Galois Groups of Difference Equations of Order Two on Elliptic Curves

This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups. For instance, our results combined with a result from trans...

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Veröffentlicht in:Symmetry, Integrability and Geometry: Methods and Applications
Datum:2015
Hauptverfasser: Dreyfus, T., Roques, J.
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Zitieren:Galois Groups of Difference Equations of Order Two on Elliptic Curves / T. Dreyfus, J. Roques // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 35 назв. — англ.

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author Dreyfus, T.
Roques, J.
author_facet Dreyfus, T.
Roques, J.
citation_txt Galois Groups of Difference Equations of Order Two on Elliptic Curves / T. Dreyfus, J. Roques // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 35 назв. — англ.
collection DSpace DC
container_title Symmetry, Integrability and Geometry: Methods and Applications
description This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups. For instance, our results combined with a result from transcendence theory due to Schneider allow us to identify a large class of discrete Lamé equations with difference Galois group GL₂(C).
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 003, 23 pages Galois Groups of Difference Equations of Order Two on Elliptic Curves? Thomas DREYFUS † and Julien ROQUES ‡ † Université Paul Sabatier - Institut de Mathématiques de Toulouse, 18 route de Narbonne, 31062 Toulouse, France E-mail: tdreyfus@math.univ-toulouse.fr URL: https://sites.google.com/site/thomasdreyfusmaths/ ‡ Institut Fourier, Université Grenoble 1, CNRS UMR 5582, 100 rue des Maths, BP 74, 38402 St Martin d’Hères, France E-mail: Julien.Roques@ujf-grenoble.fr URL: www-fourier.ujf-grenoble.fr/~jroques/ Received August 06, 2014, in final form January 08, 2015; Published online January 13, 2015 http://dx.doi.org/10.3842/SIGMA.2015.003 Abstract. This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups. For instance, our results combined with a result from transcendence theory due to Schneider allow us to identify a large class of discrete Lamé equations with difference Galois group GL2(C). Key words: linear difference equations; difference Galois theory; elliptic curves 2010 Mathematics Subject Classification: 39A06; 12H10 1 Introduction Let E ⊂ P2 be the elliptic curve defined by the projectivization of the Weierstrass equation y2 = 4x3 − g2x− g3 with g2, g3 ∈ C. (1.1) We denote by E(C) the group of C-points of E . Its abelian group law is denoted by ⊕. In this paper, we study the difference Galois groups of linear difference equations of order two on E(C) of the form: y(z ⊕ 2h) + a(z)y(z ⊕ h) + b(z)y(z) = 0, (1.2) where y is an unknown function of the variable z ∈ E(C), h is a fixed non torsion point of E(C) and a, b are given rational functions on E . This equation can be seen as a difference equation over C. Indeed, if Λ ⊂ C is a lattice of periods of E and if ℘ is the corresponding Weierstrass function, then C/Λ is identified with E(C) via the factorization through C/Λ of ϕ : z ∈ C 7→ (℘(z) : ℘′(z) : 1) ∈ E(C). Pulling back the equation (1.2) via ϕ, which is a group morphism from (C,+) to (E(C),⊕), we obtain the following difference equation on C: y(z + 2h) + a(z)y(z + h) + b(z)y(z) = 0, (1.3) ?This paper is a contribution to the Special Issue on Algebraic Methods in Dynamical Systems. The full collection is available at http://www.emis.de/journals/SIGMA/AMDS2014.html mailto:tdreyfus@math.univ-toulouse.fr https://sites.google.com/site/thomasdreyfusmaths/ mailto:Julien.Roques@ujf-grenoble.fr www-fourier.ujf-grenoble.fr/~jroques/ http://dx.doi.org/10.3842/SIGMA.2015.003 http://www.emis.de/journals/SIGMA/AMDS2014.html 2 T. Dreyfus and J. Roques where y is an unknown function of the variable z ∈ C, a := a ◦ ϕ and b := b ◦ ϕ are Λ-periodic elliptic functions and h ∈ ϕ−1(h). So, we have the following relations between the equations (1.2) and (1.3): z = ϕ(z), h = ϕ(h) and y(z) = y(z). These equations are discrete counterparts of differential equations on elliptic curves, a famous example of which is Lamé differential equation y′′(z) = (A℘(z) +B)y(z), where A,B ∈ C. The main results of this paper allow us to compute the difference Galois groups of some equations such as the discrete Lamé equation ∆2 hy = (A℘(z) +B)y, where ∆hy(z) = y(z + h)− y(z) h . (1.4) For instance, the following theorem is a consequence of our main results combined with a re- sult from transcendence theory due to Schneider in [31] (see also Bertrand and Masser’s pa- pers [3, 21]). Theorem. Assume that E is defined over Q (i.e., g2, g3 ∈ Q) and that h,A,B ∈ Q with A 6= 0. Then, the difference Galois group of equation (1.4) is GL2(C). To be precise, the base field for the difference Galois groups considered in the present paper is not the field of Λ-periodic meromorphic functions over C, but the field constituted of the meromorphic functions over C which are Λ′-periodic for some sub-lattice Λ′ of Λ. The galoisian aspects of the theory of difference equations have attracted the attention of many authors in the past years, e.g., [1, 4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 30, 34, 35]. The calculation of the difference Galois groups of finite difference or q-difference equations of order two on P1 has been considered by Hendricks [18, 19] and by the second author [29]. The work of Hendricks served as a basis for the present work, but, to the best of our knowledge, the present paper is the first to consider the difference Galois groups of difference equations on a non rational variety. The study of dynamical systems on elliptic curves appears in several areas of mathematics (e.g., discrete dynamical systems, QRT maps). In particular, it is very likely that the equations considered in the present paper will arise as linearizations of discrete dynamical systems, in connection with discrete Morales–Ramis theories [5, 6]. In this context, the difference Galois groups are used to obtain non-integrability results. This paper is organized as follows. Section 2 contains reminders and complements on difference Galois theory (for equations of arbitrary order) with a special emphasis on difference equations on elliptic curves. We insist on the fact that the base difference field for the difference Galois groups considered in the present paper is not the field of Λ-periodic elliptic functions but the field of elliptic functions which are Λ′-periodic for some sub-lattice Λ′ of Λ. In Section 3, we introduce some notations related to the special functions used in this paper (theta functions, Weierstrass ℘-functions) and we collect some useful results. In Section 4, we study the relations between the irreducibility of the difference Galois group of equation (1.3) and the solutions of an associated Riccati-type equation. We then study this Riccati equation assuming that we have a priori informations on the divisors of the coefficients a and b. In Section 5, we show that there is a similar relation between the imprimitivity of the Galois group and some Riccati-type equation. Section 6 is devoted to the calculation of some difference Galois groups, including those of the discrete Lamé equations mentioned above. 2 Difference Galois theory: reminders and complements 2.1 Generalities on difference Galois theory For details on what follows, we refer to [35, Chapter 1]. Galois Groups of Difference Equations of Order Two on Elliptic Curves 3 A difference ring (R,φ) is a ring R together with a ring automorphism φ : R→ R. An ideal of R stabilized by φ is called a difference ideal of (R,φ). If R is a field then (R,φ) is called a difference field. The ring of constants Rφ of the difference ring (R,φ) is defined by Rφ := {f ∈ R |φ(f) = f}. A difference ring morphism (resp. difference ring isomorphism) from the difference ring (R,φ) to the difference ring (R̃, φ̃) is a ring morphism (resp. ring isomorphism) ϕ : R → R̃ such that ϕ ◦ φ = φ̃ ◦ ϕ. A difference ring (R̃, φ̃) is a difference ring extension of a difference ring (R,φ) if R̃ is a ring extension of R and φ̃|R = φ; in this case, we will often denote φ̃ by φ. Two difference ring extensions (R̃1, φ̃1) and (R̃2, φ̃2) of a difference ring (R,φ) are isomorphic over (R,φ) if there exists a difference ring isomorphism ϕ from (R̃1, φ̃1) to (R̃2, φ̃2) such that ϕ|R = IdR. We now let (K,φ) be a difference field. We assume that its field of constants C := Kφ is algebraically closed and that the characteristic of K is 0. Consider a linear difference system φY = AY with A ∈ GLn(K). (2.1) According to [35, Section 1.1], there exists a difference ring extension (R,φ) of (K,φ) such that 1) there exists U ∈ GLn(R) such that φ(U) = AU (such a U is called a fundamental matrix of solutions of (2.1)); 2) R is generated, as a K-algebra, by the entries of U and det(U)−1; 3) the only difference ideals of (R,φ) are {0} and R. Such a difference ring (R,φ) is called a Picard–Vessiot ring for (2.1) over (K,φ). It is unique up to isomorphism of difference rings over (K,φ). It is worth mentioning that Rφ = C; see [35, Lemma 1.8]. Remark 2.1. Picard–Vessiot rings are not domains in general: they are finite direct sums of domains cyclically permuted by φ; see [35, Corollary 1.16]. The corresponding difference Galois group G over (K,φ) of (2.1) is the group of K-linear ring automorphisms of R commuting with φ: G := {σ ∈ Aut(R/K) |φ ◦ σ = σ ◦ φ}. The choice of the base field is by no way innocent. The bigger the base field is, the smaller the Galois group is. A straightforward computation shows that, for any σ∈G, there exists a unique C(σ)∈GLn(C) such that σ(U) = UC(σ). According to [35, Theorem 1.13], one can identify G with an algebraic subgroup of GLn(C) via the faithful representation σ ∈ G 7→ C(σ) ∈ GLn(C). If we choose another fundamental matrix of solutions U , we find a conjugate representation. Remark 2.2. Given an nth order difference equation anφ n(y) + · · ·+ a1φ(y) + a0y = 0, (2.2) 4 T. Dreyfus and J. Roques with a0, . . . , an ∈ K and a0an 6= 0, we can consider the equivalent linear difference system φY = AY, with A =  0 1 0 . . . 0 0 0 1 . . . ... ... ... . . . . . . 0 0 0 . . . 0 1 − a0 an − a1 an . . . . . . −an−1 an  ∈ GLn(K). (2.3) By “Galois group of the difference equation (2.2)” we mean “Galois group of the difference system (2.3)”. We shall now introduce a property relative to the difference base field, which appears in [35, Lemma 1.19]. Definition 2.3. We say that the difference field (K,φ) satisfies property (P) if the following properties hold: • the field K is a C1-field1; • if L is a finite field extension of K such that φ extends to a field endomorphism of L then L = K. The following result is due to van der Put and Singer. We recall that two difference systems φY = AY and φY = BY with A,B ∈ GLn(K) are isomorphic over K if and only if there exists T ∈ GLn(K) such that φ(T )A = BT . Theorem 2.4. Assume that (K,φ) satisfies property (P). Let Kφ = C. Let G ⊂ GLn(C) be the difference Galois group over (K,φ) of φ(Y ) = AY, with A ∈ GLn(K). (2.4) Then, the following properties hold: • G/G◦ is cyclic, where G◦ is the identity component of G; • there exists B ∈ G(K) such that (2.4) is isomorphic to φY = BY over K. Let G̃ be an algebraic subgroup of GLn(C) such that A ∈ G̃(K). The following properties hold: • G is conjugate to a subgroup of G̃; • any minimal element in the set of algebraic subgroups H̃ of G̃ for which there exists T ∈ GLn(K) such that φ(T )AT−1 ∈ H̃(K) is conjugate to G; • G is conjugate to G̃ if and only if, for any T ∈ G̃(K) and for any proper algebraic subgroup H̃ of G̃, one has that φ(T )AT−1 /∈ H̃(K). Proof. The proof of [35, Propositions 1.20 and 1.21] in the special case where K := C(z) and φ is the shift φ(f(z)) := f(z + h) with h ∈ C×, extends mutatis mutandis to the present case. � 1Recall that K is a C1-field if every non-constant homogeneous polynomial P over K has a non-trivial zero provided that the number of its variables is more than its degree. Galois Groups of Difference Equations of Order Two on Elliptic Curves 5 2.2 Difference equations on elliptic curves Let Λ ⊂ C be a lattice. Without loss of generality, we can assume that Λ = Z + Zτ, with =(τ) > 0, where =(·) denotes the imaginary part. For any lattice Λ′ ⊂ C, we let MΛ′ be the field of Λ′-periodic meromorphic functions. We denote by K the field defined by K := ⋃ Λ′ sub-lattice of Λ MΛ′ = ⋃ k≥1 MkΛ . Let h ∈ C such that h mod Λ is not a torsion point of C/Λ. We endow K with the non-cyclic field automorphism φ defined by φ(f)(z) := f(z + h). Then, (K,φ) is a difference field. Proposition 2.5. The field of constants of (K,φ) is Kφ = C. Proof. Consider f ∈ Kφ. Let Λ′ be a sub-lattice of Λ such that f ∈ MΛ′ . Note that f is Λ′-periodic (because f ∈ MΛ′) and h-periodic (because φ(f) = f), so f is a (Λ′ + hZ)-periodic meromorphic function. But Λ′+hZ has an accumulation point because h mod Λ is not a torsion point of C/Λ. Therefore, f is constant. � Proposition 2.6. The difference field (K,φ) satisfies property (P) (see Definition 2.3). Proof. Since K = ⋃ k≥1 MkΛ is the increasing union of the fields MkΛ, the fact that K is a C1- field follows from Tsen’s theorem [20] (according to which the function field of any algebraic curve over an algebraically closed field, e.g., MkΛ, is C1). Let L be a finite extension of K such that φ extends to a field endomorphism of L. We have to prove that L = K. The primitive element theorem ensures that there exists u ∈ L such that L = K(u). Let Λ′ be a sub-lattice of Λ such that • u is algebraic over MΛ′ , • φ(u) ∈ MΛ′(u). Then, MΛ′(u) is a finite extension of MΛ′ and φ induces an automorphism of MΛ′(u). Using the equivalence of categories between between smooth projective curves and function fields of dimension 1 [16, Corollary 6.12], we see that there exists a commutative diagram of the form X f // ϕ �� X ϕ �� C/Λ′ z 7→z+h // C/Λ′ where ϕ : X → C/Λ′ is a morphism of smooth projective curves, whose induced morphism of function fields “is” the inclusion MΛ′ ⊂ MΛ′(u), and where f is an endomorphism of X, whose induced morphism on function fields “is” φ : MΛ′(u)→ MΛ′(u). Considering this commutative diagram, we see that f has degree 1 and that, if ϕ is ramified above y ∈ C/Λ′, then ϕ is also 6 T. Dreyfus and J. Roques ramified above y − h. So, the set of ramification values of ϕ is stable by z 7→ z − h. This set being finite, it has to be empty. So, ϕ is unramified. Hurwitz’s formula implies that X has genus 1, i.e., that X is an elliptic curve. So, there exist a lattice Λ′′ ⊂ C and an isomorphism ψ : MΛ′(u)→ MΛ′′ . There exists (a, b) ∈ C××C such that aΛ′′ ⊂ Λ′ and such that the restriction ψ|MΛ′ is given, for all f ∈ MΛ′ , by ψ(f)(z) = f(az + b). (Indeed, ψ|MΛ′ : MΛ′ → MΛ′′ is a field morphism from the function field of the elliptic curve C/Λ′ to the function field of the elliptic curve C/Λ′′. So, ψ|MΛ′ is induced by a morphism from the elliptic curve C/Λ′′ to the elliptic curve C/Λ′. Now, our claim follows from the fact that the morphisms from C/Λ′′ to C/Λ′ are of the form z mod Λ′′ 7→ az + b mod Λ′ for some (a, b) ∈ C× × C such that aΛ′′ ⊂ Λ′.) The commutative diagram MΛ′ � � // v(z)7→v(az+b) $$ MΛ′(u) ψ �� $$ MΛ′′ w(z)7→w( z−b a ) //MaΛ′′ shows that the fields MΛ′(u) and MaΛ′′ are MΛ′-isomorphic. But the extension MaΛ′′ /MΛ′ is Galois (indeed, this is equivalent to the fact that the corresponding morphism of smooth projective curves C/Λ′ → C/aΛ′′ is Galois, and this is easily seen from the explicit description of the morphisms between these curves). Therefore, any MΛ′-morphism from MaΛ′′ to K(u) must leave MaΛ′′ globally invariant. But, MaΛ′′ and MΛ′(u) are MΛ′-isomorphic subfields of K(u). So MΛ′(u) ⊂ MaΛ′′ , and therefore u ∈ MaΛ′′ ⊂ K and L = K(u) ⊂ K. � Corollary 2.7. The conclusions of Theorem 2.4 are valid for (K,φ). 3 Theta functions and Weierstrass ℘-function 3.1 Theta functions We recall that Λ = Z + τZ ⊂ C with =(τ) > 0. Let θ be the Jacobi theta function defined by θ(z) = ∑ m∈Z (−1)meiπm(m−1)τe2iπmz. We shall now recall some basic facts about this function. We refer to [22, Chapter I] for details and proofs. Remark 3.1. The classical theta function is defined by ϑ(z, τ) = ∑ m∈Z eiπm 2τ+2iπmz. Actually, this is the function studied in [22, Chapter I]. But, there is a simple relation between θ and ϑ, namely θ(z) = ϑ(z + 1−τ 2 , τ). So that any statement for ϑ can be immediately translated into a statement for θ. We recall that θ is a 1-periodic entire function such that θ(z + τ) = −e−2iπzθ(z). Moreover, we have the following formula, known as Jacobi’s triple product formula: θ(z) = ∞∏ m=1 ( 1− e2iπτm )( 1− e2iπ((m−1)τ+z) )( 1− e2iπ(mτ−z)). Galois Groups of Difference Equations of Order Two on Elliptic Curves 7 For any integer k ≥ 1, we let θk be the function given by θk(z) := θ(z/k). This k-periodic entire function satisfies the following functional equation: θk(z + kτ) = −e−2iπz/kθk(z). (3.1) It follows from Jacobi’s triple product formula that the zeroes of θk are simple and that its set of zeroes is kΛ. Let Θk be the set of entire functions of the form c ∏ ξ∈C θk(z − ξ)nξ with c ∈ C× and (nξ)ξ∈C ∈ N(C) with finite support. We denote by Θquot k the set of meromorphic functions over C that can be written as quotient of two elements of Θk. We define the divisor divk(f) of f ∈ Θquot k as the following formal sum of points of C/kΛ: divk(f) := ∑ λ∈C/kΛ ordλ(f)[λ], where ordλ(f) is the (z − ξ)-adic valuation of f , for an arbitrary ξ ∈ λ (it does not depend on the chosen ξ ∈ λ). For any λ ∈ C/kΛ and any ξ ∈ λ, we set [ξ]k := [λ]. Moreover, we will write∑ λ∈C/kΛ nλ[λ] ≤ ∑ λ∈C/kΛ mλ[λ] if, for all λ ∈ C/kΛ, nλ ≤ mλ. We also introduce the weight ωk(f) of f defined by ωk(f) := ∑ λ∈C/kΛ ordλ(f)λ ∈ C/kΛ and its degree degk(f) given by degk(f) := ∑ λ∈C/kΛ ordλ(f) ∈ Z. If f = c ∏ ξ∈C θk(z − ξ)nξ ∈ Θquot k , then divk(f) = ∑ ξ∈C nξ[ξ]k, ωk(f) = ∑ ξ∈C nξξ mod kΛ and degk(f) = ∑ ξ∈C nξ. The interest of Θquot k in our context is given by the following classical result. Proposition 3.2. We have M×kΛ ⊂ Θquot k . 8 T. Dreyfus and J. Roques Proof. This inclusion means that any kΛ-periodic meromorphic function can be written, up to some multiplicative constant in C×, as a quotient of product of functions of the form θk(z − ξ). This is classical, see [22, Chapter I, Section 6]. � We now state a couple of lemmas, which will be used freely in the rest of the paper. Lemma 3.3. Any f = c ∏ ξ∈C θk(z − ξ)nξ ∈ Θquot k is k-periodic and satisfies f(z + kτ) = (−1)degk(f)e2iπωe−2iπ degk(f)z/kf(z), (3.2) where ω = ∑ ξ∈C nξξ is a representative of ωk(f)2. Conversely, any non zero k-periodic meromor- phic function f over C such that f(z + kτ) = ce−2iπnz/kf(z), (3.3) for some c ∈ C× and n ∈ Z, belongs to Θquot k . Proof. The fact that any f ∈ Θquot k is k-periodic and satisfies the functional equation (3.2) follows from the fact that θk is k-periodic and satisfies the functional equation (3.1). Conversely, consider a non zero k-periodic meromorphic function f over C satisfying an equation of the form (3.3). Using the functional equation (3.1), we see that the k-periodic meromorphic function g(z) = f(z)θk(z−ξ) θk(z)nθk(z) , where ξ ∈ C is such that e−2iπξ/k = (−1)nc, satisfies g(z + kτ) = g(z). So g belongs to M×kΛ ⊂ Θquot k , whence the result. � Lemma 3.4. If f ∈ Θk is such that degk(f) = 0 then f is constant. Proof. Consider f ∈ Θk. There exists c ∈ C× and (nξ)ξ∈C ∈ N(C) with finite support such that f(z) = c ∏ ξ∈C θk(z − ξ)nξ . Then, degk(f) = ∑ ξ∈C nξ is equal to 0 by hypothesis. Thus, for all ξ ∈ C, nξ = 0 and hence f = c is constant. � 3.2 Weierstrass ℘-function For details on what follows, we refer to [33, Chapter VI]. Recall that ℘(z) := 1 z2 + ∑ λ∈Λ\{0} 1 (z + λ)2 − 1 λ2 ∈ MΛ denotes the Weierstrass elliptic function associated to the lattice Λ. For any integer k ≥ 1, we denote by ℘k ∈ MkΛ the Weierstrass function defined by ℘k(z) := ℘(z/k) ∈ MkΛ . This kΛ-periodic meromorphic function is an even function, its poles are of order two and its set of poles is kΛ. Therefore, its derivative ℘′k is an odd function, its poles are of order three and its set of poles is kΛ. Any kΛ-periodic elliptic function is a rational function in ℘k and ℘′k, that is MkΛ = C(℘k, ℘ ′ k). 2It follows from this formula that f belongs to MkΛ if and only if degk(f) = ∑ ξ∈C nξ = 0 and ω = ∑ ξ∈C nξξ ∈ Z. Galois Groups of Difference Equations of Order Two on Elliptic Curves 9 Lemma 3.5. Assume that f ∈ MkΛ, seen has a meromorphic function over C/kΛ, has at most N poles counted with multiplicities (or, equivalently, that f = p/q with p, q ∈ Θk such that degk p,degk q ≤ N). Then, there exist A = P/Q and B = R/S with P,Q ∈ C[X] of degree at most 2N and R,S ∈ C[X] of degree at most 2N + 3 such that f = A(℘k) + ℘′kB(℘k). Proof. Using the fact that f(z) belongs to MkΛ if and only if f(kz) belongs to MΛ, it is easily seen that it is sufficient to prove the lemma for k = 1. In what follows, we see the Λ- periodic elliptic functions as meromorphic functions on C/Λ. Let A,B ∈ C(X) be such that f = A(℘) + ℘′B(℘). It follows from the formula A(℘(z)) = f(z) + f(−z) 2 that A(℘) has at most 2N poles counted with multiplicities in C/Λ. But, if A = P/Q with gcd(P,Q) = 1 then A(℘) has at least degQ poles counted with multiplicities in C/Λ (namely, the zeroes of Q(℘)). So degQ ≤ 2N . Using the fact that elliptic functions have the same numbers of zeroes and poles, the same argument applied to 1/A(℘) shows that degP ≤ 2N . Using the formula B(℘(z)) = f(z)− f(−z) 2℘′(z) , similar arguments show that degR ≤ 2N + 3 and degS ≤ 2N + 3. � 4 Irreducibility of the difference Galois group We let φ2(y) + aφ(y) + by = 0 with a ∈ MΛ and b ∈ M×Λ (4.1) be a difference equation of order 2 with coefficients in MΛ and we denote by φY = AY with A = ( 0 1 −b −a ) ∈ GL2(MΛ) the associated difference system. For the notations MΛ, φ, K, etc, we refer to Sections 2 and 3. We let G ⊂ GL2(C) be the difference Galois group over (K,φ) of equation (4.1). According to Corollary 2.7, G is an algebraic subgroup of GL2(C) such that the quotient G/G◦ of G by its identity component G◦ is cyclic. A direct inspection of the classification, up to conjugation, of the algebraic subgroups of GL2(C) given in [23, Theorem 4] shows that G satisfies one of the following properties: • The group G is reducible (i.e., conjugate to some subgroup of the group of upper-triangular matrices in GL2(C)). If G is reducible, we distinguish the following sub-cases: – the group G is completely reducible (i.e., is conjugate to some subgroup of the group of diagonal matrices in GL2(C)); – the group G is not completely reducible. • The group G is irreducible (i.e., not reducible) and imprimitive (see Section 5 for the definition). 10 T. Dreyfus and J. Roques • The groupG is irreducible and is not imprimitive, and, in this case, there exists an algebraic subgroup µ of C× such that G = µSL2(C). Therefore, G = {M ∈ GL2(C) | det(M) ∈ H} where H = det(G) ⊂ C×. In order to determine H, one can use the fact that H = det(G) is the difference Galois group of φy = (detA)y = by (this follows for instance from Tannakian duality [35, Section 1.4]). Our first task, undertaken in the present section, is to study the reducibility of G. The imprimitivity of G will be considered in Section 5. 4.1 Riccati equation and irreducibility The non linear difference equation (φ(u) + a)u = −b (4.2) is called the Riccati equation associated to equation (4.1). A straightforward calculation shows that u is a solution of this equation if and only if φ− u is a right factor of φ2 + aφ+ b, whence its link with irreducibility. In what follows, we denote by I2 the identity matrix of GL2(C). Lemma 4.1. The following statements hold: 1. If (4.2) has one and only one solution in K then G is reducible but not completely reducible. 2. If (4.2) has exactly two solutions in K then G is completely reducible but not an algebraic subgroup of C×I2. 3. If (4.2) has at least three solutions in K then it has infinitely many solutions in K and G is an algebraic subgroup of C×I2. 4. If none of the previous cases occur then G is irreducible. Proof. The proof of this lemma is identical to that of [19, Theorem 4.2], to whom we refer for more details. (1) We assume that (4.2) has one and only one solution u ∈ K. A straightforward calculation shows that φ(T )AT−1 = ( u ∗ 0 b/u ) for T := ( 1− u 1 −u 1 ) ∈ GL2(K). We deduce from this and from Corollary 2.7 that G is reducible. Moreover, if G was completely reducible then, in virtue of Corollary 2.7, φ(T )AT−1 would be diagonal for some T := (ti,j)1≤i,j≤2 ∈ GL2(K). Equating the entries of the antidiagonal of φ(T )AT−1 with 0, we find that − t21 t22 ,− t11 t12 ∈ K are solutions of the Riccati equation. Since det(T ) 6= 0, these solutions are distinct, whence a contradiction. (2) Assume that (4.2) has exactly two solutions u1, u2 ∈ K. We have φ(T )AT−1 = ( u1 0 0 u2 ) for T := 1 u1 − u2 ( −u2 1 −u1 1 ) ∈ GL2(K). We deduce from this and from Corollary 2.7 that G is completely reducible. Moreover, if G was an algebraic subgroup of C×I2 then, according to Corollary 2.7, there would exist u ∈ K and T = (ti,j)1≤i,j≤2 ∈ GL2(K) such that φ(T )AT−1 = uI2. Galois Groups of Difference Equations of Order Two on Elliptic Curves 11 This equality implies that t21 and t22 are non zero and that, for all c, d ∈ C with ct2,2 +dt1,2 6= 0, −ct21 + dt11 ct22 + dt12 ∈ K is a solution of (4.2). It is easily seen that we get in this way infinitely many solutions of the Riccati equation, this is a contradiction. (3) Assume that (4.2) has at least three solutions u1, u2, u3 ∈ K. The proof of assertion (2) of the present lemma shows that φY = AY is isomorphic over K to φY = ( ui 0 0 uj ) Y for all 1 ≤ i < j ≤ 3. Therefore, there exists T ∈ GL2(K) such that φ(T ) ( u1 0 0 u2 ) = ( u1 0 0 u3 ) T. Equating the second columns in this equality, we see that there exists f ∈ K× such that either u1 = φf f u2 or u3 = φf f u2; up to renumbering, one can assume that the former case holds true. It follows that φY = AY is isomorphic over K to φY = (u1I2)Y and, according to Corollary 2.7, G is an algebraic subgroup of C×I2. We have shown during the proof of statement (2) that this implies that the Riccati equation (4.2) has infinitely many solutions in K. (4) Assume that G is reducible. According to Corollary 2.7, there exists T = (ti,j)1≤i,j≤2 ∈ GL2(K) such that φ(T )AT−1 is upper triangular. Then t22 6= 0 and − t21 t22 ∈ K is a solution of the Riccati equation (4.2). This proves claim (4). � In the proof of the previous lemma, we have shown the following result, which we state independently for ease of reference. Lemma 4.2. The following properties are equivalent: • The Riccati equation (4.2) has at least three solutions in K. • The Riccati equation (4.2) has infinitely many solutions in K. • The difference Galois group G is a subgroup of C×I2. • There exist u ∈ K× and T ∈ GL2(K) such that φ(T )AT−1 = uI2. We shall now state and prove one more lemma. Lemma 4.3. Let Λ′′ ⊂ Λ′ be sublattices of Λ such that the quotient Λ′/Λ′′ is cyclic. Assume that there exist u ∈ M×Λ′′ and T ∈ GL2(MΛ′′) such that φ(T )AT−1 = uI2. (4.3) Then, the Riccati equation (4.2) has at least two distinct solutions in MΛ′. Proof. The Galois extension MΛ′′ |MΛ′ is cyclic of order k := [MΛ′′ : MΛ′ ]. Its Galois group Gal(MΛ′′ |MΛ′) is generated by the field automorphism σ1 given by σ1(f(z)) = f(z+λ′), where λ′ ∈ Λ′ is a representative of a generator of Λ′/Λ′′. Note that the action of Gal(MΛ′′ |MΛ′) on MΛ′′ commutes with the action of φ. Applying σ1 to equation (4.3), we get φ(σ1(T ))Aσ1(T )−1 = σ1(u)I2, 12 T. Dreyfus and J. Roques so φ(S)u = σ1(u)S, with S := σ1(T )T−1 ∈ GL2(MΛ′′). It follows that there exists gσ1 ∈ M×Λ′′ (namely, one of the non zero entries of S) such that σ1(u) = φ(gσ1) gσ1 u. Consider the norm N := NMΛ′′ |MΛ′ (gσ1) = ∏ σ∈Gal(MΛ′′ |MΛ′ ) σ(gσ1) ∈ M×Λ′ . We have φ(N) = ∏ σ∈Gal(MΛ′′ |MΛ′ ) σ ( σ1(u)gσ1 u ) = ∏ σ∈Gal(MΛ′′ |MΛ′ ) σ(gσ1) = N, so N = c ∈ (Kφ)× = C×. Up to replacing gσ1 by gσ1c −1/k, we may assume that NMΛ′′ |MΛ′ (gσ1) = 1. Hilbert’s 90 theorem [32, Section X.1] ensures that there exists m ∈ M×Λ′′ such that gσ1 = m σ1(m) . For any σ = σj1 ∈ Gal(MΛ′′ |MΛ′), we set gσ := gσ1σ1(gσ1) · · ·σj−1 1 (gσ1) = m/σ(m) ∈ M×Λ′′ ; we have σ(u) = φ(gσ) gσ u. It follows that ũ := φ(m) m u is invariant under the action of Gal(MΛ′′ |MΛ′) and hence belongs to M×Λ′ . We have φ ( T ′ ) A ( T ′ )−1 = ũI2, with T ′ := mT ∈ GL2(MΛ′′). Applying σ ∈ Gal(MΛ′′ |MΛ′) to this equality, we get φ ( σ(T ′) ) A ( σ(T ′) )−1 = ũI2. It follows that the matrix Cσ := T ′σ(T ′)−1 ∈ GL2(MΛ′′) satisfies φ(Cσ) = Cσ and, hence, that its entries belong to Kφ = C. Moreover, σ 7→ Cσ is a 1-cocyle for the natural action of Gal(MΛ′′ |MΛ′) on GL2(C) but this action is trivial so σ 7→ Cσ is a group morphism from Gal(MΛ′′ |MΛ′) to GL2(C). Since Gal(MΛ′′ |MΛ′) is cyclic, Galois Groups of Difference Equations of Order Two on Elliptic Curves 13 this implies that there exists P ∈ GL2(C) such that, for all σ ∈ Gal(MΛ′′ |MΛ′), the matrix Dσ := P−1C−1 σ P ∈ GL2(C) is diagonal. We have, for all σ ∈ Gal(MΛ′′ |MΛ′), σ(T ′′) = DσT ′′, where T ′′ = (t′′i,j)1≤i,j≤2 := P−1T ′ ∈ GL2(MkΛ). It follows that u1 := −t′′11 t′′12 and v1 := −t′′21 t′′22 are invariant by the action of Gal(MΛ′′ |MΛ′) and hence belong to MΛ′ . But u1 and v1 are solutions of the Riccati equation (4.2) (this was already used in the proof of assertion (2) of Lemma 4.1). So u1 and v1 are solutions in MΛ′ of the Riccati equation (4.2). � We now come to the main result of this subsection. Theorem 4.4. The following statements hold: 1. The Galois group G is reducible if and only if the Riccati equation (4.2) has at least one solution in M2Λ. 2. The Galois group G is completely reducible if and only if the Riccati equation (4.2) has at least two solutions in M2Λ. Proof. In virtue of Lemma 4.1, it is sufficient to prove that: (a) If the Riccati equation (4.2) has a unique solution in K, then it belongs to MΛ. (b) If the Riccati equation (4.2) has exactly two solutions in K, then they belong to M2Λ. (c) If the Riccati equation (4.2) has at least three solutions in K, then the Riccati equa- tion (4.2) has at least two solutions in M2Λ. (a) Assume that the Riccati equation (4.2) has a unique solution u in K. Since u(z), u(z+1) and u(z + τ) are solutions of (4.2), we get u(z) = u(z + 1) = u(z + τ) and hence u ∈ MΛ. (b) Assume that the Riccati equation (4.2) has exactly two solutions in K and let u ∈ K be one of these solutions. Since u(z), u(z + 1) and u(z + 2) are solutions of (4.2), we get u(z + 2) = u(z). Similarly, we have u(z + 2τ) = u(z). So u ∈ M2Λ. (c) What follows is inspired by [19, Theorem 4.2], but is a little bit subtler. Assume that the Riccati equation (4.2) has at least three solutions in K. According to Lemma 4.2, there exist u ∈ K and T = (ti,j)1≤i,j≤2 ∈ GL2(K) such that φ(T )AT−1 = uI2. (4.4) Let k ∈ N∗ be such that the entries of T and u belong to MkΛ. Consider the following field extensions: MΛ ⊂ L ⊂ MkΛ, with L := MZ+kτZ . Applying Lemma 4.3 to the extension MkΛ |L and to the equation (4.4), we get that the Riccati equation (4.2) has two distinct solution u1 and v1 in L. If both of them belong to M2Λ then the proof is completed. Otherwise, up to renumbering, we can assume that u1 6∈ M2Λ, i.e., that u1 is not 2τ -periodic. Then u1(z), u2(z) := u1(z + τ) and u3(z) := u1(z + 2τ) 14 T. Dreyfus and J. Roques are distinct solutions in L of the Riccati equation. For all integers i, j ∈ {1, 2, 3} with i < j we set Ti,j := 1 ui−uj ( −uj 1 −ui 1 ) ∈ GL2(L) and we have φ(Ti,j)A(Ti,j) −1 = ( ui 0 0 uj ) (this was already used in the proof of assertion (2) of Lemma 4.1). Therefore, φ ( T1,3(T1,2)−1 )(u1 0 0 u2 ) = ( u1 0 0 u3 ) T1,3(T1,2)−1. Equating the second columns in this equality, we see that there exists f ∈ L× such that either u1 = φf f u2 or u3 = φf f u2; up to renumbering, we may assume that the former equality holds true. Then, we have φ ( T̃ ) AT̃−1 = u1I2 (4.5) with u1 ∈ L× and T̃ := ( 1 0 0 f ) T1,2 ∈ GL2(L). Applying Lemma 4.3 to the extension L|MΛ and to the equation (4.5), we see that the Riccati equation (4.2) has 2 distinct solutions in MΛ. This concludes the proof. � 4.2 On the solutions of the Riccati equation We refer to Section 3.1 for the notations (divk, degk, ωk, etc.) used in this subsection. Let k ≥ 1 be an integer. Consider p1 ∈ Θk ∪ {0} and p2, p3 ∈ Θk such that a = p1 p3 and b = p2 p3 . We let u ∈ MkΛ be a potential solution of the Riccati equation (4.2). Proposition 4.5. We have u = φ(r) r p q for some p, q, r ∈ Θk such that (i) divk(p) ≤ divk(p2), (ii) divk(q) ≤ divk(φ −1(p3)), (iii) degk(p) = degk(q), (iv) ωk(p/q) = degk(r)h mod kΛ. Proof. In what follows, the greatest common divisors (gcd) has to be understood in the ring O(C) of entire functions3. Let p4, p5 ∈ Θk, with gcd(p4, p5) = 1, be such that u = p4/p5. Let r ∈ Θk be a greatest common divisor of φ−1(p4) and p5 and consider p := p4 φ(r) ∈ Θk and q := p5 r ∈ Θk. 3According to [17], any finitely generated ideal of O(C) is principal, whence the existence of the greatest common divisor of any couple of elements of O(C). Such a greatest common divisor is unique up to multiplication by an unit of O(C). Galois Groups of Difference Equations of Order Two on Elliptic Curves 15 By construction, we have u = φr r p q with gcd(p, φ(q)) = gcd(φ(r)p, rq) = 1. Then, the Riccati equation (4.2) becomes p3 φr r p q φ ( φr r p q ) + p1 φr r p q = −p2, i.e., p3φ 2(r)pφ(p) + p1φ(r)pφ(q) = −p2rqφ(q). It is now easily seen that p divides p2 and that q divides φ−1(p3) in O(C). In terms of divisors, this is exactly (i) and (ii). According to Lemma 3.3, we have p q (z + kτ) = (−1)degk(p/q)e2iπω/ke−2iπ degk(p/q)z/k p q (z) for some representative ω of ωk(p/q), and φ(r) r (z + kτ) = e−2iπ degk(r)h/kφ(r) r (z). Therefore u(z + kτ) = (−1)degk(p/q)e2iπω/ke−2iπ degk(p/q)z/ke−2iπ degk(r)h/ku(z). But u ∈ MkΛ, so u(z + kτ) = u(z) and, hence, (−1)degk(p/q)e2iπω/ke−2iπ degk(p/q)z/ke−2iπ degk(r)h/k = 1. Hence degk(p/q) = 0 and ω = degk(r)h mod kΛ. This proves (iii) and (iv). � We will see in Section 6.1 that Proposition 4.5 is a useful theoretic tool in order to deter- mine the difference Galois groups of families of equations, such as the discrete Lamé equations mentioned in the introduction. We shall now conclude this section with a few words about Proposition 4.5. Remark 4.6. How to use Proposition 4.5 in order to decide whether G is irreducible? Theo- rem 4.4 ensures that G is irreducible if and only if the Riccati equation (4.2) has a solution u ∈ M2Λ; we let p, q, r be as in Proposition 4.5. Assertions (i) and (ii) of Proposition 4.5, show that there are finitely many explicit possibilities for the divisors div2(p) and div2(q). But deg2(r) is entirely determined by these divisors in virtue of (iv) of Proposition 4.5. So, we can compute an integer N ≥ 0 such that if the Riccati equation (4.2) has a solution u ∈ M2Λ, then u = p0/q0 with p0, q0 ∈ Θ2 such that deg2(p0) ≤ N and deg2(q0) ≤ N . Lemma 3.5 ensures that u = A(℘2) + ℘′2B(℘2) for some A = P/Q and B = R/S with P,Q ∈ C[X] of degree at most 2N and R,S ∈ C[X] of degree at most 2N + 3. 16 T. Dreyfus and J. Roques So, in order to determine whether or not the Riccati equation (4.2) has at least one solution in M2Λ, we are lead to the following question: do there exist A = P/Q and B = R/S with P,Q ∈ C[X] of degree at most 2N and R,S ∈ C[X] of degree at most 2N + 3 such that u = A(℘2) +℘′2B(℘2) is a solution of the Riccati equation (4.2)? Substituting u = A(℘2) +℘′2B(℘2) in the Riccati equation (4.2) and using the addition formula: ℘2(z) + ℘2(h) + ℘2(z + h) = 1 4 ( ℘′2(z)− ℘′2(h) ℘2(z)− ℘2(h) )2 , we are lead to decide whether multivariate polynomials, whose indeterminates are the coefficients of P , Q, R and S, have a common complex solution. This can be decided by using Gröbner bases. Note however that, in order to make this method an effective tool, we have to know the divisors of a and b, and to be able to deduce degk(r) from assertion (iv) of Proposition 4.5. 5 Imprimitivity of the difference Galois group We want to determine whether G is imprimitive, that is whether G is conjugate to a subgroup of {( α 0 0 β ) |α, β ∈ C× }⋃{( 0 γ δ 0 ) | γ, δ ∈ C× } . Theorem 5.1. Assume that G is irreducible and that a 6= 0. Then, G is imprimitive if and only if there exists u ∈ M2Λ such that( φ2(u) + ( φ2 ( b a ) − φ(a) + φ(b) a )) u = −φ(b)b a2 . (5.1) Proof. Arguing exactly as in [19, Theorem 4.6], we get that G is imprimitive if and only if equation (5.1) has a solution in K. But this is a Riccati-type equation, with φ replaced by φ2. Therefore, the assertions (a), (b) and (c) given at the beginning of the proof of Theorem 4.4 allow us to conclude. � Remark 5.2. If a = 0 then G is imprimitive in virtue of Corollary 2.7. Note that Proposition 4.5 can be used in order to find restrictions on the solutions of the above Riccati-type equation, but with φ replaced by φ2. 6 Applications We recall that h ∈ C is such that h mod Λ is not a torsion point of C/Λ, i.e., that the corre- sponding point h of E(C) is not a torsion point. 6.1 A discrete version of Lamé equation Let us consider the difference equation ∆2 hy = (A℘(z) +B)y, where ∆hy(z) = y(z + h)− y(z) h (6.1) and A,B ∈ C. This is a discrete version of the so-called Lamé differential equation y′′(z) = (A℘(z) +B)y(z). Galois Groups of Difference Equations of Order Two on Elliptic Curves 17 Theorem 6.1. Assume that E is defined over Q (i.e., g2, g3 ∈ Q) and that h,A,B ∈ Q with A 6= 0. Then, the difference Galois group over (K,φ) of equation (6.1) is GL2(C). A straightforward calculation shows that equation (6.1) can be rewritten as follows: φ2y − 2φy + ( −Ah2℘(z)−Bh2 + 1 ) y = 0. We will deduce Theorem 6.1 from the following theorem combined with a transcendence result due to Schneider. Theorem 6.2. Consider a ∈ C× and b(z) = α℘(z) + β with (α, β) ∈ C× × C. Let z0 ∈ C be such that ℘(z0) = −β/α.4 If Zh∩(`z0 +Λ) = {0} for all ` ∈ {−8, . . . , 8} (this holds in particular if Zh ∩ (Zz0 + Λ) = {0}) then the difference Galois group over (K,φ) of φ2y + aφy + by = 0 is GL2(C). Proof of Theorem 6.2. For the notations, divk, [·]k, etc, we refer to Section 3.1. Note that div1(b) = [z0]1 + [−z0]1 − 2[0]1. So, we can write a = p1 p3 and b = p2 p3 for some p1, p2, p3 ∈ Θ1 with div1(p2) = [z0]1 + [−z0]1 and div1(p3) = 2[0]1. We claim that G is irreducible, i.e., in virtue of Theorem 4.4, that the Riccati equation (φ(u) + a)u = −b (6.2) does not have any solution in M2Λ. Suppose to the contrary that it has a solution u ∈ M2Λ. Proposition 4.5 ensures that there exist p, q, r ∈ Θ2 such that u = φ(r) r p q and (i) div2(p) ≤ ∑ `1,`2∈{0,1} [`1 + `2τ − z0]2 + [`1 + `2τ + z0]2, (ii) div2(q) ≤ ∑ `1,`2∈{0,1} 2[`1 + `2τ + h]2, (iii) deg2(p) = deg2(q), (iv) ω2(p/q) = deg2(r)h mod 2Λ. Properties (i) and (ii) above imply that ω2 (p/q) = `z0 − deg2(q)h mod Λ for some ` ∈ {−4, . . . , 4}. We infer from this and from (iv) that (deg2(r) + deg2(q))h = `z0 mod Λ. 4Any non constant elliptic function f(z) has at least one zero (otherwise, 1/f(z) would be an entire elliptic function and hence would be constant). In particular, ℘(z) + β/α has a least one zero in C. 18 T. Dreyfus and J. Roques The assumption on z0 ensures that deg2(r) = deg2(q) = 0. It follows from (iii) that deg2(p) = 0 and hence u is a constant. But it is easily seen that equation (6.2) does not have any constant solution; this proves our claim. We claim that G is not imprimitive, i.e., in virtue of Theorem 5.1, that( φ2(u) + φ2(b) a − a+ φ(b) a ) u = −φ(b)b a2 (6.3) does not have any solution in M2Λ. Suppose to the contrary that it has a solution u ∈ M2Λ. Equation (6.3) is of the form: u ( φ2(u) + p1 p3 ) = p2 p3 , for some p1, p2, p3 ∈ Θ1 with div1(p2) = 2[−2h]1 + [z0]1 + [−z0]1 + [z0 − h]1 + [−z0 − h]1 and div1(p3) = 2[−2h]1 + 2[−h]1 + 2[0]1. We apply Proposition 4.5 with φ replaced by φ2 to obtain the existence of p, q, r ∈ Θ2 such that u = φ2(r) r p q , where (v) div2(p) ≤ ∑ `1,`2∈{0,1} 2[`1 + `2τ − 2h]2 + [`1 + `2τ + z0]2 + [`1 + `2τ − z0]2 + [`1 + `2τ + z0 − h]2 + [`1 + `2τ − z0 − h]2, (vi) div2(q) ≤ ∑ `1,`2∈{0,1} 2[`1 + `2τ ]2 + 2[`1 + `2τ + h]2 + 2[`1 + `2τ + 2h]2, (vii) deg2(p) = deg2(q), (viii) ω2(p/q) = 2 deg2(r)h mod 2Λ. We claim that (v′) div2(p) ≤ ∑ `1,`2∈{0,1} [`1 + `2τ + z0]2 + [`1 + `2τ − z0]2, (vi′) div2(q) ≤ ∑ `1,`2∈{0,1} 2[`1 + `2τ ]2. Indeed, otherwise, arguing as for the proof of the irreducibility of G, we see that (v), (vi) and (viii) would lead to a relation of the form (2 deg2(r) + d)h = `z0 mod Λ for some integer ` ∈ {−8, . . . , 8} and some integer d > 0 and this would contradict our assump- tion on z0. Then, (viii) shows that 2 deg2(r)h = `z0 mod Λ Galois Groups of Difference Equations of Order Two on Elliptic Curves 19 for some integer ` ∈ {−4, . . . , 4} and hence deg2(r) = 0. Therefore, u = p/q with p, q ∈ Θ2 satisfying (v′) and (vi′) above. Now remark that φ2(u) + φ2(b) a − a+ φ(b) a does not have poles in Λ. But any element of Λ is a pole of order 2 of the right hand side of equation (6.3), so any element of Λ is a pole of order at least 2 of u. It follows that (vi′) is an equality. Then, using (vii), we see that (v′) is also an equality. So div2(u) = div2(b) and hence u = cb for some c ∈ C×. We now plug u = cb into equa- tion (6.3) and we get: c (( c+ 1 a ) φ2(b)− a+ φ(b) a ) = −φ(b) a2 . Since −2h is a pole of φ2(b) but not of φ(b), we get c = −1/a and the above equation simplifies as follows: −1 a ( −a+ φ(b) a ) = −φ(b) a2 . This gives 1 = 0, whence a contradiction. Therefore, G is irreducible and not imprimitive. So, as explained at the beginning of Section 4, G = {M ∈ GL2(C) | det(M) ∈ H} where H ⊂ C× is the Galois group of φy = by, which is easily seen to be the multiplicative group (C×, ·). This concludes the proof. � Proof of Theorem 6.1. In virtue of Theorem 6.2, it is sufficient to prove that Zh∩(Zz0+Λ) = {0}. Consider m1,m2 ∈ Z and λ ∈ Λ such that m1h = m2z0 +λ. We have ℘(z0) = −Bh2+1 Ah2 ∈ Q. It follows that either m2z0 + λ ∈ Λ or ℘(m2z0 + λ) ∈ Q. (Indeed, suppose that m2z0 + λ 6∈ Λ. Using equation (1.1), we see that ℘′(z0) ∈ Q. Therefore, ϕ(z0) belongs to E(Q), the map ϕ being defined in the introduction. Using the fact that ϕ is a group morphism and that E(Q) is a subgroup of E(C), we get ϕ(mz0) ∈ E(Q). Therefore, ℘(mz0 + λ) = ℘(mz0) ∈ Q.) In the former case, we get m1h ∈ Λ and hence m1 = 0. In the later case, it follows from the work of Schneider [31] (for a reference in english, see Baker’s book [2, Theorem 6.2]; see also Bertrand and Masser’s papers [3, 21]) that m2z0 + λ and hence m1h are transcendental numbers, which is excluded. � 6.2 A family of examples with Galois groups between SL2(C) and GL2(C) Theorem 6.3. Let us consider b ∈ C×, and a(z) := α℘(z) + β with (α, β) ∈ C× × C. Let z0 ∈ C be such that ℘(z0) = −β/α. If Zh ∩ (`z0 + Λ) = {0} for all ` ∈ {−16, . . . , 16} (this holds in particular if Zh ∩ (Zz0 + Λ) = {0}) then the difference Galois group over (K,φ) of φ2y+ aφy+ by = 0 is µ2k SL2(C) if b is a primitive kth root of the unity and GL2(C) otherwise, where µ2k is the group of complex kth roots of the unity. The proof will be given after the following corollary. Corollary 6.4. Assume that E is defined over Q (i.e., g2, g3 ∈ Q). Consider b ∈ Q× and a(z) := α℘(z) + β with α, β ∈ Q and α 6= 0. Then, the difference Galois group over (K,φ) of φ2y+ aφy+ by = 0 is µ2k SL2(C) if b is a primitive kth root of the unity and GL2(C) otherwise. Proof. Similar to deduction of Theorem 6.1 from Theorem 6.2. � 20 T. Dreyfus and J. Roques Proof of Theorem 6.3. Note that div1(a) = [z0]1 + [−z0]1 − 2[0]1. So, we can write a = p1 p3 and b = p2 p3 for some p1, p2, p3 ∈ Θ1 with div1(p2) = 2[0]1 and div1(p3) = 2[0]1. We claim that G is irreducible, i.e., in virtue of Theorem 4.4, that the Riccati equation (φ(u) + a)u = −b (6.4) does not have any solution in M2Λ. Suppose to the contrary that it has a solution u ∈ M2Λ. Proposition 4.5 ensures that there exist p, q, r ∈ Θ2 such that u = φ(r) r p q and (i) div2(p) ≤ ∑ `1,`2∈{0,1} 2[`1 + `2τ ]2, (ii) div2(q) ≤ ∑ `1,`2∈{0,1} 2[`1 + `2τ + h]2, (iii) deg2(p) = deg2(q), (iv) ω2(p/q) = deg2(r)h mod 2Λ. Properties (i) and (ii) above imply that ω2 (p/q) = −hdeg2(q) mod Λ. We infer from this and from (iv) that (deg2(r) + deg2(q))h = 0 mod Λ. This yields deg2(r) = deg2(q) = 0. It follows from (iii) that deg2(p) = 0 and hence u is a constant. But it is easily seen that equation (6.4) does not have any constant solution; this proves our claim. We claim that G is not imprimitive, i.e., in virtue of Theorem 5.1, that (we recall that b is constant)( φ2(u) + b φ2(a) − φ(a) + b a ) u = − b 2 a2 (6.5) does not have any solution in M2Λ. Suppose to the contrary that it has a solution u ∈ M2Λ. Equation (6.5) is of the form: u ( φ2(u) + p1 p3 ) = p2 p3 , for some p1, p2, p3 ∈ Θ1 with div1(p2) = 4[0]1 + 2[−h]1 + [z0 − 2h]1 + [−z0 − 2h]1 Galois Groups of Difference Equations of Order Two on Elliptic Curves 21 and div1(p3) = 2[−h]1 + 2[z0]1 + 2[−z0]1 + [z0 − 2h]1 + [−z0 − 2h]1. Proposition 4.5 ensures that there exist p, q, r ∈ Θ2 such that u = φ2(r) r p q , and (v) div2(p) ≤ ∑ `1,`2∈{0,1} 4[`1 + `2τ ]2 + 2[`1 + `2τ − h]2 + [`1 + `2τ + z0 − 2h]2 + [`1 + `2τ − z0 − 2h]2, (vi) div2(q) ≤ ∑ `1,`2∈{0,1} 2[`1 + `2τ + h]2 + 2[`1 + `2τ + z0 + 2h]2 + 2[`1 + `2τ − z0 + 2h]2 + [`1 + `2τ + z0]2 + [`1 + `2τ − z0]2, (vii) deg2(p) = deg2(q), (viii) ω2(p/q) = 2 deg2(r)h mod 2Λ. We claim that (v′) div2(p) ≤ ∑ `1,`2∈{0,1} 4[`1 + `2τ ]2, (vi′) div2(q) ≤ ∑ `1,`2∈{0,1} [`1 + `2τ + z0]2 + [`1 + `2τ − z0]2. Otherwise, arguing as for the proof of the irreducibility of G, we see that (v), (vi) and (viii) would lead to a relation of the form (2 deg2(r) + d)h = `z0 mod Λ for some integer ` ∈ {−16, . . . , 16} and some integer d > 0 and this would contradict our assumption on z0. Then, (viii) shows that 2 deg2(r)h = `z0 mod Λ for some integer ` ∈ {−4, . . . , 4} and hence deg2(r) = 0. So u = p/q with p, q ∈ Θ2 satisfying (v′) and (vi′) above. In particular, −h is not a zero of u. But −h (which is a pole of φ(a)) is a pole of φ2(u) + b φ2(a) − φ(a) + b a . So −h is a pole of the left hand side of (6.5). This a contradiction because −h is not a pole of the right hand side of (6.5). Therefore, G is irreducible and not imprimitive. So, as explained at the beginning of Section 4, G = {M ∈ GL2(C) | det(M) ∈ H} where H ⊂ C× is the Galois group of φy = by, which is easily seen to be µk if b is a kth root of the unity and C× otherwise. � 22 T. Dreyfus and J. Roques Acknowledgements Our original interest in difference equations on elliptic curves arose from discussions with Jean- Pierre Ramis some years ago. We thank Jean-Pierre Ramis and Michael Singer for interesting discussions. We thank the referees for their careful reading and useful suggestions. The first author is founded by the labex CIMI. The second author is partially funded by the French ANR project QDIFF (ANR-2010-JCJC-010501). 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[35] van der Put M., Singer M.F., Galois theory of difference equations, Lecture Notes in Math., Vol. 1666, Springer-Verlag, Berlin, 1997. http://dx.doi.org/10.1016/S0019-3577(08)80004-3 http://dx.doi.org/10.1016/j.crma.2011.08.021 http://dx.doi.org/10.1093/imrn/rnu060 http://arxiv.org/abs/1304.2649 http://arxiv.org/abs/#2 http://arxiv.org/abs/0711.4034 http://arxiv.org/abs/1207.0107 http://dx.doi.org/10.2140/pjm.2008.235.303 http://dx.doi.org/10.1016/j.ansens.2002.10.001 http://arxiv.org/abs/math.QA/0210221 http://dx.doi.org/10.1007/BF01571618 http://dx.doi.org/10.1007/978-0-387-09494-6 http://dx.doi.org/10.5802/afst.1164 http://arxiv.org/abs/math.QA/0507098 http://dx.doi.org/10.1007/BFb0096118 1 Introduction 2 Difference Galois theory: reminders and complements 2.1 Generalities on difference Galois theory 2.2 Difference equations on elliptic curves 3 Theta functions and Weierstrass -function 3.1 Theta functions 3.2 Weierstrass -function 4 Irreducibility of the difference Galois group 4.1 Riccati equation and irreducibility 4.2 On the solutions of the Riccati equation 5 Imprimitivity of the difference Galois group 6 Applications 6.1 A discrete version of Lamé equation 6.2 A family of examples with Galois groups between SL 2(C) and GL2(C) References
id nasplib_isofts_kiev_ua-123456789-146865
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 1815-0659
language English
last_indexed 2025-12-07T16:27:02Z
publishDate 2015
publisher Інститут математики НАН України
record_format dspace
spelling Dreyfus, T.
Roques, J.
2019-02-11T18:08:38Z
2019-02-11T18:08:38Z
2015
Galois Groups of Difference Equations of Order Two on Elliptic Curves / T. Dreyfus, J. Roques // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 35 назв. — англ.
1815-0659
2010 Mathematics Subject Classification: 39A06; 12H10
DOI:10.3842/SIGMA.2015.003
https://nasplib.isofts.kiev.ua/handle/123456789/146865
This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups. For instance, our results combined with a result from transcendence theory due to Schneider allow us to identify a large class of discrete Lamé equations with difference Galois group GL₂(C).
This paper is a contribution to the Special Issue on Algebraic Methods in Dynamical Systems. The full collection is available at http://www.emis.de/journals/SIGMA/AMDS2014.html. Our original interest in dif ference equations on elliptic curves arose from discussions with JeanPierre Ramis some years ago. We thank Jean-Pierre Ramis and Michael Singer for interesting discussions. We thank the referees for their careful reading and useful suggestions. The first author is founded by the labex CIMI. The second author is partially funded by the French ANR project QDIFF (ANR-2010-JCJC-010501).
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Galois Groups of Difference Equations of Order Two on Elliptic Curves
Article
published earlier
spellingShingle Galois Groups of Difference Equations of Order Two on Elliptic Curves
Dreyfus, T.
Roques, J.
title Galois Groups of Difference Equations of Order Two on Elliptic Curves
title_full Galois Groups of Difference Equations of Order Two on Elliptic Curves
title_fullStr Galois Groups of Difference Equations of Order Two on Elliptic Curves
title_full_unstemmed Galois Groups of Difference Equations of Order Two on Elliptic Curves
title_short Galois Groups of Difference Equations of Order Two on Elliptic Curves
title_sort galois groups of difference equations of order two on elliptic curves
url https://nasplib.isofts.kiev.ua/handle/123456789/146865
work_keys_str_mv AT dreyfust galoisgroupsofdifferenceequationsofordertwoonellipticcurves
AT roquesj galoisgroupsofdifferenceequationsofordertwoonellipticcurves