𝐹₁1([3,234],[3];𝑧⋅(𝑧−2)³(1−2𝑧)³), (2.120) which is associated with the elliptic curve𝑦³−(1−𝑡)²⋅𝑡²=0. (2.121) Note that, similarly to the main example of the paper, there exist many rational transformations²⁵ satisfying (2.19) that cannot be reduced to iterates of (2.119), for instance,𝑇(𝑧)=−27⋅𝑧⋅(1−𝑧)(𝑧²−𝑧+1)³(𝑧³+3𝑧²−6𝑧+1)³=−27𝑧−378𝑧²−3888𝑧³−34074𝑧⁴−271620𝑧⁵−2032209𝑧⁶+⋯. (2.122) One verifies immediately that (2.122) actually verifies (2.19) with (2.118). Not surprisingly, the two rational transformations (2.119) and (2.122) commute.

Another simple example with rational symmetries corresponds to Ω=(𝐷_𝑧+𝐴(𝑧))⋅𝐷_𝑧 with1𝐴(𝑧)=−2⋅3𝑧−1=1𝑧(1−𝑧)2⋅𝑑ln(𝑧⋅(1−𝑧)²).𝑑𝑧 (2.123) It has the simple (genus zero) hypergeometric solution²⁶:ℱ(𝑧)=𝑧^1/2⋅₂𝐹₁1([1,23],[2];𝑧)=arctanh(𝑧^1/2). (2.124) The linear differential operator Ω is covariant under the change of variable 𝑧→1/𝑧 and 𝑧→𝑅(𝑧), where²⁷𝑅(𝑧)=4𝑧(1+𝑧)². (2.125) One can easily check that (2.123) and (2.125) satisfy the functional equation (2.19). One also verifies that (2.123) and 𝑧→1/𝑧 or the iterates of (2.125) satisfy the functional equation (2.19). The solution of the adjoint operator are (1−𝑧)⋅𝑧^1/2 and𝐹(𝑧)=𝑧⋅(1−𝑧)⋅₂𝐹₁1([1,23],[2];𝑧)=𝑧^1/2⋅(1−𝑧)⋅arctanh(𝑧^1/22)=𝑧−3𝑧²−2𝑧15³−2𝑧35⁴−2𝑧63⁵−2𝑧99⁶−2𝑧143⁷+⋯. (2.126) One verifies, again, that (2.126) and (2.125) commute, (2.126) corresponding to the “infinitesimal composition” of (2.125) (see (2.56)).

A first natural generalization amounts to keeping the remarkable factorization (2.6) which will, in fact, reduce the covariance of a second-order operator to the covariance of a first-order operator.²⁸ Such a situation occurs for Gauss hypergeometric functions₂𝐹₁([𝑎,𝑏],[1+𝑎];𝑧) solution of the (𝑎,𝑏)-symmetric linear differential operator𝑧⋅(1−𝑧)⋅𝐷²_𝑧+(𝑐−(𝑎+𝑏+1)⋅𝑧)⋅𝐷_𝑧−𝑎⋅𝑏, (2.127) as soon as²⁹𝑐=1+𝑎. For instanceℱ(𝑧)=𝑧^𝑎⋅₂𝐹₁([𝑎,𝑏],[1+𝑎];𝑧), (2.128) is an integral of a simple algebraic function and is solution with the constant function of the second-order operatorΩ=𝐷_𝑧+(𝑎−𝑏−1)𝑧+1−𝑎𝑧⋅(1−𝑧)⋅𝐷_𝑧=(𝐷_𝑧+𝑑ln((1−𝑧)^𝑏⋅𝑧^1−𝑎)𝑑𝑧)⋅𝐷_𝑧, (2.129) yielding a new 𝐴(𝑧):𝐴(𝑧)=(1−𝑎)+(𝑎−𝑏−1)𝑧=(1−𝑧)⋅𝑧1−𝑎𝑧−𝑏.1−𝑧 (2.130) The adjoint of (2.129) has the simple solution 𝑧^1−𝑎⋅(1−𝑧)^𝑏:𝐹(𝑧)=𝑧⋅(1−𝑧)^𝑏⋅₂𝐹₁([𝑎,𝑏],[1+𝑎];𝑧). (2.131)

Due to the (𝑎,𝑏)-symmetry of (2.127) we have a similar result for 𝑐=1+𝑏. The function ℱ(𝑧)=𝑧^𝑏⋅₂𝐹₁([𝑎,𝑏],[1+𝑏];𝑧) is solution of (2.129) where 𝑎 and 𝑏 have been permuted:(𝐷_𝑧+(𝑏−𝑎−1)𝑧+1−𝑏𝑧⋅(1−𝑧))⋅𝐷_𝑧, (2.132) yielding another 𝐴(𝑧)𝐴(𝑧)=(1−𝑏)+(𝑏−𝑎−1)𝑧,(1−𝑧)⋅𝑧 (2.133) The adjoint of (2.132) has the solution (1−𝑧)^𝑎⋅𝑧^1−𝑏 together with the hypergeometric function:𝐹(𝑧)=𝑧⋅(1−𝑧)^𝑎⋅₂𝐹₁([𝑎,𝑏],[1+𝑏];𝑧), (2.134) where one recovers the previous result (2.126).

We are seeking for (Gauss hypergeometric) second-order differential equations³⁰ 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 [11] second-order differential operators). Let us denote by 𝐷 the common denominator of the two rational numbers 𝑎=𝑁_𝑎/𝐷 and 𝑏=𝑁_𝑏/𝐷, function (2.128) is associated to a period of the algebraic curve𝑦^𝐷=(1−𝑡)^𝑁𝑏⋅𝑡^{𝐷−𝑁𝑎}. (2.135) 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, with₂𝐹₁1([3,167],[61];𝑧)=6⋅𝑧^−1/6⋅^𝑧₀𝑡^−5/6(1−𝑡)^−1/3⋅𝑑𝑡, (2.136) which corresponds to the genus two curve:𝑦⁶−(1−𝑡)²⋅𝑡⁵=0. (2.137) Again one introduces 𝐴(𝑧)1𝐴(𝑧)=6⋅𝑑ln((1−𝑧)²𝑧⁵)=1𝑑𝑧6⋅5−7𝑧𝑧⋅(1−𝑧) (2.138) and seeks for 𝑅(𝑧) as series expansions analytical at 𝑧=0. One gets actually, order by order, a one-parameter family𝑅_𝑎1(𝑧)=𝑎₁2⋅𝑧−7𝑎₁⋅(𝑎₁−1)⋅𝑧²+1𝑎637¹⋅(𝑎₁−1)⋅(17𝑎₁−87)⋅𝑧³−2𝑎84721¹⋅(𝑎₁−1)⋅(113𝑎²₁−856𝑎₁+3438)⋅𝑧⁴−1𝑎38548055¹⋅(𝑎₁−1)⋅(3674𝑎³₁+121194𝑎²₁−552261𝑎₁+2095059)⋅𝑧⁵++⋯1+𝜖_𝑛𝑁(𝑛)⋅𝑎₁⋅(𝑎₁−1)⋅𝑃_𝑛(𝑎₁)⋅𝑧^𝑛+⋯, (2.139) where 𝜖_𝑛=0 for 𝑛 odd and 𝜖_𝑛=1 for 𝑛 even, and 𝑁(𝑛) is a (large) integer depending on 𝑛, and 𝑃_𝑛(𝑎₁) 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:𝑅_𝑎1(𝑅_𝑏1(𝑧))=𝑅_𝑏1(𝑅_𝑎1(𝑧))=𝑅_𝑎1𝑏1(𝑧). (2.140)

As far as the “algorithmic complexity” of this series (2.139) is concerned it is worth noticing that the degree growth [18] of the series coefficients is actually linear and not exponential as we could expect [19] at first sight. Even if this series is transcendental, it is not a “wild” series.

Seeking for selected values of 𝑎₁ such that the previous series (2.139) reduces to a rational function one can try to reproduce the simple calculations (2.48), (2.49), 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 (2.48), (2.49).

It is also worth noticing that if we slightly change 𝐴(𝑧) into1𝐴(𝑧)=𝑁⋅𝑑ln((1−𝑧)²𝑧⁵)=1𝑑𝑧𝑁⋅5−7𝑧,𝑧⋅(1−𝑧) (2.141) the algebraic curve (2.137) becomes 𝑦^𝑁−(1−𝑡)²⋅𝑡⁵=0 which has, for instance genus five for 𝑁=11, but genus zero for 𝑁=7. For any of these cases of (2.141) one can easily get, order by order, a one-parameter series 𝑅_𝑎1 totally similar to (2.139) with, again, polynomials 𝑃_𝑛(𝑎₁) of degree 𝑛−2.

The first coefficient 𝑎₂ is in general𝑎₂2=−2𝑁−5⋅𝑎₁⋅(𝑎₁−1). (2.142)

For the genus zero case, 𝑁=7𝑎₂2=−9⋅𝑎₁⋅(𝑎₁−1),𝑎₃1=−1296⋅𝑎₁⋅(𝑎₁−1)⋅(127−𝑎₁𝑎),₄1=−134136⋅𝑎₁⋅(𝑎₁−1)⋅(254𝑎²₁+185𝑎₁𝑎+7499),…,_𝑛1=−𝑁(𝑛)⋅𝑎₁⋅(𝑎₁−1)⋅𝑃_𝑛(𝑎₁), (2.143) which corresponds to the solution27⋅^𝑧₀𝑧^−5/7⋅(1−𝑧)^−2/7⋅𝑑𝑡=𝑧^2/7⋅₂𝐹₁2([7,279],[7];𝑧). (2.144) Using the parametrization of the genus zero curve 𝑦=−(𝑢+1)²⋅𝑢⁵(𝑢+1)⁷−𝑢⁷𝑢,𝑡=−⁷(𝑢+1)⁷−𝑢⁷, (2.145) one can actually perform the integration (2.144) of 𝑑𝑡/𝑦 and get an alternative form of the hypergeometric function (2.144):^𝑧₀𝑧^−5/7⋅(1−𝑧)^−2/7⋅𝑑𝑡=^𝑢₀𝜌(𝑢)⋅𝑑𝑢=^𝑣₀𝑣1−𝑣⁷𝑢⋅𝑑𝑣,where∶𝑧=−⁷(𝑢+1)⁷−𝑢⁷,𝜌(𝑢)=(𝑢+1)⁴⋅𝑢(𝑢+1)⁷−𝑢⁷,𝑢and∶𝑣=𝑣1+𝑢,𝑧=⁷𝑣⁷.−1 (2.146) Except transformations like 𝑣→𝜔⋅𝑣 (with 𝜔⁷=1) which have no impact on 𝑧, it seems difficult to find rational symmetries in this genus zero case.

For 𝑁=11 (genus five) the first successive coefficients read:𝑎₂2=−17⋅𝑎₁⋅(𝑎₁𝑎−1),₃1=−8092⋅𝑎₁⋅(𝑎₁−1)⋅(143𝑎₁𝑎+367),₄1=−206346⋅𝑎₁⋅(𝑎₁−1)⋅(1186𝑎²₁+2473𝑎₁𝑎+5011),…,_𝑛1=−𝑁(𝑛)⋅𝑎₁⋅(𝑎₁−1)⋅𝑃_𝑛(𝑎₁). (2.147) The “infinitesimal composition” function 𝐹(𝑧) (see (2.56), (2.57), and (2.58)) reads,𝐹(𝑧)=𝜕𝑅_𝑎1𝜕𝑎₁|_𝑎1=12=𝑧−𝑧17²−15z238³−5𝑧119⁴−37𝑧1190⁵−888𝑧36295⁶−2183𝑧108885⁷−4366𝑧258213⁸−58941𝑧4045337⁹−1807524𝑧141586795¹⁰−46543743𝑧4106017055¹¹−5305986702𝑧521464165985¹²+⋯ (2.148) and again we can actually check that this is actually the series expansion of the hypergeometric function𝑧⋅(1−𝑧)⋅₂𝐹₁([1,1511],[1711];𝑧), (2.149) solution of Ω^∗ the adjoint of the Ω linear differential operator corresponding to this (genus 5) 𝑁=11 case:Ω^∗=𝐷_𝑧⋅(𝐷_𝑧+1⋅117𝑧−5𝑧⋅(1−𝑧)). (2.150)

We have similar results for (2.128), (2.129), (2.130). As far as these one-parameter families of transformations 𝑅_𝑎1 are concerned, the only difference between the generic cases corresponding to arbitrary genus and genus one cases like (2.118) is that in the generic higher genus case, only a finite number of values of the parameter 𝑎₁ can correspond to rational functions. Note that this higher genus result generalizes to the arbitrary genus Gauss hypergeometric functions (2.128) and associated operators (2.129) and function (2.130). In this general case one can also get order by order a one-parameter family of transformations 𝑅_𝑎1 satisfying a commutation relation (2.141).

Note that 𝑅(𝑧)=1/𝑧 is actually a solution of (2.19) for this genus-two example (2.139). Along this line of selected 𝑅(𝑧) solutions of (2.19) many interesting subcases of this general case (2.128), (2.129), (2.130) 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 Appendices C and D). Beyond this framework we still have one-parameter families (see (2.141)) but we cannot expect an infinite number of rational (and probably algebraic) transformations to be particular cases of such families of transcendental transformations.

3. Conclusion

We have shown that several selected Gauss hypergeometric linear differential operators associated to elliptic curves, and factorised into order-one linear 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 [11] of quite high orders [20–24] that, in fact, factor into globally nilpotent operators of smaller orders³¹ 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 [25–27], 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 [28, 29] and, beyond, some knowledge of mirror symmetries [30–34] of Calabi-Yau manifolds, these mirror symmetries generalizing³² the Hauptmodul structure for elliptic curves.

Appendices

A. Comment on the Rota-Baxter-Like Functional Equation (2.19)

We saw, several times, that the Rota-Baxter-like functional equation (2.19) is such that for a given 𝐴(𝑧) one gets a one-parameter family of analytical functions 𝑅(𝑧) obtained order by order by series expansion (see (2.32), (2.139)). Conversely for a given 𝑅(𝑧), for instance, 𝑅(𝑧)=−4𝑧/(1−𝑧)², let us see if 𝑅(𝑧) can come from a unique 𝐴(𝑧). Assume that there are two 𝐴(𝑧) satisfying (2.26) with the same 𝑅(𝑧)=−4𝑧/(1−𝑧)². We will denote 𝛿(𝑧) the difference of these two 𝐴(𝑧), and we will also introduce Δ(𝑧)=𝑧⋅𝛿(𝑧). It is a straightforward calculation to see that Δ(𝑧) verifiesΔ(𝑧)=1+𝑧1−𝑧⋅Δ(−4𝑧(1−𝑧)²), (A.1) which has, beyond Δ(𝑧)=0, at least one solution analytical at 𝑧=0 that we can get order by order:2Δ(𝑧)=1+5𝑧+22𝑧75²+394𝑧1625³+262634𝑧1243125⁴+⋯. (A.2) It is straightforward to show from (A.1), from similar arguments we introduced for (2.57) on the functional equations (2.58) that Δ(𝑧) is a transcendental function.

B. Miscellaneous Nonlinear ODEs on 𝑃(𝑧)

From (2.70) one can get𝐹^′𝑃(𝑃(𝑧))=1+𝑧⋅⁽²⁾𝑃⁽¹⁾,𝐹^′′𝑃(𝑃(𝑧))=⁽²⁾(𝑃⁽¹⁾)²𝑃+𝑧⋅⁽³⁾(𝑃⁽¹⁾)²−𝑧⋅(𝑃⁽²⁾)²(𝑃⁽¹⁾)³, (B.1) and from (2.64), the linear second-order ODE on 𝐹(𝑧), one deduces the third-order nonlinear ODE³³ on the (at first sight nonholonomic) function 𝑃(𝑧):𝑧⋅(5𝑃²−6𝑃+3)⋅(𝑃⁽¹⁾)⁴−𝑃⋅(5𝑃−3)⋅(𝑃−1)⋅(𝑃⁽¹⁾)³−𝑧⋅(𝑃−1)⋅𝑃⋅(5𝑃−3)⋅𝑃⁽²⁾⋅(𝑃⁽¹⁾)²+4𝑃²⋅(𝑃−1)²⋅(𝑃⁽²⁾+𝑧⋅𝑃⁽³⁾)⋅𝑃⁽¹⁾−4𝑧⋅(𝑃⁽²⁾)²⋅𝑃²⋅(𝑃−1)²=0, (B.2) where the 𝑃^(𝑛)'s denote the 𝑛th derivative of 𝑃(𝑧). This third order nonlinear ODE has a rescaling symmetry 𝑧→𝜌⋅𝑧, for any 𝜌, and, also, an interesting symmetry, namely an invariance by 𝑧→𝑧^𝛼, for any³⁴ value of 𝛼.

In a second step, using differential algebra tools, and, more specifically, the fact that 𝑃(𝑄(𝑧))=𝑄(𝑃(𝑧))=𝑧 together with the linear ODE for 𝑄(𝑧), one finds the simpler second-order nonlinear ODE for 𝑃(𝑧):𝑃⁽²⁾−14⋅5𝑃−3(𝑃−1)⋅𝑃⋅(𝑃⁽¹⁾)²+34⋅1𝑧⋅𝑃⁽¹⁾=0 (B.3) or𝑃⁽²⁾3−(4⋅1𝑃+12⋅1𝑃−1)⋅(𝑃⁽¹⁾)²+34⋅1𝑧⋅𝑃⁽¹⁾=0. (B.4) Note that, more generally, the second-order nonlinear ODE𝑃⁽²⁾3−(4⋅1𝑃+12⋅1𝑃−1)⋅(𝑃⁽¹⁾)²+𝜂𝑧⋅𝑃⁽¹⁾=0, (B.5) yields (B.2) for any value of the constant 𝜂. The change of variable z→𝑧^𝛼, changes the parameter 𝜂 into 1+𝛼⋅(𝜂−1). In particular the involution 𝑧↔1/𝑧 changes 𝜂=3/4 into 𝜂=5/4.

This nonlinear ODE, looking like Painlevé V, is actually invariant by the change of variable 𝑃→−4𝑃/(1−𝑃)². It is, also, invariant by any rescaling 𝑧→𝜆𝑧, like the particular degenerate³⁵ subcase of Painlevé V 𝑦^′′1−(+12𝑦𝑦−1)⋅𝑦^′2+1𝑧⋅𝑦^′=0. (B.6)

With (2.70) we recover the “Gauss-Manin” idea of Painlevé functions being seen as deformations of elliptic functions:𝑧⋅𝑑𝑃(𝑧)𝑑𝑧=𝑃(𝑧)⋅(1−𝑃(𝑧))^1/2⋅₂𝐹₁1([4,125],[4];𝑃(𝑧)). (B.7) or−2𝑧⋅𝑑arctanh((1−𝑃(𝑧))^1/2)=𝑑𝑧²𝐹₁1([4,125],[4];𝑃(𝑧)). (B.8) In fact, recalling 𝑄(𝑃(𝑧))=𝑧, one also has the relation𝑃(𝑧)⋅₂𝐹₁1([4,125],[4];𝑃(𝑧))⁴=𝑧, (B.9) yielding with (B.7) the simple nonlinear order-one differential equation𝑧³⋅(𝑃^′)⁴−(1−𝑃)²⋅𝑃³=0, (B.10) already seen with (2.83), and that we can write in a separate way:𝑑𝑃(1−𝑃)^1/2⋅𝑃^3/4=𝑑𝑧𝑧^3/4. (B.11) Note that 𝑃(𝑧^{4⋅(1−𝜂)}) is actually solution of (B.5).

Equation (B.10) has (B.9) as a solution but in general the Puiseux series solutions 𝑃_𝜇(𝑧) of the functional equation (𝜇 is a constant):𝑃_𝜇(𝑧)^1/4⋅₂𝐹₁1([4,125],[4];𝑃_𝐴(𝑧))=𝜇+𝑧^1/4𝑃or_𝜇(𝑧)=𝑃((𝜇+𝑧^1/4)⁴𝑃),_𝜇(𝑧)=𝑃(𝜇⁴)+4⋅𝜇³⋅𝑃^′(𝜇⁴)⋅𝑧^1/4+⋯. (B.12) It is a straightforward exercise of differential algebra to see that the order-one nonlinear differential equation (B.10) implies (B.3). In particular not only (B.9) is solution of (B.3) but also all the Puiseux series solutions (B.12) of (B.10). More generally the solutions of the functional equation:𝑃_𝜇,𝜆(𝑧)^1/4⋅₂𝐹₁1([4,125],[4];𝑃_𝜇,𝜆(𝑧))=𝜇+𝜆⋅𝑧^1/4 (B.13) verify (B.3). This corresponds to the fact that𝑧³⋅(𝑃^′)⁴−𝜆⁴⋅(1−𝑃)²⋅𝑃³=0, (B.14) yields (B.2) which is scaling symmetric (𝑧→𝜌⋅𝑧) when (B.10) is not. More generally𝑧^4𝜂⋅(𝑃^′)⁴−𝜆⁴⋅(1−𝑃)²⋅𝑃³=0 (B.15) yields (B.2) for any value of the parameters 𝜂 and 𝜆. Finally, one also has that the solution of the functional equation𝑃_𝜂(𝑧)^1/4⋅₂𝐹₁1([4,125],[4];𝑃_𝜂(𝑧))=𝜇+𝜆⋅𝑧^1−𝜂 (B.16) is solution of (B.2), but also of (B.5) and even of (B.15).

Equation (B.5) with 𝜂=1/2 (instead of 𝜂=3/2 in (B.3)) has a solution, analytical at 𝑧=0:11+𝑥+2𝑥²+7𝑥40³+1𝑥20⁴+121𝑥9600⁵+7𝑥2400⁶+211𝑥332800⁷+41𝑥312000⁸+⋯. (B.17) This series has a singularity at −1/4⋅𝑧²_𝑠, where 𝑧_𝑠 is given by (2.85). The radius of convergence of (B.17) corresponds to this singularity, namely, 𝑅=1/4⋅𝑧²_𝑠. This singularity result can be understood from the fact that, at 𝜂=1/2, 𝑃(𝑧²) is actually solution of (B.5).

In fact, we have the following solutions of (B.5) for various selected values of 𝜂. For 𝜂=0, 𝑃(𝑧⁴) is solution of (B.5). For 𝜂=2/3, 𝑃(𝑧^4/3) is solution of (B.5), and, more generally, 𝑃(𝑧^{4⋅(1−𝜂)}) is solution of (B.5).

C. Gauss Hypergeometric ODEs Related to Elliptic Curves

It is not necessary to recall the close connection between Gauss hypergeometric functions and elliptic curves, or even modular curves [35, 36] and Hauptmoduls. This is very clear on the Goursat-type relation₂𝐹₁([2𝑎,2𝑎+13],[4𝑎+23];𝑥)=(1−𝑥+𝑥²)^−𝑎⋅₂𝐹₁𝑎([3,𝑎+13],[4𝑎+56];274⋅(𝑥−1)²⋅𝑥²(1−𝑥+𝑥²)³), (C.1) which generalizes the simpler quadratic Gauss relation:₂𝐹₁([𝑎,𝑏],[𝑎+𝑏+12];𝑥)=₂𝐹₁𝑎([2,𝑏2],[𝑎+𝑏+12];4𝑥(1−𝑥)). (C.2) On (C.1) one recognizes (the inverse of) the Klein modular invariant ³⁶ for the pull-back of the hypergeometric function on the rhs.

Many values of [[𝑎,𝑏],[𝑐]] are known to correspond to elliptic curves like [[1/2,1/2],[1]] (complete elliptic integrals of the first and second kind) or modular forms: [[1/12,5/12],[1]], [[2/3,2/3],[1]], [[2/3,2/3],[3/2]], and they can even be simply related:(𝑧+27)27^1/3⋅₂𝐹₁2([3,231],[1];−27𝑧)=𝜇(𝑧)⋅₂𝐹₁1([,512𝑧12],[1];1728(𝑧+27)(𝑧+3)³), (C.3) where:𝜇(𝑧)=((𝑧+27)(𝑧+3)³)729^−1/12. (C.4)

Once we have a hypergeometric function corresponding to an elliptic curve for some values of (𝑎,𝑏,𝑐), one can find other values of (𝑎,𝑏,𝑐) also corresponding to elliptic curves₂𝐹₁([𝑎,𝑏],[𝑐];𝑥)→𝑥^1−𝑐⋅₂𝐹₁([1+𝑎−𝑐,1+𝑏−𝑐],[2−𝑐];𝑥). (C.5)

In order to provide simple examples of linear differential ODEs we will restrict ourselves (just for heuristic reasons) to Gauss hypergeometric second-order differential equations.

Let us recall the Euler integral representation of the Gauss hypergeometric functions:₂𝐹₁([𝑎,𝑏],[𝑐];𝑧)=Γ(𝑐)⋅Γ(𝑏)Γ(𝑐−𝑏)¹₀𝑑𝑤𝑤𝑤^𝑏⋅(1−𝑤)^{𝑐−1−𝑏}⋅(1−𝑧𝑤)^−𝑎=Γ(𝑐)⋅Γ(𝑎)Γ(𝑐−𝑎)¹₀𝑑𝑤𝑤𝑤^𝑎⋅(1−𝑤)^{𝑐−1−𝑎}⋅(1−𝑧𝑤)^−𝑏=Γ(𝑐)Γ(𝑎)Γ(𝑐−𝑎)⋅𝑧^−𝑎𝑧₀𝑑𝑢𝑢𝑢^𝑎𝑢⋅(1−𝑧)^{𝑐−1−𝑎}⋅(1−𝑢)^−𝑏. (C.6)

On the last line of (C.6), the selected role of 𝑐=1+𝑎 is quite clear.

Recall that the corresponding second-order differential operator is invariant under the permutation of 𝑎 and 𝑏 which is not obvious³⁷ 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 [11, 37–40] that the corresponding second-order differential operator is necessarily globally nilpotent [11, 37–40]. 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 𝑦^𝐷=(1−𝑢)^𝑁𝑏⋅𝑢^{𝐷−𝑁𝑎}𝑢⋅(1−𝑧)^{−𝑁𝑐+𝐷+𝑁𝑎}. (C.7) Recalling the main example of the paper, one associates with ₂𝐹₁([1/4,1/2],[5/4];𝑧)₂𝐹₁1([2,145],[4];𝑧)=Γ(5/4)Γ(1/2)Γ(3/4)⋅¹₀𝑑𝑤𝑤⋅𝑤^1/2⋅(1−𝑤)^−1/4⋅(1−𝑧𝑤)^−1/4=Γ(5/4)Γ(1/2)Γ(3/4)⋅𝑧^−1/2⋅^𝑧₀𝑢^−1/2𝑢⋅(1−𝑧)^−1/4⋅(1−𝑢)^−1/4⋅𝑑𝑢 (C.8) the 𝑧-pencil of elliptic curves³⁸𝑦⁴−𝑢²𝑢⋅(1−𝑢)⋅(1−𝑧)=0, (C.9) where we associated (see (2.2)) to ₂𝐹₁([1/2,1/4],[5/4];𝑧) the elliptic curve𝑦⁴−𝑢³⋅(1−𝑢)²=0. (C.10)

C.1. Miscellaneous Examples

In the more general (2.128), (2.129), (2.130), (resp. (2.132), (2.133)) framework, one can find many interesting subcases. (i)The previous 𝑅(𝑧)=1/𝑧 involution is solution of the functional relation (2.19) when 𝑎=2𝑏 if 𝑐=1+𝑏, or 𝑏=2𝑎 if 𝑐=1+𝑎.(ii)The involution 𝑅(𝑧)=1−𝑧 is solution of the functional relation (2.19) when 𝑎+𝑏=1 if 𝑐=1+𝑏, or 𝑐=1+𝑎.(iii)The infinite-order transformation: 𝑧𝑅(𝑧)=𝑡⋅1+(𝑡−1)⋅𝑧,𝑅^(𝑛)(𝑧)=𝑡^𝑛⋅𝑧1+(𝑡^𝑛,−1)⋅𝑧 (C.11) is solution of the functional relation (2.19) when 𝑎=1+𝑏 if 𝑐=1+𝑏, or 𝑏=1+𝑎𝑐=1+𝑎.(iv)The scaling transformation 𝑅(𝑧)=𝑡⋅𝑧 is solution of the functional relation (2.19) when 𝑎=0 and 𝑐=1+𝑏 (resp., 𝑏=0 and 𝑐=1+𝑎).(v)We also have a quite degenerate situation for 𝑏=1 or 𝑎=1 when 𝑐=2 with the infinite-order transformation 𝑅(𝑧)=1−𝑡⋅(1−𝑧),𝑅^(𝑛)(𝑧)=1−𝑡^𝑛⋅(1−𝑧), (C.12) solution of (2.19).(vi)The two order-three transformations 𝑅(𝑧)=𝑧−1𝑧1,𝑅(𝑅(𝑧))=,1−𝑧 (C.13) are solutions of the functional relation (2.19) for 𝑎=2/3,𝑏=1/3,𝑐=4/3, or 𝑎=1/3,𝑏=1/3,𝑐=4/3.

D. Ising Model Susceptibility: 𝜒⁽²⁾ and Elliptic Curves

The two-particle contribution of the susceptibility of the Ising model [25–27] is given by a double integral. This double integral on two angles 𝜒⁽²⁾ reduces to a simple integral³⁹ (because the two angles are opposite):𝜒⁽²⁾=^𝜋₀𝑑𝜃⋅𝑦²⋅1+𝑥²1−𝑥²⋅(𝑥⋅sin(𝜃)1−𝑥²)², (D.1) where1𝑥=𝐴−𝐵,𝐴=2𝑤−cos(𝜃),𝐵²=𝐴²−1,𝑦²=1𝐴².−1 (D.2) Denoting 𝐶=cos(𝜃) we can rewrite the integral 𝜒⁽²⁾ as𝜒⁽²⁾=¹₀𝑑𝐶(1−𝐶²)^1/2⋅𝑥²⋅𝑦²⋅1+𝑥²(1−𝑥²)³, (D.3) that we want to see as:¹₀𝑑𝐶𝑧=^𝑤₀𝑑𝑞𝑍. (D.4) The variable 𝑧 reads:1𝑧−1(1−𝐶²)^1/2⋅𝑥²⋅𝑦²⋅1+𝑥²(1−𝑥²)³=0, (D.5) which after simplifications gives𝐴²(𝐶²−1)⋅𝑧²+(𝐴²−1)⁵=0, (D.6) that is(12𝑤−𝐶)²⋅(𝐶²−1)⋅𝑧²1+((2𝑤−𝐶)²−1)⁵=0. (D.7) In terms of the variable 𝑞=𝑤⋅𝐶 one can rewrite (see (D.4)) the integral (D.3) as an incomplete integral:256⋅(1−2𝑞)²(𝑞²−𝑤²)⋅𝑍²𝑤⁴+(2𝑞−1+2𝑤)⁵(2𝑞−1−2𝑤)⁵=0. (D.8) This 𝑤-pencil of algebraic curves is actually a 𝑤-pencil of genus one curves, seen as algebraic curves in 𝑍 and 𝑞.

Acknowledgments

One of us (J. M. Maillard) thanks D. Bertrand, A. Enge, M. Hindry, D. Loeffler, J. Nekovár, J. Oesterlé, and J. Watkins for fruitful discussions on modular curves, modular forms, and modular functions. He also thanks D. Mouhanna for large discussions on the renormalization group and A. Ramani and N. Witte for detailed discussions on the Painlevé equations.

Endnotes

1. The renormalization group approach of important problems like first-order phase transitions, commensurate-incommensurate phase transitions, or off-critical problems is more problematic.
2. In contrast with functional renormalization group [41–43].
3. One simply verifies that these transformations reduce to the previous 𝑇_𝑁∶𝑡→𝑡^𝑁 in the 𝑧=1 limit (no magnetic field).
4. For instance the fixed points of (1.3) are not isolated fixed points but lie on (an infinite number) of genus zero curves.
5. In well-suited Boltzmann weight variables like 𝑥 and 𝑧 in (1.3), and not in (bad) variables like 𝐾, the coupling constants or the temperature.
6. 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...).
7. For which the partition function or other physical quantities are algebraic functions.
8. See for instance (2.18) in [44].
9. Suggesting an understanding [4, 45] of the quite rich structure of infinite number of the singularities of the 𝜒^(𝑛)'s in the complex plane from a Hauptmodul approach [4, 45]. Furthermore the notion of Heegner numbers is closely linked to the isogenies mentioned here [4]. An exact value of the 𝑗-function 𝑗(𝜏) corresponding one of the first Heegner number is, e.g., 𝑗(1+𝑖)=12³.
10. It should be recalled that the mirror symmetry found with Calabi-Yau manifolds [30–34] 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.
11. The fact that the renormalization group must be reversible has apparently been totally forgotten by most of the authors who just see a semigroup corresponding to forward iterations converging to the critical points (resp. manifolds).
12. Corresponding to Atkin-Lehner polynomials and Weber's functions.
13. Conversely, and more precisely, writing 1728𝑧²/(𝑧+256)³=1728𝑧^′/(𝑧^′+16)³ gives the Atkin-Lehner [46] involution 𝑧⋅𝑧^′=2¹², together with the quadratic relation 𝑧²−𝑧𝑧^′2−48𝑧𝑧^′−4096𝑧^′=0
14. They are not only Fuchsian, the corresponding linear differential operators are globally nilpotent or𝐺-operators [11].
15. Where 𝑗 is typically the 𝑗-function [44, 47].
16. Such formula is actually valid for Ω_𝐴=(𝐷_𝑧+𝐴(𝑧))⋅𝐷_𝑧 for any 𝐴(𝑧). Denoting 𝒮_𝑁 symmetric 𝑁th power of Ω_𝐴 one has 𝒮_𝑁=(𝐷_𝑧+𝐴(𝑧))⋅𝒮_𝑁−1.
17. The Rota-Baxter relation of weight Θ reads: 𝑅(𝑥)𝑅(𝑦)+Θ𝑅(𝑥𝑦)𝑅(𝑅(𝑥)𝑦+𝑥𝑅(𝑦)).
18. For 𝐴(𝑧) given we get a one-parameter family of 𝑅(𝑧) solution of (2.19). Conversely, for 𝑅(𝑧) given one can ask if there are several 𝐴(𝑧) such that (2.19) is verified. This is sketched in Appendix A.
19. Using the command “dchange” with PDEtools in Maple.
20. Note that the result for 𝜔^∗₁ is nothing but transformation (2.14) on 𝜔_𝑘 for 𝑘=−1. Also note that the two transformations, performing the change of variable 𝑧→−4𝑧/(1−𝑧)² and taking the adjoint, do not commute: (𝜔^∗₁)^(𝑅)≠((𝜔₁)^(𝑅))^∗.
21. Denoted 𝐽𝑎𝑐𝑜𝑏𝑖𝑆𝑁 in Maple: 𝑃(𝑧)=(𝐽𝑎𝑐𝑜𝑏𝑖𝑆𝑁(𝑧^1/4,𝐼))⁴.
22. As a (nonholonomic) elliptic function 𝑃(𝑧) provides elementary examples [48] of nonlinear ODEs with the Painlevé property (like the Weierstrass P-function).
23. It is the absolute value of the inverse of the image of the 𝑛-th iterate of 𝑆⁽¹⁾_−1/4 of −1.
24. If this previous singularity argument was valid we would have had singularities as close as possible to 𝑧=0 (namely, 𝑧_𝑠/(−4)^𝑛), yielding a zero radius of convergence. Similarly combining 𝑇^∗(𝑧) and the inverse of 𝑇(𝑧) we would have obtained an infinite number of singularities on the circle of radius |𝑧_𝑠|.
25. Note a (small) misprint in formula (64) page 174 of Vidunas [12].
26. Of the 𝑐=1+𝑏 type (see below).
27. The change of variable (2.125) can be parametrized with hyperbolic tangents: 𝑧→𝑧^′ with 𝑧=tanh(𝑢)²,𝑧^′=tanh(2𝑢)². Note that 𝑧→4⋅𝑧/(1−𝑧)² is parametrized by 𝑧=tan(𝑢)²𝑧^′=tan(2𝑢)² but 𝑧→−4⋅𝑧/(1−𝑧)² is not parametrized by trigonometric functions.
28. Thus avoiding the full complexity (and subtleties) of the covariance of ODEs by algebraic transformations like modular transformations (1.8).
29. See for instance (C.6) in Appendix C.
30. More generally in our models of lattice statistical mechanics (or enumerative combinatorics etc.) we are seeking for (high order) globally nilpotent [11] 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 [25–27] is clearly, and straightforwardly, associated with an elliptic curve.
31. Experimentally [21] and as could be expected from Dwork's conjecture [11], one often finds for these small order factors hypergeometric second-order operators and sometimes selected Heun functions [49] (or their symmetric products).
32. For instance equation (1.9) of [31]. Do note that the periods of certain K3 families (and hence the original Calabi-Yau family) can be described by the squares of the periods of the elliptic curves [31]. The mirror maps of some K3 surface families are always reciprocals of some McKay-Thompson series associated to the Monstruous Moonshine list of Conway and Norton, with the mirror maps of these examples being always automorphic functions for genus zero [32, 33].
33. Using differential algebra tools one can verify that (2.84) implies (B.2).
34. Beyond diffeomorphisms of the circle: the parameter 𝛼 can be a complex number.
35. Having the movable-poles solutions: (𝛼^𝛽+𝑧^𝛽)²/(𝛼^𝛽−𝑧^𝛽)².
36. Taking for𝑥 the elliptic lambda function.
37. For instance for 2𝐹₁([1/4,1/2],[5/4];𝑧) it changes an Euler integral with Γ(5/4)/Γ(1/4)Γ(1)=1/4 into an Euler integral with Γ(5/4)/Γ(3/4)Γ(1/2)=(1/4)⋅((2𝜋)^1/2/Γ(3/4)²).
38. The algebraic curves (C.9) are genus one curves for any value of 𝑧, except 𝑧=1, where the curve becomes the union of two rational curves (𝑢²−𝑢+𝑦²)(𝑢²−𝑢−𝑦²)=0.
39. The prefactors in front of the integrals are not relevant for our discussion here.