On the differential transcendence of the elliptic hypergeometric functions
Abstract.
We apply the differential Galois theory for difference equations in order to prove a criterion ensuring that any nonzero solution of a given order two difference equation is differentially transcendental. We then apply our result to the elliptic analogue of the hypergeometric equation.
Key words and phrases:
Linear difference equations, difference Galois theory, elliptic curves, differential algebra2010 Mathematics Subject Classification:
39A06,12H05Contents
Introduction
The elliptic hypergeometric functions form a common analogue of classical hypergeometric functions and hypergeometric functions, which have been a focus of intense study in the last 200 years within the theory of special functions and are ubiquitous in physics and mathematics. The general theory of these elliptic hypergeometric functions was initiated by Spiridonov in [spiridonov2016elliptic] and has been a dynamic field of research, see for instance [van2007hyperbolic, fokko2009basic, magnus2009elliptic, rains2010transformations, rosengren2002elliptic]. In the intervening years a number of remarkable analogues of known properties and applications of classical and hypergeometric functions have been discovered for the elliptic hypergeometric functions; see [spiridonov2016elliptic] for more details.
In this work we develop a criterion to decide differential transcendence for elliptic hypergeometric functions. More precisely, our main result is that for “generic” values of the parameters, in a sense made precise in Section 4, the elliptic hypergeometric functions are differentially transcendental, i.e. they do not satisfy any polynomial differential equations with elliptic function coefficients, see Definition 2.3. Our algebraic proof of this result is based on differential Galois theory for difference equations [HS], which associates a geometric object to such a difference equation, the Galois group, that encodes the polynomial differential equations that may be satisfied by the solutions. There is a Galois correspondence that implies in particular that the larger the group, the fewer the polynomial differential relations that exist among the solutions. As a preliminary result, we prove in Theorem 2.4 a criterion that ensures that the Galois group is large enough to force every nonzero solution to be differentially transcendental. Then we apply Theorem 2.4 to the elliptic hypergeometric function solution of equation (4.2) discovered in [spiridonov2016elliptic] by interpreting the latter as a secondorder linear difference equation over an elliptic curve.
Our strategy here is in the tradition of other applications of differential Galois theory for difference equations of [HS] to questions about shift difference equations [arreche2017computation], difference equations, [dreyfus2016functional], deterministic finite automata and Mahler functions [DHR], lattice walks in the quarter plane [dreyfus2018nature, dreyfus2017differential, dreyfus2017walks], and shift, dilation, and Mahler difference equations in general [arreche2017galois]. In order to apply our criterion in practice, one needs to check that there are no telescoper relations of a certain kind and that a certain Riccati equation has no solutions. In recent years, the algorithmic solution of these two problems has attracted the attention of many researchers independently of the question of differential transcendence, see for example [petkovvsek1992hypergeometric, Hen97, Hen98, roques2018algebraic, dreyfus2015galois] for the Riccati equations, see also [tietze1905funktionalgleichungen, nishioka2018differential], and [Abramovtelesc, Chen_Singer12] for the telescopers. We hope that our results will motivate the development of new algorithms to handle the remaining cases.
The paper is organized as follows. In Section 1, we recall some facts about the difference Galois theory developed in [VdPS97]. To a difference equation is associated an algebraic group. The larger the group, the fewer the algebraic relations that exist among the solutions of the difference equation. In Section 2, we recall some facts about the differential Galois theory for difference equations of [HS]. Here the Galois group is a linear differential algebraic group, that is, a group of matrices defined by a system of algebraic differential equations in the matrix entries. This group encodes the polynomial differential relations among the solutions of the difference equation. In this section we prove a criterion to ensure that every nonzero solution of a given secondorder difference equation is differentially transcendental; see Theorem 2.4. In Section 3 we restrict ourselves to the situation where the coefficients of the difference equation are elliptic functions. We recall some results from [dreyfus2015galois], where the authors explain how to compute the difference Galois group of [VdPS97] for order two equations with elliptic coefficients. This computation was inspired by Hendricks’ algorithm, see [Hen97]. In Section 4, we follow [spiridonov2016elliptic] in defining the elliptic analogue of the hypergeometric equation (4.2) and, under a certain genericity assumption, we prove that its nonzero solutions are differentially transcendental, see Theorem 4.3.
1. Difference Galois theory
For details on what follows, we refer to [VdPS97, Chapter 1]. Unless otherwise stated, all rings are commutative with identity and contain the field of rational numbers. In particular, all fields are of characteristic zero.
A ring (or difference ring) is a ring together with a ring automorphism . If is a field then is called a field. When there is no possibility of confusion the ring will be simply denoted by . There are natural notions of ideals, ring extensions, algebras, morphisms, etc. We refer to [VdPS97, Chapter 1] for the definitions.
The ring of constants of the ring is defined by
We now let be a field. We assume that the field of constants is algebraically closed and that the characteristic of is .
We consider a difference equation of order two with coefficients in :
(1.1) 
and the associated difference system:
(1.2) 
By [VdPS97, 1.1], there exists a ring extension of such that

there exists such that (such a is called a fundamental matrix of solutions of (1.2));

is generated, as a algebra, by the entries of and ;

the only ideals of are and .
Note that the last assumption implies . Such an is called a PicardVessiot ring, or PV ring for short, for (1.2) over . It is unique up to isomorphism of algebras. Note that a PV ring is not always an integral domain, but it is a direct sum of integral domains transitively permuted by .
The corresponding Galois group of (1.2) over , or Galois group for short, is the group of automorphisms of :
A straightforward computation shows that, for any , there exists a unique such that . According to [VdPS97, Theorem 1.13], one can identify with an algebraic subgroup of via the faithful representation
If we choose another fundamental matrix of solutions , we find a conjugate representation. In what follows, by “Galois group of the difference equation (1.1)”, we mean “Galois group of the difference system (1.2)”.
We shall now introduce a property relative to the base field , which appears in [VdPS97, Lemma 1.19].
Definition 1.1.
We say that the field satisfies the property if:

the field is a field ^{1}^{1}1Recall that is a field if every nonconstant homogeneous polynomial over has a nontrivial zero provided that the number of its variables is more than its degree.;

and the only finite field extension of such that extends to a field endomorphism of is .
Example 1.2.
The following are natural examples of difference fields that satisfy property :

S: Shift case with , , . See [Hen97].

Q: difference case. , , , . See [Hen98].

M: Mahler case. , , . See [roques2018algebraic].

E: Elliptic case. See Section 3, and [dreyfus2015galois].
The following result is due to van der Put and Singer. We recall that two difference systems and with are isomorphic over if and only if there exists such that . Note that if and only if .
Theorem 1.3.
Assume that satisfies property . Then the following properties relative to hold:

is cyclic, where is the identity component of ;

there exists such that (1.2) is isomorphic to over .
Let be an algebraic subgroup of such that . The following properties hold:

is conjugate to a subgroup of ;

any minimal element (with respect to inclusion) in the set of algebraic subgroups of for which there exists such that is conjugate to ;

is conjugate to if and only if, for any and for any proper algebraic subgroup of , one has that .
Proof.
The proof of [VdPS97, Propositions 1.20 and 1.21] in the special case where and is the shift with , extends mutatis mutandis to the present case. ∎
This theorem is at the heart of many algorithms to compute Galois groups, see for example [Hen97, Hen98, dreyfus2015galois, roques2018algebraic].
2. Parametrized Difference Galois theory
2.1. General facts
A ring is a ring endowed with a ring automorphism and a derivation (this means that is additive and satisfies the Leibniz rule ) such that . If is a field, then is called a field. When there is no possibility of confusion, we write instead of . There are natural notions of ideals, ring extensions, algebras, morphisms, etc. We refer to [HS, Section 6.2] for the definitions.
If is a field, and if belong to some field extension of , then denotes the algebra generated over by and denotes the field generated over by .
We now let be a field. We assume that the field of constants is algebraically closed and that is of characteristic .
In order to apply the Galois theory developed in [HS], we need to work with a base field such that is closed.^{2}^{2}2The field is called closed if, for every (finite) set of polynomials with coefficients in , if the system of equations has a solution with entries in some field extension , then it has a solution with entries in . Note that when the derivation is trivial, i.e. , then a field is closed if and only if it is algebraically closed. To this end, the following lemma will be useful.
Lemma 2.1 ([Dhr, Lemma 2.3]).
Suppose that is algebraically closed and let be a closure of (the existence of such a is proved in [Kol74]). Then the ring is an integral domain whose fraction field is a field extension of such that .
We still consider the difference equation (1.1) and the associated difference system (1.2). By [HS, 6.2.1], there exists a ring extension of such that

there exists such that ;

is generated, as an algebra, by the entries of and ;

the only ideals of are and .
Such an is called a PicardVessiot ring, or PV ring for short, for (1.2) over . It is unique up to isomorphism of algebras. Note that a PV ring is not always an integral domain, but it is the direct sum of integral domains that are transitively permuted by .
The corresponding Galois group of (1.2) over , or Galois group for short, is the group of automorphisms of :
In what follows, by “Galois group of the difference equation (1.1)”, we mean “Galois group of the difference system (1.2)”.
A straightforward computation shows that, for any , there exists a unique such that . By [HS, Proposition 6.18], the faithful representation
identifies with a linear differential algebraic group , that is, a subgroup of defined by a system of polynomial equations over in the matrix entries. If we choose another fundamental matrix of solutions , we find a conjugate representation.
Let be a PV ring for (1.2) over and let be a fundamental matrix of solutions. Then the algebra generated by the entries of and is a PV ring for (1.2) over . We can (and will) identify with a subgroup of by restricting the elements of to .
Proposition 2.2 ([Hs], Proposition 2.8).
The group is a Zariskidense subgroup of .
2.2. Differential transcendence criteria
The aim of this section is to develop a galoisian criterion for the differential transcendence of the nonzero solutions of (1.1).
Definition 2.3.
Let be a field extension. We say that is differentially algebraic over if there exists such that are algebraically dependent over . Otherwise, we say that is differentially transcendental over .
Recall that be a field satisfying property such that is algebraically closed and such that has characteristic .
Let be a closure of . According to Lemma 2.1, is an integral domain and is a field extension of such that . Let be a PV ring for (1.2) over and let be a PV ring for (1.2) over . We also consider a PV ring for (1.2) over .
Our differential transcendence criterion is the following.
Theorem 2.4.
Assume that is irreducible and that the Galois group of over is . Then any nonzero solution of (1.1) in any field extension of is differentially transcendental over .
Note that the irreducibility of may be tested algorithmically in many contexts, see [Hen97, Hen98, dreyfus2015galois, roques2018algebraic]. More precisely, the group is irreducible if and only if there does not exist satisfying the Riccati equation . The following lemma gives a more tractable version of the second assumption.
Lemma 2.5 (Proposition 2.6, [Dhr]).
The Galois group of over is a proper subgroup of if and only if there exist a nonzero linear differential operator with coefficients in and such that
The following lemma will be used in the proof of Theorem 2.4.
Lemma 2.6.
Proof of Lemma 2.6.
Since any two PV rings for (1.1) over are isomorphic, it is sufficient to prove the lemma for some PV ring, not necessarily for itself. Let be a nonzero differentially algebraic solution of (1.1) in . We consider the localization of at , where are indeterminates over . This ring has a natural structure of algebra such that and is a fundamental matrix of solutions of with coefficients in . If we let be a maximal ideal of , then the quotient is a PV ring for over and the image of in this quotient is differentially algebraic. Let us prove that it is nonzero. Otherwise the image of the fundamental solution in the PV ring would have a zero first column and therefore would not be inversible, leading to a contradiction. This concludes the proof. ∎
Proof of Theorem 2.4.
Assume to the contrary that Equation (1.1) has a nonzero differentially algebraic solution in a field extension of . According to Lemma 2.6, there exists a nonzero differentially algebraic solution of (1.1) in .
By [Hen97, Lemma 4.1] combined with Theorem 1.3, one of the following three cases holds

is reducible.

is irreducible and imprimitive.

contains .
Since is irreducible by assumption, only the last two cases may occur. Then we split our study in two cases depending on whether is imprimitive or not.
Let us first assume that is imprimitive. It follows from Theorem 1.3 and [Hen97, Section 4.3] that (1.1) is equivalent over to
(2.1) 
for some . More precisely, let
be the system associated to (2.1). Then there exists such that . Let . Since if and only if , we obtain that satisfies (2.1) with . Let us prove that is non zero. If , then implies and then is solution of the Riccati equation , which contradicts the irreducibility of by [dreyfus2015galois, Lemma 13].
Since is differentially algebraic over , we have that , and hence also , are differentially algebraic over . By [HS, Proposition 6.26], this implies that the Galois group of (2.1) over is a strict subgroup of . By Lemma 2.5 there exist a nonzero and such that
(2.2) 
Taking the determinant in allows us to deduce the existence of such that , and therefore the Galois groups for and are the same. Consequently, by Lemma 2.5 and the assumption on the Galois group of over , for any nonzero and any , we have . This is in contradiction with (2.2).
Assume now that is not imprimitive, so it contains . By [DHR, Proposition 2.10], we deduce that
Let be as small as possible such that there exists with , and suppose that this has smallest possible total degree . For , let with corresponding matrix . For all , we find
Since is differentially closed, there exists such that and . Since for such a , we have that
and we find that must be homogeneous of degree , for otherwise the total degree would not be minimal. We may further assume that the degree of in is as small as possible. Again since is differentially closed, there exists such that but . But then
for some nonzero homogeneous polynomial of total degree in which the degree of is strictly smaller than . This contradiction concludes the proof. ∎
3. Difference equations over elliptic curves
In this section we will be mainly interested in difference equations
(3.1) 
with , where

denotes the field of meromorphic functions over the elliptic curve for some such that , i.e. the field of meromorphic functions on satisfying ;

is the automorphism of defined by
for some such that and .
Note that this choice ensures that is non cyclic.
3.1. The base field
The difference Galois groups of linear difference equations over elliptic curves have been studied in [dreyfus2015galois]. In loc.cit. the elliptic curves are given by quotients of the form for some lattice . However, in the present work, we are mainly interested in difference equations on elliptic curves given by quotients of the form for some such that . The translation between elliptic curves of the form and elliptic curves of the form is standard, namely by using the fact that if with and then the map induces an isomorphism .
We shall now recall some constructions and results from [dreyfus2015galois], restated in the “ context” via the above identification between and . For we denote by the Riemann surface of , and we let be a coordinate function on each such that for every . We will write and .
We let denote the field of meromorphic functions on satisfying , or equivalently the field of meromorphic functions on the elliptic curve . The power map induces an inclusion of function fields for each . We denote by the field defined by
We endow with the noncyclic field automorphism defined by
(3.2) 
where is such that and , and defines a compatible system of th roots of such that for every (cf. [Hen98, Section 2]). Then is a difference field and we have the following properties.
Proposition 3.1 ([dreyfus2015galois], Proposition 5).
The field of constants of is .
Proposition 3.2 ([dreyfus2015galois], Proposition 6).
The difference field satisfies property (see Definition 1.1).
Remark 3.3.
The field equipped with the automorphism does not satisfy property . This is why we work over instead of .
Corollary 3.4.
The conclusions of Theorem 1.3 are valid for .
3.2. Theta functions
We shall now recall some basic facts and notations about theta functions extracted from [dreyfus2015galois, Section 3] (but stated in the “ context”, see the beginning of the previous section). For the proofs, we refer to [Mumford, Chapter I]. We still consider such that . We consider the infinite product
The theta function defined by
(3.3) 
satisfies
(3.4) 
Let be the set of holomorphic functions on of the form
with and with finite support. We denote by the set of meromorphic functions on that can be written as a quotient of two elements of . We have
We define the divisor of as the following formal sum of points of :
where is the adic valuation of , for an arbitrary (it follows from (3.4) that this valuation does not depend on the chosen ). For any and any , we set
Moreover, we will write
if for all . We also introduce the weight of defined by
and its degree given by
Example 3.5.
Consider defined above. Then it follows from (3.3) that , since has a zero of multiplicity one at each point of the subgroup . However, since , we have that
where denotes a primitive th root of unity and is the th power of an arbitrary choice of th root of .
Similarly, for any we have that , where denotes the power map and denotes the induced pullback map on divisors.
3.3. Irreducibility of the Galois groups
One of the assumptions of Theorem 2.4 concerns the irreducibility of the Galois group. The main tool used in this paper in order to study the irreducibility of the Galois group of (3.1) over is the following result.
Theorem 3.6 (Proposition 17 in [dreyfus2015galois]).
4. Application to the elliptic hypergeometric functions
4.1. The elliptic hypergeometric functions
We shall now introduce the elliptic hypergeometric functions following [spiridonov2016elliptic]. Consider such that , , and . Consider
We have
For satisfying the balancing condition , we set
where denotes the positively oriented unit circle and . For , we follow [spiridonov2016elliptic] by setting , , and introducing new parameters
We denote . Note that we still have the balancing condition
(4.1) 
Definition 4.1.
The elliptic hypergeometric function is the meromorphic function on defined by the following formula
4.2. The elliptic hypergeometric equation
The elliptic hypergeometric function satisfies the following equation
(4.2) 
where
It is easily seen that , so that the previous equation has coefficients in .
From now on, we denote by the Galois group of (4.3) over (with respect to some PV ring).
4.3. Irreducibility of the Galois group of the elliptic hypergeometric function
Theorem 4.2.
Assume that every multiplicative relation among the is induced by (4.1), in the sense that if there are integers such that
then and for some . Then is irreducible.
Proof.
To the contrary, assume that is reducible. According to Theorem 3.6, the following Riccati equation has a solution :
(4.4) 
First, note that is a solution of (4.4) if and only if with . Then to simplify the expression of the divisors of and , we may replace them by , , and consider the Riccati equation satisfied by . Consider and such that
In view of the explicit expressions for and , we see that we may take and such that