DIFFERENTIAL TRANSCENDENCE CRITERIA FORSECOND-ORDER LINEAR DIFFERENCE EQUATIONSAND ELLIPTIC HYPERGEOMETRIC FUNCTIONSCARLOS E. ARRECHE, THOMAS DREYFUS, AND JULIEN ROQUES

Abstract. We develop general criteria that ensure that any non-zero
solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, suchas shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are
obtained as an application of differential Galois theory for difference
equations. We apply our criteria to prove a new result to the effect thatmost elliptic hypergeometric functions are differentially transcendental.

Contents
1. Introduction
2. Difference Galois theory
3. Parametrized Difference Galois theory
3.1. General facts
3.2. Differential transcendence criteria
4. Difference equations over elliptic curves
4.1. The base field
4.2. Theta functions
4.3. Irreducibility of the σ-Galois groups
5. Application to the elliptic hypergeometric functions
5.1. The elliptic hypergeometric functions
5.2. The elliptic hypergeometric equation
5.3. Irreducibility of the σ-Galois group of the elliptic
hypergeometric function
5.4. Differential transcendence of the elliptic hypergeometricfunctions
References

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Date: September 20, 2019.
2010 Mathematics Subject Classification. 39A06,12H05.
Key words and phrases. Linear difference equations, difference Galois theory, elliptic
curves, differential algebra.
The first author was partially supported by the National Science Foundation underGrant No. CCF-1815108.
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CARLOS E. ARRECHE, THOMAS DREYFUS, AND JULIEN ROQUES

1. Introduction
The differential Galois theory for difference equations developed in [HS08]
provides a theoretical tool to understand the differential-algebraic properties
of solutions of linear difference equations. Given a σδ-field K (i.e., K isequipped with an automorphism σ and a derivation δ such that σ ◦δ = δ ◦σ),one considers an nth order linear difference equation of the form
(1.1)

an σ n (y) + an−1 σ n−1 (y) + · · · + a1 σ(y) + a0 y = 0,

where ai ∈ K for i = 0, . . . , n, an a0 6= 0, and y is an indeterminate. The
theory of [HS08] associates with (1.1) a geometric object G, called the differential Galois group, that encodes the polynomial differential equations
satisfied by the solutions of (1.1). Traditionally, the following special cases
have attracted special attention: K = C(z), and σ is one of the following:a shift operator σ : z 7→ z + r, where 0 6= r ∈ C; the q-dilation operator
σ : z 7→ qz, where q ∈ C∗ is not a root of unity; and the Mahler operator σ : z 7→ z p , where p ∈ N≥2 . 1 More recently, the elliptic case has also
attracted a lot of interest: here K = Mer(E) is the field of meromorphic
functions on the elliptic curve E = C∗ /pZ , where p ∈ C∗ is such that |p| =6 1— or equivalently, K is the field of (multiplicatively) p-periodic meromorphic functions f (z) on C∗ such that f (pz) = f (z) — and σ : f (z) 7→ f (qz),where q ∈ C∗ is such that pZ ∩ q Z = {1} (or equivalently, q represents a
non-torsion point of E). In each one of these four cases of interest there isa corresponding choice of derivation δ that makes K into a σδ-field.
The main contribution of this work (Theorem 3.4) is the development of
a new set of criteria for second-order equations(1.2)

σ 2 (y) + aσ(y) + by = 0,

which guarantee that any non-zero solution y of (1.2) must be differentiallytranscendental over K, i.e., for any m ∈ N there is no non-zero polynomial
P ∈ K[y0 , y1 , . . . , ym ] such that P (y, δ(y), . . . , δ m (y)) = 0. These criteria
apply uniformly under mild conditions on the base σδ-field K (see Definition 2.1), which are satisfied in the four particular cases mentioned above:
shift, q-dilation, Mahler, and elliptic. Moreover, the verification of the criteria only requires one to check whether the following auxiliary equations
associated with (1.2) admit any solutions in K: if there is no u ∈ K such
that
(1.3)

uσ(u) + au + b = 0,

and there are no g ∈ K and linear differential operator L ∈ C[δ] such thatδ(b)= σ(g) − g,
(1.4)
L
b
then every non-zero solution of (1.2) must be differentially transcendental
over K. Therefore, although we do apply the differential Galois theory for
difference equations [HS08] in the proof that our criteria are correct, theactual verification of the criteria does not involve any prior knowledge of
1. In the Mahler case the base field must be taken to be K = C({z 1/` }`∈N ) withσ(z 1/` ) = z p/` in order for σ to be an automorphism of K and not merely an (injective)
endomorphism.

DIFF. TRANSC. CRITERIA & ELLIPTIC HYPERGEOMETRIC FUNCTIONS

3

this theory at all. Moreover, in each of the four cases of interest mentioned
above there are effective algorithms to decide whether the Riccati equation(1.3) and the telescoping problem (1.4) admit solutions, for which we provide case-by-case references below. Hence these user-friendly criteria are ofpractical import to non-experts seeking to decide differential transcendenceof solutions of second-order difference equations in many settings that arise
in applications.
Indeed, we illustrate the practical applicability of our criteria in the elliptic case by proving differential transcendence of “most” elliptic hypergeometric functions. The elliptic hypergeometric functions form a common
analogue of classical hypergeometric functions and q-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 [Spi16] and has been a dynamic field of research,see for instance [vdB+ 07, FR09, M+ 09, Rai10, Ros02]. In the intervening
years a number of remarkable analogues of known properties and applications of classical and q-hypergeometric functions have been discovered forthe elliptic hypergeometric functions; see [Spi16] for more details.
The theoretical part of our strategy to prove differential transcendence for
elliptic hypergeometric functions is in the tradition of other applications of
the differential Galois theory for difference equations of [HS08] to questions
about shift difference equations [Arr17], q-difference equations, [DHR16],
deterministic finite automata and Mahler functions [DHR18], lattice walks
in the quarter plane [DHRS18, DR17, DHRS17], and shift, q-dilation, andMahler difference equations in general [AS17]. The Galois correspondence of[HS08] implies in particular that if the differential Galois group G is “large”
then there are “few” differential-algebraic relations among the solutions of(1.1). However, this theoretical strategy is only practical in the presence ofalgorithmic decision procedures that ensure that G is indeed large enoughto force any solution of (1.1) to be differentially transcendental. The criteria
developed here in Theorem 3.4 serve to fulfill precisely this purpose. To put
the novelty and usefulness of these criteria in context, let us briefly recall
the state of the art in each of the four particular cases of interest mentioned
above.
In the shift case, a complete algorithm to compute the differential Galois
group G for (1.2) is developed in [Arr17], based on the earlier algorithm of[Hen98] to compute the non-differential Galois group H of (1.2) [vdPS97].
Even in this case, it is still useful to have the isolated criteria of Theorem 3.4 to decide differential transcendence only, without having to compute the whole Galois group G of (1.2). An algorithm for deciding whether
the Riccati equation (1.3) admits a solution in K has been developed in[Hen98], and to decide whether there is a telescoper (1.4) one can apply[HS08, Cor. 3.4].
The situation in the q-dilation and Mahler cases is similar. One knows
how to compute the differential Galois group G for first-order equations(1.1) with n = 1 by solving an associated telescoping problem (see for example [HS08, Corollary 3.4] in the q-dilation case and [DHR18, Prop. 3.1]

4

CARLOS E. ARRECHE, THOMAS DREYFUS, AND JULIEN ROQUESin the Mahler case), but there is no general algorithm to compute G for
higher-order equations (1.1) with n ≥ 2. The general criteria developedin [DHR16, DHR18] for differential transcendence of solutions of (1.1) are
valid for arbitrary n, but these criteria require prior knowledge of the (nondifferential) Galois group H of (1.1) [vdPS97]. At present this group H can
only be computed in general when n ≤ 2 by [Hen97] in the q-dilation caseand [Roq18] in the Mahler case. Even when n = 2, the criteria given herein Theorem 3.4 strictly generalize those of [DHR16, DHR18], and require
no knowledge of (differential) Galois theory of difference equations for theirapplication. Algorithms for deciding whether the Riccati equation (1.3)
admits solutions in K have been developed in the the q-dilation [Hen97]
and Mahler [Roq18] cases. Algorithms for deciding whether the telescoping
problem (1.4) can be solved have also been developed in [HS08, Cor. 3.4] inthe q-dilation case and in [DHR18, Prop. 3.1] in the Mahler case.
In the elliptic case, the recent algorithm developed in [DR15] computes
the (non-differential) Galois group H of (1.2) associated by the theory of[vdPS97], but there are no general algorithms to compute the differential
Galois group G for (1.1) for any order n. In spite of the relative dearthof algorithms in this case, the authors of [DHRS18] were still successful
in proving differential transcendence of some first-order (inhomogeneous)
elliptic difference equations arising in connection with generating series forwalks in the quarter plane. The criteria of Theorem 3.4 are the first to
provide a test for differential transcendence that applies to second-orderdifference equations in the elliptic case. An algorithm for deciding whether
the Riccati equation (1.3) admits solutions in K has been developed in[DR15], and an algorithm to decide whether the telescoping problem (1.4)
can be solved has also been developed in [DHRS18, Prop. B.8].
The paper is organized as follows. In Section 2, we recall some facts
about the difference Galois theory developed in [vdPS97]. To a difference
equation (1.1) is associated an algebraic group. The larger the group, thefewer the algebraic relations that exist among the solutions of the difference
equation. In Section 3, we recall some facts about the differential Galois
theory for difference equations of [HS08]. Here the Galois group is a linear
differential algebraic group, that is, a group of matrices defined by a systemof 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 our differential transcendence criteria
for second-order difference equations (1.2) in Theorem 3.4. In Section 4
we restrict ourselves to the situation where the coefficients of the difference
equation are elliptic functions. We recall some results from [DR15], 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 5, we follow [Spi16]
in defining the elliptic analogue of the hypergeometric equation (5.4) and,
under a certain genericity assumption, we prove that its nonzero solutions
are differentially transcendental, see Theorem 5.7.

DIFF. TRANSC. CRITERIA & ELLIPTIC HYPERGEOMETRIC FUNCTIONS

5

2. Difference Galois theory
For details on what follows, we refer to [vdPS97, Chapter 1]. Unlessotherwise stated, all rings are commutative with identity and contain thefield of rational numbers. In particular, all fields are of characteristic zero.
A σ-ring (or difference ring) (R, σ) is a ring R together with a ring automorphism σ : R → R. If R is a field then (R, σ) is called a σ-field. When
there is no possibility of confusion the σ-ring (R, σ) will be simply denotedby R. There are natural notions of σ-ideals, σ-ring extensions, σ-algebras,
σ-morphisms, etc. We refer to [vdPS97, Chapter 1] for the definitions.
The ring of σ-constants Rσ of the σ-ring (R, σ) is defined by
Rσ := {f ∈ R | σ(f ) = f }.
We now let (K, σ) be a σ-field. We assume that the field of constants
C := Kσ is algebraically closed and that the characteristic of K is 0.
We consider a difference equation of order two with coefficients in K:(2.1)

σ 2 (y) + aσ(y) + by = 0 with a ∈ K and b ∈ K∗

and the associated difference system:


0
1
(2.2)
σY = AY with A =∈ GL2 (K).
−b −aBy [vdPS97, §1.1], there exists a σ-ring extension (R, σ) of (K, σ) suchthat1) there exists U ∈ GL2 (R) such that σ(U ) = AU (such a U is called afundamental matrix of solutions of (2.2));
2) R is generated, as a K-algebra, by the entries of U and det(U )−1 ;3) the only σ-ideals of (R, σ) are {0} and R.
Note that the last assumption implies Rφ = C. Such an R is called a σPicard-Vessiot ring, or σ-PV ring for short, for (2.2) over (K, σ). It is unique
up to isomorphism of (K, σ)-algebras. Note that a σ-PV ring is not always
an integral domain, but it is a direct sum of integral domains transitively
permuted by σ.
The corresponding σ-Galois group Gal(R/K) of (2.2) over (K, σ), or σGalois group for short, is the group of (K, σ)-automorphisms of R:
Gal(R/K) := {φ ∈ Aut(R/K) | σ ◦ φ = φ ◦ σ}.
A straightforward computation shows that, for any φ ∈ Gal(R/K), there
exists a unique C(φ) ∈ GL2 (C) such that φ(U ) = U C(φ). According to[vdPS97, Theorem 1.13], one can identify Gal(R/K) with an algebraic subgroup G of GL2 (C) via the faithful representation
ρ : Gal(R/K) → GL2 (C)φ
7→ C(φ).
If we choose another fundamental matrix of solutions U , we find a conjugate representation. In what follows, by “σ-Galois group of the difference
equation (2.1)”, we mean “σ-Galois group of the difference system (2.2)”.

6

CARLOS E. ARRECHE, THOMAS DREYFUS, AND JULIEN ROQUESWe shall now introduce a property relative to the base σ-field (K, σ),which appears in [vdPS97, Lemma 1.19].
Definition 2.1. We say that the σ-field (K, σ) satisfies the property (P) if:— the field K is a C 1 -field 2;— and the only finite field extension L of K such that σ extends to a
field endomorphism of L is L = K.
Example 2.2. The following are natural examples of difference fields thatsatisfy property (P):∗S: Shift case with K = C(z), σ : f (z) 7→[ f (z + h), h ∈ C . See [Hen97].
Q: q-difference case. K = C(z 1/∗ ) =C(z 1/` ), σ : f (z) 7→ f (qz), q ∈ C∗ ,|q| =6 1. See [Hen98].
`∈N∗
M: Mahler case. K = C(z 1/∗ ), σ : f (z) 7→ f (z p ), p ∈ N≥2 . See [Roq18].
E: Elliptic case. See Section 4, and [DR15].
The following result is due to van der Put and Singer. We recall thattwo difference systems σY = AY and σY = BY with A, B ∈ GL2 (K)are isomorphic over K if and only if there exists T ∈ GL2 (K) such thatσ(T )A = BT . Note that σ(Y ) = AY if and only if σ(T Y ) = BT Y .
Theorem 2.3. Assume that (K, σ) satisfies property (P). Then the following properties relative to G = ρ(Gal(R/K)) hold:— G/G◦ is cyclic, where G◦ is the identity component of G;— there exists B ∈ G(K) such that (2.2) is isomorphic to σY = BY
over K.
e be an algebraic subgroup of GL2 (C) such that A ∈ G(K).
e
Let G
The following
properties hold:e— G is conjugate to a subgroup of G;— any minimal element (with respect to inclusion) in the set of algee of G
e for which there exists T ∈ GL2 (K) such that
braic subgroups Heσ(T )AT −1 ∈ H(K)is conjugate to G;
e if and only if, for any T ∈ G(K)e
— G is conjugate to G
and for any−1eee
proper algebraic subgroup H of G, one has that σ(T )AT ∈/ H(K).
Proof. The proof of [vdPS97, Propositions 1.20 and 1.21] in the special casewhere K := C(z) and σ is the shift σ(f (z)) := f (z + h) with h ∈ C∗ , extends
mutatis mutandis to the present case.
This theorem is at the heart of many algorithms to compute σ-Galois
groups, see for example [Hen97, Hen98, DR15, Roq18].
3. Parametrized Difference Galois theory
3.1. General facts. A (σ, δ)-ring (R, σ, δ) is a ring R endowed with a ring
automorphism σ and a derivation δ : R → R (this means that δ is additiveand satisfies the Leibniz rule δ(ab) = aδ(b) + δ(a)b) such that σ ◦ δ = δ ◦ σ. If
R is a field, then (R, σ, δ) is called a (σ, δ)-field. When there is no possibilityof confusion, we write R instead of (R, σ, δ). There are natural notions
2. Recall that K is a C 1 -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.

DIFF. TRANSC. CRITERIA & ELLIPTIC HYPERGEOMETRIC FUNCTIONS

7

of (σ, δ)-ideals, (σ, δ)-ring extensions, (σ, δ)-algebras, (σ, δ)-morphisms, etc.
We refer to [HS08, Section 6.2] for the definitions.
If K is a δ-field, and if y1 , . . . , yn belong to some δ-field extension of K,
then K{y1 , . . . , yn }δ denotes the δ-algebra generated over K by y1 , . . . , yn
and Khy1 , . . . , yn iδ denotes the δ-field generated over K by y1 , . . . , yn .
We now let (K, σ, δ) be a (σ, δ)-field. We assume that the field of
σ-constants C := Kσ is algebraically closed and that K is of characteristic 0.
In order to apply the (σ, δ)-Galois theory developed in [HS08], we need towork with a base (σ, δ)-field L such that C = Lσ is δ-closed. 3 To this end,
the following lemma will be useful.
Lemma 3.1 ([DHR18, Lemma 2.3]). Suppose that C is algebraically closed
and let Ce be a δ-closure of C (the existence of such a Ce is proved in [Kol74]).
Then the ring Ce ⊗C K is an integral domain whose fraction field L is ae(σ, δ)-field extension of K such that Lσ = C.
We still consider the difference equation (2.1) and the associated difference
system (2.2). By [HS08, § 6.2.1], there exists a (σ, δ)-ring extension (S, σ, δ)of (L, σ, δ) such that
1) there exists U ∈ GL2 (S) such that σ(U ) = AU ;2) S is generated, as an L-δ-algebra, by the entries of U and det(U )−1 ;3) the only (σ, δ)-ideals of S are {0} and S.
Such an S is called a (σ, δ)-Picard-Vessiot ring, or (σ, δ)-PV ring for short,for (2.2) over (L, σ, δ). It is unique up to isomorphism of (L, σ, δ)-algebras.
Note that a (σ, δ)-PV ring is not always an integral domain, but it is the
direct sum of integral domains that are transitively permuted by σ.
The corresponding (σ, δ)-Galois group Galδ (S/L) of (2.2) over (L, σ, δ),or (σ, δ)-Galois group for short, is the group of (L, σ, δ)-automorphisms of S:
Galδ (S/L) = {φ ∈ Aut(S/L) | σ ◦ φ = φ ◦ σ and δ ◦ φ = φ ◦ δ}.
In what follows, by “(σ, δ)-Galois group of the difference equation (2.1)”, wemean “(σ, δ)-Galois group of the difference system (2.2)”.
A straightforward computation shows that, for any φ ∈ Galδ (S/L), there
e such that φ(U ) = U C(φ). By [HS08, Propoexists a unique C(φ) ∈ GL2 (C)sition 6.18], the faithful representationeρδ : Galδ (S/L) → GL2 (C)φ
7→ C(φ)identifies Galδ (S/L) with a linear differential algebraic group Gδ , that is, ae defined by a system of δ-polynomial equations over Ce
subgroup of GL2 (C)in the matrix entries. If we choose another fundamental matrix of solutions
U , we find a conjugate representation.
3. The field Ce is called δ-closed if, for every (finite) set of δ-polynomials F with coeffie if the system of δ-equations F = 0 has a solution with entries in some δ-field
cients in C,e then it has a solution with entries in C.
e Note that when the derivation δ isextension L|C,
trivial, i.e. δ = 0, then a field is δ-closed if and only if it is algebraically closed.

8

CARLOS E. ARRECHE, THOMAS DREYFUS, AND JULIEN ROQUESLet S be a (σ, δ)-PV ring for (2.2) over L and let U ∈ GL2 (S) be afundamental matrix of solutions. Then the L-σ-algebra R generated by theentries of U and det(U )−1 is a σ-PV ring for (2.2) over L. We can (and will)
identify Galδ (S/L) with a subgroup of Gal(R/L) by restricting the elementsof Galδ (S/L) to R.
Proposition 3.2 ([HS08], Proposition 2.8). The group Galδ (S/L) is a
Zariski-dense subgroup of Gal(R/L).
3.2. Differential transcendence criteria. The aim of this section is todevelop a galoisian criterion for the differential transcendence of the nonzero
solutions of (2.1).
Definition 3.3. Let F/K be a (σ, δ)-field extension. We say that f ∈ F isdifferentially algebraic over K if there exists n ∈ N such that f, . . . , δ n (f ) arealgebraically dependent over K. Otherwise, we say that f is differentiallytranscendental over K.
Recall that K be a (σ, δ)-field satisfying property (P) such that C = Kσis algebraically closed and such that K has characteristic 0.
Let Ce be a δ-closure of C. According to Lemma 3.1, Ce ⊗C K is an integral
domain and L := Frac(Ce ⊗C K) is a (σ, δ)-field extension of K such thate Let S be a (σ, δ)-PV ring for (2.2) over L and let R ⊂ S be a σ-PV
Lσ = C.
e for (2.2) over K.
ring for (2.2) over L. We also consider a σ-PV ring R
Our differential transcendence criteria are given in our main result below.
Theorem 3.4. Consider the second-order difference equation (2.1):
σ 2 (y) + a(y) + by = 0,
where a ∈ K and b ∈ K∗ and K satisfies property (P). Assume the following:(1) there is no u ∈ K such that uσ(u) + au + b = 0; and(2) there are no g ∈ K and non-zero linear differential operator L ∈ C[δ]such that


δ(b)
L
= σ(g) − g.
bThen any non-zero solution of (2.1) in any (σ, δ)-field extension F of K isdifferentially transcendental over K.
Note that the first criterion of Theorem 3.4 is equivalent to the irreducibile
ity of Gal(R/K),
and may be tested algorithmically in many contexts, see[Hen97, Hen98, DR15, Roq18]. The following lemma similarly relates theesecond criterion to a different largeness condition on Gal(R/K).
Lemma 3.5 (Proposition 2.6, [DHR18]). The (σ, δ)-Galois group of σy = bye if and only if there exist a nonzero
over L is a proper subgroup of GL1 (C)linear differential operator L with coefficients in C and g ∈ K such that


δ(b)
L
= σ(g) − g.
b
The following lemma will be used in the proof of Theorem 3.4.

DIFF. TRANSC. CRITERIA & ELLIPTIC HYPERGEOMETRIC FUNCTIONS

9

Lemma 3.6. Assume that (2.1) has a nonzero differentially algebraic solution in a (σ, δ)-field extension F of K. Then (2.1) has a nonzero differentially algebraic solution in S.
Proof of Lemma 3.6. Since any two (σ, δ)-PV rings for (2.1) over L
are isomorphic, it is sufficient to prove the lemma for some (σ, δ)PV ring, not necessarily for S itself. Let f be a nonzero differentially algebraic solution of (2.1) in F. We consider the localization T
of Lhf, σ(f )iδ {X1,2 , X2,2 }δ at f X2,2 − σ(f )X1,2 , where X1,2 , X2,2 are δindeterminates over Lhf, σ(f. Thisring 
has a natural
structureof L )iδ


X1,2
X1,2
f
X1,2
(σ, δ)-algebra such that σ=Aandis a funX2,2X2,2
σ(f ) X2,2
damental matrix of solutions of σY = AY with coefficients in T . If we let
M be a maximal (σ, δ)-ideal of T , then the quotient T /M is a (σ, δ)-PVring for σY = AY over L and the image of f in this quotient is differentially algebraic. Let us prove that it is nonzero. Otherwise the image ofthe fundamental solution in the (σ, δ)-PV ring T /M would have a zero firstcolumn and therefore would not be inversible, leading to a contradiction.
This concludes the proof.
Proof of Theorem 3.4. Assume to the contrary that (2.1) has a nonzero differentially algebraic solution in a (σ, δ)-field extension F of K. According
to Lemma 3.6, there exists a nonzero differentially algebraic solution f of(2.1) in S.
By [Hen97, Lemma 4.1] combined with Theorem 2.3, one of the following
three cases holdse— Gal(R/K)
is reducible.
e— Gal(R/K)
is irreducible and imprimitive.
e— Gal(R/K) contains SL2 (C).
By [DR15, Lemma 13], the assumption that there is no solution in K for
the Riccati equation uσ(u) + au + b = 0 is equivalent to the irreducibilityeof Gal(R/K).
Hence only the last two cases may occur. Then we split ourestudy in two cases depending on whether Gal(R/K)
is imprimitive or not.
e
Let us first assume that Gal(R/K)
is imprimitive. It follows from Theorem 2.3 and [Hen97, Section 4.3] that (2.1) is equivalent over K to
(3.1)

σ 2 (y) + ry = 0

for some r ∈ K∗ . More precisely, letσY = BY with B =0 1∈ GL2 (K),−r 0

be the system associated to (3.1). Then there exists T ∈ GL2 (K) such thatσ(T )A = BT . Let T = (ti,j ). Since σY = AY if and only if σ(T Y ) = BT Y ,we obtain that t1,1 f + t1,2 σ(f ) satisfies (3.1) with (t1,1 , t1,2 ) 6= (0, 0). Let us
prove that t1,1 f + t1,2 σ(f ) is non zero. If t1,1 f + t1,2 σ(f ) = 0, then f 6= 0
implies t1,1 t1,2 6= 0 and then σ(f )/f is solution of the Riccati equationuσ(u) + au + b = 0, which contradicts the first assumption of Theorem 3.4.
Since f is differentially algebraic over K, we have that σ(f ), and hence
also t1,1 f + t1,2 σ(f ), are differentially algebraic over L. By [HS08, Proposition 6.26], this implies that the (σ 2 , δ)-Galois group of (3.1) over L is a

10

CARLOS E. ARRECHE, THOMAS DREYFUS, AND JULIEN ROQUESe By Lemma 3.5 there exist a nonzero D ∈ C[δ]proper subgroup of GL1 (C).
and h ∈ K such that
(3.2)

2
D( δ(r)r ) = σ (h) − h = σ(σ(h) + h) − (σ(h) + h).

Taking the determinant in σ(T )A = BT allows us to deduce the existenceof p ∈ K∗ such that b = σ(p)p r, and therefore the (σ, δ)-Galois groups for
σ(y) = ry and σ(y) = by are the same. Consequently, by Lemma 3.5 andthe assumption on the (σ, δ)-Galois group of σy = by over L, for any nonzero
D ∈ C[δ] and any g ∈ K, we have D( δ(r)r ) 6= σ(g)−g. This is in contradiction
with (3.2).
eAssume now that Gal(R/K)
is not imprimitive, so it contains SL2 (C). By[DHR18, Proposition 2.10], we deduce that



c 0
∗
e
Gm :=
c∈C
⊂ Galδ (S/L).
0 c
Let n ∈ N be as small as possible such that there exists
0 6= P ∈ L[X0 , . . . , Xn ] with P (f, δ(f ), . . . , δ n (f )) = 0, and suppose that
this P has smallest possible total degree d ∈ N. For c ∈ Ce∗ , let φc ∈ Gm
with corresponding matrix ( 0c 0c ). For all c ∈ Ce∗ , we findφc P (f, δ(f ), . . . , δ n (f )) = P (φc (f ), φc (δ(f )), . . . , φc (δ n (f )))= P (cf, δ(cf ), . . . , δ n (cf )) = 0.
Since Ce is differentially closed, there exists c ∈ Ce∗ such that δ(c) = 0 and
cd 6= 1. Since δ i (cf ) = cδ i (f ) for such a c, we have that
cd P (f, δ(f ), . . . , δ n (f )) − P (cf, cδ(f ), . . . , cδ n (f )) = 0,
and we find that P must be homogeneous of degree d, for otherwise the total
degree d would not be minimal. We may further assume that the degree dn
of Xn in P is as small as possible. Again since Ce is differentially closed,
there exists c ∈ Ce such that δ 2 (c) = 0 but δ(c) 6= 0. But then0 = P (cf, δ(cf ), . . . , δ n (cf )) = P (cf, cδ(f )+δ(c)f, . . . , cδ n (f )+δ(c)δ n−1 (f ))= cd P (f, δ(f ), . . . , δ n (f )) + Q(f, δ(f ), . . . , δ n (f )) = Q(f, δ(f ), . . . , δ n (f ))for some nonzero homogeneous polynomial Q ∈ L[X0 , . . . , Xn ] of total degree d in which the degree of Xn is strictly smaller than dn . This contradiction concludes the proof.

4. Difference equations over elliptic curves
In this section we will be mainly interested in difference equations(4.1)

σ 2 (y) + aσ(y) + by = 0,

with a, b ∈ Mp , where— Mp denotes the field of meromorphic functions over the elliptic curveC∗ /pZ for some p ∈ C∗ such that |p| < 1, i.e. the field of meromorphic
functions on C∗ satisfying f (z) = f (pz);DIFF. TRANSC. CRITERIA & ELLIPTIC HYPERGEOMETRIC FUNCTIONS

11— σ is the automorphism of Mp defined byσ(f )(z) := f (qz)for some q ∈ C∗ such that |q| =6 1 and pZ ∩ q Z = {1}.
Note that this choice ensures that σ is non cyclic.
4.1. The base field. The difference Galois groups of linear difference equations over elliptic curves have been studied in [DR15]. In loc. cit. the elliptic
curves are given by quotients of the form C/Λ for some lattice Λ. However,in the present work, we are mainly interested in difference equations on elliptic curves given by quotients of the form C∗ /pZ for some p ∈ C∗ suchthat |p| < 1. The translation between elliptic curves of the form C/Λ andelliptic curves of the form C∗ /pZ is standard, namely by using the fact that if
Λ = Z + τ Z with =(τ ) > 0 and p = e2πiτ then the map C → C∗ : w 7→ e2πiw
induces an isomorphism C/Λ ' C∗ /pZ .
We shall now recall some constructions and results from [DR15], restatedin the “C∗ /pZ context” via the above identification between C/Λ and C∗ /pZ .
For k ∈ N∗ we denote by C∗k the Riemann surface of z 1/k , and we let zk be
d = z for every d ∈ N∗ . Wea coordinate function on each C∗k such that zdk
k∗∗will write C1 = C and z1 = z.
We let Mp,k denote the field of meromorphic functions on C∗k satisfyingf (pzk ) = f (zk ), or equivalently the field of meromorphic functions on theelliptic curve C∗k /pZ . The d-power map C∗dk → C∗k : ξ 7→ ξ d induces an
inclusion of function fields Mp,k ,→ Mp,dk for each k, d ∈ N∗ . We denote by
K the field defined by[K :=Mp,k .
k≥1

We endow K with the non-cyclic field automorphism σ defined by(4.2)

σ(f )(zk ) := f (qk zk )where q1 = q ∈ C∗ is such that |q| 6= 1 and pZ ∩ q Z = {1}, and qk ∈ C∗k
d = q fordefines a compatible system of k-th roots of q1 = q such that qdk
k∗every d ∈ N (cf. [Hen98, Section 2]). Then (K, σ) is a difference field andwe have the following properties.
Proposition 4.1 ([DR15], Proposition 5). The field of constants of (K, σ)
is Kσ = C.
Proposition 4.2 ([DR15], Proposition 6). The difference field (K, σ) satisfies property (P) (see Definition 2.1).
Remark 4.3. The field Mp = Mp,1 equipped with the automorphism σ does
not satisfy property (P). This is why we work over (K, σ) instead of (Mp , σ).
Corollary 4.4. The conclusions of Theorem 2.3 are valid for (K, σ).
4.2. Theta functions. We shall now recall some basic facts and notations
about theta functions extracted from [DR15, Section 3] (but stated in the“C∗ /pZ context”, see the beginning of the previous section). For the proofs,

12

CARLOS E. ARRECHE, THOMAS DREYFUS, AND JULIEN ROQUES

we refer to [Mum07, Chapter I]. We still consider p ∈ C∗ such that |p| < 1.
We consider the infinite productY(1 − zpj ).
(z; p)∞ =j≥0

The theta function defined byθ(z; p) = (z; p)∞ (pz −1 ; p)∞(4.3)
satisfies(4.4)

θ(pz; p) = θ(z −1 ; p) = −z −1 θ(z; p).

Let Θk be the set of holomorphic functions on C∗k of the formYcθ(ξzk )nξ
ξ∈C∗k∗with c ∈ C∗ and (nξ )ξ∈C∗k ∈ N(Ck ) with finite support. We denote by Θquotk
the set of meromorphic functions on C∗k that can be written as a quotientof two elements of Θk . We have.
Mp,k ⊂ Θquotk
as the following formal sum of
We define the divisor divk (f ) of f ∈ Θquotkpoints of C∗k /pZ :X
divk (f ) :=ordλ (f )[λ],λ∈C∗k /pZ

where ordλ (f ) is the (zk − ξ)-adic valuation of f , for an arbitrary ξ ∈ λ (itfollows from (4.4) that this valuation does not depend on the chosen ξ ∈ λ).
For any λ ∈ C∗k /pZ and any ξ ∈ λ, we set[ξ]k := [λ].
Moreover, we will writeXXnλ [λ] ≤λ∈C∗k /pZmλ [λ]

λ∈C∗k /pZ

if nλ ≤ mλ for all λ ∈ C∗k /pZ . We also introduce the weight ωk (f ) of f
defined byYωk (f ) :=λordλ (f ) ∈ C∗k /pZ
λ∈C∗k /pZ

and its degree degk (f ) given byX

degk (f ) :=ordλ (f ) ∈ Z.

λ∈C∗k /pZ

Example 4.5. Consider θ = θ(z; p) defined above. Then it follows from (4.3)
that div1 (θ) = [1], since θ(z; p) has a zero of multiplicity one at each pointof the subgroup pZ ⊂ C∗ . However, since z = zkk , we have thatdivk (θ) =k−1 h p iXζki k pj ,i,j=0DIFF. TRANSC. CRITERIA & ELLIPTIC HYPERGEOMETRIC FUNCTIONS

13p
where ζk ∈ C∗k denotes a primitive k-th root of unity and k pj is the j-th√
power of an arbitrary choice k p of k-th root of p.
Similarly, for any f (z) ∈ Mp = Mp,1 we have that divk (f ) = ϕ∗k (div1 (f )),where ϕk : C∗k /pZ → C∗ /pZ denotes the k-power map and ϕ∗k denotes theinduced pull-back map on divisors.
4.3. Irreducibility of the σ-Galois groups. One of the criteria of Theorem 3.4 concerns the non-existence of a solution in K of a difference Riccati
equation. The main tool used in this paper to address this is the following
result.
Theorem 4.6 (Proposition 17 in [DR15]). Let G be the σ-Galois group of(4.1) over K. The following statements are equivalent:— the group G is reducible;— the following Riccati equation has a solution in Mp,2 :(4.5)

uσ(u) + au + b = 0.

Moreover, if p1 ∈ Θ2 ∪ {0} and p2 , p3 ∈ Θ2 are such thatp2
p1
and b = ,a=p3p3then any solution u ∈ Mp,2 of (4.5) is of the formu=

σ(r0 ) r1
r0 r2

for some r0 , r1 , r2 ∈ Θ2 such that(i) div2 (r1 ) ≤ div2 (p2 ),(ii) div2 (r2 ) ≤ div2 (σ −1 (p3 )),(iii) deg2 (r1 ) = deg2 (r2 ),deg (r )(iv) ω2 (r1 /r2 ) = q2 2 0 mod pZ .
5. Application to the elliptic hypergeometric functions
5.1. The elliptic hypergeometric functions. We shall now introduce
the elliptic hypergeometric functions following [Spi16]. Consider p, q ∈ C∗
such that |p| < 1, |q| < 1, and q Z ∩ pZ = {1}. Define(z; p, q)∞ =Y(1 − zpj q k )(pq/z; p, q)∞.
(z; p, q)∞andΓ(z; p, q) =and

Γ(qz; p, q) = θ(z; p)Γ(z; p, q).

j,k≥0

We haveΓ(pz; p, q) = θ(z; q)Γ(z; p, q)

For t1 , . . . , t8 ∈ C∗ such that |tj | < 1 for each 1 ≤ j ≤ 8 and satisfying
8Y
the balancing condition
tj = p2 q 2 , we set
j=1

Z Q8
(5.1)

V (t1 , . . . , t8 ; p, q) = κT

j=1 Γ(tj z; p, q)Γ(tj /z; p, q) dz,
Γ(z 2 ; p, q)Γ(z −2 ; p, q)z

14

CARLOS E. ARRECHE, THOMAS DREYFUS, AND JULIEN ROQUES(q;q)∞. For
where T denotes the positively oriented unit circle and κ = (p;p)∞4πiz ∈ C∗ , we follow [Spi16] by setting t6 = cz, t7 = c/z, and introducing newparameters
√
ε6 ε8
q
c
ε8
for j = 1, . . . , 5, ε8 = , ε7 = , c =
.
(5.2)
εj =
ctj
t8
q
p2

We denote ε = (ε1 , . . . , ε8 ). Note that we still have the balancing condition
8
Y

(5.3)

ε j = p2 q 2 .

j=1

Definition 5.1. The elliptic hypergeometric function fε (z) is defined by
the following formula
V (q/cε1 , . . . , q/cε5 , cz, c/z, cε8 ; p, q).
fε (z) :=2
Γ(c z/ε8 ; p, q)Γ(z/ε8 ; p, q)Γ(c2 /zε8 ; p, q)Γ(1/zε8 ; p, q)
Remark 5.2. As explained in [Spi16], the function V (t; p, q) definedin (5.1) can be extended by analytic continuation, so thatQ∗1≤j<k≤8 (tj tk ; p, q)∞ V (t; p, q) is holomorphic for t1 , . . . , t8 ∈ C . We should
also mention for completeness that, as explained in [Spi16], in Definition 5.1 it is initially necessary topimpose the constraints (expressed interms of the old parametrization) |pq| < |tj | < 1 for j = 1, . . . , 5 andp|pq| < |q ±1 tj | < 1 for j = 6, 7, 8, which can then be relaxed by analytic
continuation. These important but subtle considerations will not play a rolein what follows.
5.2. The elliptic hypergeometric equation. The elliptic hypergeometric function fε (z) satisfies the following equation(5.4)

A(z)(y(qz) − y(z)) + A(z −1 )(y(q −1 z) − y(z)) + νy(z) = 0,

where
8

(5.5)

A(z) =Y1
θ(εj z; p)θ(z 2 ; p)θ(qz 2 ; p)and ν =j=1

6Y

θ(εj ε8 /q; p).

j=1

It is easily seen that A(pz) = A(z), so that the previous equation has coefficients in Mp,1 .
Replacing z by qz in (5.4), we obtain the following equation:(5.6)
with a = ν−A(qz)−A(qA(qz)

σ 2 (y) + aσ(y) + by = 0,
−1 z −1 )

−1 −1

z )
, b = A(qA(qz)∈ Mp,1 .

Remark 5.3. Note that the new parameters ε1 , . . . , ε8 used in the definitionof fε (z) are not defined to be free independent parameters, since they aredefined in terms of the old parameters t1 , . . . , t8 (which are free parametersQ
save for the balancing condition 8j=1 tj = p2 q 2 ), and in fact one of theequations in the reparametrization (5.2) is equivalent to ε8 = ε7 q.
On the other hand, the elliptic hypergeometric equation (5.4) is defined forarbitrary parameters ε1 , . . . , ε8 ∈ C∗ , subject only to the balancing condition(5.3), which is equivalent to imposing that the coefficients A(z) and A(z −1 )actually belong to the field of elliptic functions Mp,1 .

DIFF. TRANSC. CRITERIA & ELLIPTIC HYPERGEOMETRIC FUNCTIONS

15

For this reason, we prove two related but distinct results on differential
transcendence: (A) differential transcendence of solutions of the elliptic hypergeometric equation (5.4), where we think of the εj as free parameters
subject only to the balancing condition (5.3) and without imposing the additional constraint ε8 = ε7 q; and (B) differential transcendence of the elliptic
hypergeometric functions fε (z) where the εj are defined in terms of the tj
as in (5.2), and where in particular we do impose the additional constraint
ε8 = ε7 q.
Note that in case (B) above the balancing condition (5.3) for the remaining
independent parameters ε1 , . . . , ε7 becomes


6
Y

(5.7)
εj  ε27 = p2 q.
j=1

In the next lemma we show that in case (B) there are no universal relations among the parameters ε1 , . . . , ε7 induced from the reparametrization(5.2), save for formal algebraic consequences of the balancing condition (5.7).
This result ensures that the hypothesis in case (B) of Theorem 5.6 and Theorem 5.7 below are not vacuous.
Lemma 5.4. Assume that case (B) holds. Every multiplicative relation
among the ε1 , . . . , ε7 , p, q is induced by (5.7), in the sense that if there areintegers α1 , . . . , α7 , m, n such that7Y
αεj j = pm q n ,j=1

then α1 = · · · = α6 = α = n and m = α7 = 2α for some α ∈ Z.
Proof.
√ Let us begin to write c and the εj in terms of the tj . We havec = t6 t7 , and
√
√
c2 p4
ε8
c
t6 t7
4
4
ε6 =
= cp t8 = p t6 t7 t8 , ε7 =
=
=
.
ε8
q
qt8
qt8
Assume now that there are integers α1 , . . . , α8 , m, n such that7Y
αεj j = pm q n .
j=1
7
Y
α
Let us write this equality in term of tj . The relation
εj j = pm q n gives
j=1


(5.8)

5
Y



q αj
 p4α6 (t6 t7 )(α6 +α7 )/2 tα8 6 −α7 q −α7 = pm q n .
αj /2 tαj
(t
t
)
6
7
j
j=1



Using the balancing condition

8Y

tj = p2 q 2 , we obtain the existence of an

j=1

integer α such thatα1 = · · · = α5 = α.

16

CARLOS E. ARRECHE, THOMAS DREYFUS, AND JULIEN ROQUES

Furthermore, regarding the terms in q, p, tj , j = 6, 7, and t8 respectively,
we find(5.9) n − 5α + α7 = −2α,m − 4α6 = −2α,− 5α/2 + α6 /2 + α7 /2 = −α,α6 − α7 = −α.

If we put the equality of the fourth relation α6 = α7 − α into the third, we
obtain 2α = α7 . With α6 = α7 − α, we find α = α6 . Finally from the first
and the second equality, we deduce n = α and m = 2α.
Remark 5.5. Assume that case (B) holds. With Lemma 5.4 and the relation
ε8 = qε7 it follows that if there are integers α1 , . . . , α8 , m, n such that8Y
αεj j = pm q n
j=1

then α1 = · · · = α6 = α = n − α8 and α7 + α8 = 2α = m for some α ∈ Z.
5.3. Irreducibility of the σ-Galois group of the elliptic hypergeometric function. From now on, we denote by G the σ-Galois group of(5.6) over K (with respect to some σ-PV ring).
Theorem 5.6. Assume one of the two hypotheses (A) or (B) below.
(A) Every multiplicative relation among the ε1 , . . . , ε8 , p, q is induced by(5.3), in the sense that if there are integers α1 , . . . , α8 , m, n such that8Y
αεj j = pm q n
j=1

then α1 = · · · = α8 =: α and m = n = 2α for some α ∈ Z.
(B) ε8 = ε7 q and every multiplicative relation among the ε1 , . . . , ε7 , p, q is
induced by (5.7), in the sense that if there are integers α1 , . . . , α8 , m, n such
that
8
Y
α
εj j = p m q n
j=1

then α1 = · · · = α6 = α = n − α8 and α7 + α8 = 2α = m for some α ∈ Z.
Then G is irreducible.
Proof. To the contrary, assume that G is reducible. According to Theorem 4.6, the following Riccati equation has a solution u ∈ Mp,2 :(5.10)uσ(u) + au + b = 0.

First, note that u ∈ Mp,2 is a solution of (5.10) if and only ifv(σ(v) + σ −1 (a)) + σ −1 (b) = 0 with v = σ −1 (u) ∈ K. Then to simplify the expression of the divisors of a and b, we may replace them by
−1 )
−1 )
, σ −1 (b) = A(zσ −1 (a) = ν−A(z)−A(zA(z)A(z) , and consider the Riccati equation
satisfied by v. Consider p1 ∈ Θ2 ∪ {0} and p2 , p3 ∈ Θ2 such thatp1p2
σ −1 (a) =and σ −1 (b) = .
p3
p3
In view of the explicit expressions for σ −1 (a) and σ −1 (b), we see that we
may take p2 and p3 such that

DIFF. TRANSC. CRITERIA & ELLIPTIC HYPERGEOMETRIC FUNCTIONS

div2 (p2 ) =17

8X√   √  √   √ 
εj + − εj + pεj + − pεj
j=1

+

  q
 q
  q

3 q
X
4
pj /q + − 4 pj /q + i 4 pj /q + −i 4 pj /q
j=0

and
  q
 q
  q

8 q
X
1/εj + − 1/εj +p/εj + − p/εj
div2 (p3 ) =
j=1

+

3 hp
i h p i hp i h p i
X4
qpj + − 4 qpj + i 4 qpj + −i 4 qpj .
j=0

We note for convenience that
  q
 q
  q

8 q
X
−1
q/εj + − q/εj +qp/εj + − qp/εj
div2 (σ (p3 )) =
j=1

+

3 h
i h √ p i h√ p i h √ p i
X√ pq 4 qpj + − q 4 qpj + i q 4 qpj + −i q 4 qpj .
j=0

We now consider r0 , r1 , r2 ∈ Θ2 as in Theorem 4.6. For i = 1, 2, let
Si := {λ ∈ C∗2 /pZ | ordλ (ri ) 6= 0}
denote the support of div2 (ri ). For each j ∈ {1, . . . , 8} we let αj ∈ N denote√√the number of points in S1 of the form ± εj or ± pεj . Similarly, for eachj ∈ {1, . . . , 8} we let αj0 ∈ N denote the number of points in S2 of the formpp
± q/εj or ± qp/εj . We find that there exist `1 , `2 ∈ {0, 1, 2, 3} and γ ∈ Nsuch that8√√ Y √ αj +α0j √ − deg2 (r2 ) √4 −γ
ω2 (r1 /r2 ) = i`1 4 p`2εjqq = q deg2 (r0 )mod pZ ,j=1

where the second equality is obtained from property (iv) of Theorem 4.6.
After taking fourth powers we see that
(5.11)8Y
2αj +2α0j
εj= pm q 2 deg2 (r2 )+γ+2 deg2 (r0 )j=1

for some m ∈ Z. We now claim that v is constant.
Suppose first that we are in case (A). Since every multiplicative relation
among the ε1 , . . . , ε8 , p, q is induced by (5.3), it follows from (5.11) that
there exists α ∈ N such that 2αj + 2αj0 = α for every j ∈ {1, . . . , 8} and m =2 deg2 (r2 )+γ +2 deg2 (r0 ) = 2α. In particular, we have that 2 deg2 (r2 ) ≤ 2α.
On the other hand, it follows from properties (i) and (ii) of Theorem 4.6,
respectively, that α1 + · · · + α8 ≤ deg2 (r1 ) and α10 + · · · + α80 ≤ deg2 (r2 ). We

18

CARLOS E. ARRECHE, THOMAS DREYFUS, AND JULIEN ROQUES

note that by property (iii) of Theorem 4.6 2 deg2 (r2 ) = deg2 (r1 ) + deg2 (r2 ).
Putting together these inequalities we obtain
4α =

8
X

αj + αj0 ≤ deg2 (r1 ) + deg2 (r2 ) = 2 deg2 (r2 ) ≤ 2α.

j=1

It follows from this that α = deg2 (r1 ) = deg2 (r2 ) = 0. Hence, r1 /r2 isconstant and√ω2 (r1 /r2 ) = 1 = q deg2 (r0 ) mod pZ
by property (iv) of Theorem 4.6. Since pZ ∩ q Z = {1}, we see thatdeg2 (r0 ) = 0 also.
Now suppose we are in case (B). Since every multiplicative relation amongthe ε1 , . . . , ε8 , p, q is induced by (5.7), it follows from (5.11) that there exists
α ∈ N such that 2αj + 2αj0 = α = 2deg2 (r2 ) + γ + 2deg2 (r0 ) − (2α8 + 2α80 )for every j ∈ {1, . . . , 6}, and 2α7 + 2α70 + 2α8 + 2α80 = 2α = m. It follows
from the second set of equations that 2α8 + 2α80 ≤ 2α. From this and thefirst set of equations it then follows that 2deg2 (r2 ) ≤ 3α. On the other
hand, it follows from properties (i) and (ii) of Theorem 4.6, respectively,that α1 + · · · + α8 ≤ deg2 (r1 ) and α10 + · · · + α80 ≤ deg2 (r2 ). We note thatby property (iii) of Theorem 4.6, 2 deg2 (r2 ) = deg2 (r1 ) + deg2 (r2 ). Putting
together these inequalities we obtain
4α =

8
X

αj + αj0 ≤ deg2 (r1 ) + deg2 (r2 ) = 2 deg2 (r2 ) ≤ 3α.

j=1

It follows from this that α = deg2 (r1 ) = deg2 (r2 ) = 0. Hence, r1 /r2 isconstant and√ω2 (r1 /r2 ) = 1 = q deg2 (r0 ) mod pZ
by property (iv) of Theorem 4.6. Since pZ ∩ q Z = {1}, we see thatdeg2 (r0 ) = 0 also.
It follows from the above in either of the cases (A) or (B) that v ∈ C∗ isconstant. Therefore (5.10) can be rewritten as(5.12)

v 2 A(z) + v(ν − A(z) − A(z −1 )) + A(z −1 ) = 0,i.e.
(5.13)(v 2 − v)A(z) + vν = (v − 1)A(z −1 ).

√But since q −1 is a pole of A(z) but not of A(z −1 ) and, on the other hand,√q is a pole of A(z −1 ) but not of A(z), we obtain that v 2 −v = v−1 = vν = 0.
So we must have ν = 0. On the other hand, we see from the definitionof ν in (5.5) that ν = 0 if and only if εj ε8 = qp` for some ` ∈ Z and
j = 1, . . . , 6, which is ruled out by our hypotheses in both cases (A) and(B). This contradiction concludes the proof that G is irreducible.

5.4. Differential transcendence of the elliptic hypergeometric funcd
tions. We may equip (K, σ) with the classical derivation δ := z dz
as ine
[DHRS18, Section 3.1]. Note that δ commutes with σ. Let C be the δeclosure of C. Following Lemma 3.1, we may consider L := Frac(K ⊗C C)
σ∗eand we have L = C. Recall that fε (z) is meromorphic on C , and notethat the field of meromorphic functions on C∗ is a (σ, δ)-extension of K.

DIFF. TRANSC. CRITERIA & ELLIPTIC HYPERGEOMETRIC FUNCTIONS

19

Theorem 5.7. Assume one of the two hypotheses (A) or (B) below.
(A) Every multiplicative relation among the ε1 , . . . , ε8 , p, q is induced by(5.3), in the sense that if there are integers α1 , . . . , α8 , m, n such that8Y
αεj j = p m q n
j=1

then α1 = · · · = α8 =: α and m = n = 2α for some α ∈ Z.
(B) ε8 = ε7 q and every multiplicative relation among the ε1 , . . . , ε7 , p, q is
induced by (5.7), in the sense that if there are integers α1 , . . . , α8 , m, n such
that
8
Y
α
εj j = p m q n
j=1

then α1 = · · · = α6 = α = n − α8 and α7 + α8 = 2α = m for some α ∈ Z.
Then any non-zero solution to (5.4) is differentially transcendental
over K.
Proof. We apply the criteria of Theorem 3.4. We proved in Theorem 5.6
that G is irreducible, which by [DR15, Lemma 13] is equivalent to the nonexistence of a solution u ∈ K to the Riccati equationuσ(u) + au + b = 0.
It remains to show that there is no nonzero linear differential operator L in
δ with coefficients in C and g ∈ K such that δb
= σ(g) − g.
Lb
Let k ∈ N∗ such that g ∈ Mp,k and consider b as an element of Mp,k . Letω ∈ C∗k /pZ be a zero or a pole of b. Then it is a pole of δb. Since L hasbδb
constant coefficients, we get that ω is also a pole of L b . Therefore, ω
is a pole of σ(g) − g and hence also a pole of σ(g) or of g. Furthermore,
σ(g)−g has at least two distinct poles ω 0 , ω 00 ∈ C∗k /pZ such that ω ≡ ω 0 ≡ ω 00
mod qkZ , where qk ∈ C∗k is as in (4.2). These ω 0 and ω 00 are poles of δbb , andhence zeros or poles of b has well. We have proved that, for every ω ∈ C∗k /pZ
that is a pole or zero of b, there exists ` ∈ Z6=0 such that ωqk` is a pole orzero of b.
Let us now consider b as an element of Mp,1 . From the preceding, we
deduce that for every ω ∈ C∗ /pZ , pole or zero of b, there exists ` ∈ Z6=0 suchthat ωq ` is a pole or zero of b. We will use this to find a contradiction. Note8Y
θ(εj q −1 z −1 ; p)θ(q 2 z 2 ;p)θ(q 3 z 2 ;p)that the set of zeros or poles of b = θ(q−2 z −2 ;p)θ(q−1 z −2 ;p) ×,θ(εj qz; p)j=1

seen as an element of Mp,1 , is included in
√
√
−1 ±1
−1/2
S = {q −1 ε±1, ±q −1/2 p, ±q −3/2 , ±q −3/2 p} mod pZ .
1 , . . . , q ε8 , ±q
Let us prove that the elements of S are all distinct. To see this, notethat if any two elements of S were the same modulo pZ then we wouldfind a non-trivial multiplicative relation satisfied by at most four elements
among p, q, ε1 , . . . , ε8 . This contradicts the hypothesis in both cases (A)20

CARLOS E. ARRECHE, THOMAS DREYFUS, AND JULIEN ROQUES

and (B). Therefore, no simplifications occur and S is exactly the set ofzeros or poles of b. It suffices to show that for all ` ∈ Z6=0 , we haveS ∩ {q ` q −1 ε1 mod pZ } = ∅. Let ` ∈ Z such that S ∩ {q ` q −1 ε1 mod pZ } 6=∅. If ` 6= 0, then we again find a non-trivial multiplicative relation satisfied
by at most four elements among p, q, ε1 , . . . , ε8 . In either case (A) or case(B) this contradiction to the hypothesis concludes the proof.

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The University of Texas at Dallas, Mathematical Sciences FO 35, 800 WestCampbell Road, Richardson, TX 75024, USA
E-mail address: arreche@utdallas.edu
Institut de Recherche Mathématique Avancée, U.M.R. 7501 Université deStrasbourg et C.N.R.S. 7, rue René Descartes 67084 Strasbourg, FRANCE
E-mail address: dreyfus@math.unistra.fr
Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut
Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex,
FranceE-mail address: roques@math.univ-lyon1.fr