# Algebraic independence and difference equations over elliptic function
fields

Ehud de Shalit  Einstein Institute of Mathematics, The Hebrew Univeristy of
Jerusalem<EMAIL_ADDRESS>
(Date: July 27, 2022)

###### Abstract.

For a lattice $\Lambda$ in the complex plane, let $K_{\Lambda}$ be the field
of $\Lambda$-elliptic functions. For two relatively prime integers $p$
(respectively $q$) greater than 1, consider the endomorphisms $\psi$ (resp.
$\phi)$ of $K_{\Lambda}$ given by multiplication by $p$ (resp. $q$) on the
elliptic curve $\mathbb{C}/\Lambda$. We prove that if $f$ (resp. $g$) are
complex Laurent power series that satisfy linear difference equations over
$K_{\Lambda}$ with respect to $\phi$ (resp. $\psi$) then there is a dichotomy.
_Either_ , for some sublattice $\Lambda^{\prime}$ of $\Lambda,$ one of $f$ or
$g$ belongs to the ring
$K_{\Lambda^{\prime}}[z,z^{-1},\zeta(z,\Lambda^{\prime})]$, where
$\zeta(z,\Lambda^{\prime})$ is the Weierstrass zeta function, _or_ $f$ and $g$
are algebraically independent over $K_{\Lambda}.$ This is an elliptic analogue
of a recent theorem of Adamczewski, Dreyfus, Hardouin and Wibmer (over the
field of rational functions).

###### Key words and phrases:

Difference equations, elliptic functions, algebraic independence

###### 2000 Mathematics Subject Classification:

12H10, 14H52, 39A10

##  1\. Introduction

###  1.1. Background, over fields of rational functions

A $\phi$-field is a field $K$ equipped with an endomorphism $\phi.$ The fixed
field $C=K^{\phi}$ of $\phi$ is called _the field of constants_ of $K.$
Throughout this paper we shall only consider ground fields which are
_inversive:_ $\phi$ is an automorphism of $K,$ but for a general extension of
$K$ we do not impose this condition. Let $(K,\phi)\subset(F,\phi)$ be an
extension of $\phi$-fields (written from now on $K\subset F$), which is also
inversive, and with the same field of constants:

$C=K^{\phi}=F^{\phi}.$

Denote by $S_{\phi}(F/K)$ the collection of all $u\in F$ which satisfy a
linear homogenous $\phi$-difference equation

(1.1) $a_{0}\phi^{n}(u)+a_{1}\phi^{n-1}(u)+\cdots+a_{n}u=0,$

with coefficients $a_{i}\in K.$ The set $S_{\phi}(F/K)$ is a $K$-subalgebra of
$F$.

Suppose now that $K$ and $F$ are endowed with a second automorphism $\psi,$
commuting with $\phi,$ and that $\mathrm{tr.deg.}(K/C)\leq 1.$ Various results
obtained in recent years support the philosophy that _if $\phi$ and $\psi$ are
sufficiently independent, the $K$-algebras $S_{\phi}(F/K)$ and $S_{\psi}(F/K)$
are also “independent” in an appropriate sense._

Here are some classical examples. Let $C$ be an algebraically closed field of
characteristic 0. We consider three classes of examples where

  * •

(2S) $K=C(x),\,\,F=C((x^{-1})),\,\,\phi f(x)=f(x+h),\,\,\psi f(x)=f(x+k),$
($h,k\in C$),

  * •

(2Q) $K=C(x),\,\,F=C((x)),\,\,\phi f(x)=f(qx),\,\,\psi f(x)=f(px)$ ($p,q\in
C^{\times})$,

  * •

(2M)
$K=\bigcup_{s=1}^{\infty}C(x^{1/s}),\,\,F=\bigcup_{s=1}^{\infty}C((x^{1/s}))$
(the field of Puiseux series), $\phi f(x)=f(x^{q}),\,\,\psi f(x)=f(x^{p})$
($p,q\in\mathbb{N}$).

Note that $\phi$ and $\psi$ are indeed _auto_ morphisms. In all three
examples, the assumption that $\phi$ and $\psi$ are “sufficiently independent”
is that the group

$\Gamma=\left\langle\phi,\psi\right\rangle\subset\mathrm{Aut}(K)$

is free abelian of rank 2. The letters S,Q and M stand for “shifts”,
“$q$-difference operators” and “Mahler operators” respectively, and the
independence assumption gets translated into the additive independence of $h$
and $k$ in the case (2S), and the multiplicative independence of $p$ and $q$
in the cases (2Q) and (2M). Schäfke and Singer proved in [Sch-Si] the
following theorem, confirming the above philosophy.

######  Theorem 1.

Assume that $\Gamma$ is free abelian of rank 2. Then in any of the three cases
(2S), (2Q) or (2M)

$S_{\phi}(F/K)\cap S_{\psi}(F/K)=K.$

Some instances of this theorem were known before. The case (2Q) dates back to
the work of Bézivin and Boutabaa [Bez-Bou], and the case (2M), originally a
conjecture of Loxton and van der Poorten [vdPo], was proved by Adamczewski and
Bell in [Ad-Be]. The earlier proofs, however, used a variety of ad-hoc
techniques, and only [Sch-Si] gave a unified treatment, revealing the common
principles behind these theorems. This new approach enabled the authors to
prove a few more theorems of the same nature, dealing with power series
satisfying simultaneously a $\phi$-difference equation and a linear ordinary
differential equation. See, also, the exposition in [dS-G], where we removed
some unnecessary restrictions on the characteristic of the field $C$ and on
$|p|$ and $|q|$, in the case (2Q). In addition, this last paper deals for the
first time with a case, denoted there (1M1Q), in which $\phi$ is a
$q$-difference operator and $\psi$ a $p$-Mahler operator, and the resulting
group $\Gamma$ is generalized dihedral rather than abelian. The formulation of
the analogous result has to be cast now in the language of _difference
modules_ , the classical language of equations being inadequate when $\Gamma$
is non-abelian. Modulo this remark, however, the main result _and_ its proof
are very similar, if not identical, to the above three cases.

The next breakthrough occured in a recent paper by Adamczewski, Hardouin,
Dreyfus and Wibmer [A-D-H-W]. Building on an earlier work [A-D-H] dealing with
difference-differential systems, these authors obtained a far-reaching
strengthening of the above theorem.

######  Theorem 2.

Consider any of the three cases (2S), (2Q) or (2M). Let $f\in S_{\phi}(F/K)$
and $g\in S_{\psi}(F/K)$. If $f,g\notin K$ then $f$ and $g$ are algebraically
independent over $K$.

Letting $f=g$ one recovers Theorem 1. The key new tool which allows to upgrade
Theorem 1 to Theorem 2 is the “parametrized” Picard-Vessiot theory, as
developed in [H-S, O-W]. We shall elaborate on this theory and summarize its
main ingredients in section 3.

###  1.2. Background, over fields of elliptic functions

####  1.2.1. The case (2Ell)

In [dS1, dS2] we initiated the study of the same theme over fields of elliptic
functions. For a lattice $\Lambda\subset\mathbb{C}$ let $E_{\Lambda}$ stand
for the elliptic curve whose associated Riemann surface is
$\mathbb{C}/\Lambda$ and

$K_{\Lambda}=\mathbb{C}(\wp(z,\Lambda),\wp^{\prime}(z,\Lambda))$

its function field, the field of $\Lambda$-periodic meromorphic functions on
$\mathbb{C}.$ Fix $\Lambda_{0}$ and let

$K=\bigcup_{\Lambda\subset\Lambda_{0}}K_{\Lambda}.$

This is the function field of the universal cover of $E_{\Lambda_{0}},$ and
should be compared to the field $K$ in the case (2M), which is the function
field of the universal cover of $\mathbb{G}_{m}$. Let $p,q\in\mathbb{N}.$
Multiplication by $p$ or $q$ induces an endomorphism of $E_{\Lambda}$ for each
$\Lambda,$ and _automorphisms_ of the field $K$ given by

$\phi f(z)=f(qz),\,\,\psi f(z)=f(pz).$

For $F$ we take the field of Laurent series $\mathbb{C}((z))$ with the same
$\phi$ and $\psi.$ Via the Taylor-Maclaurin expansion at 0, $K\subset F.$ We
label this choice of $(K,F,\phi,\psi)$ by (2Ell).

####  1.2.2. The ring $S$

To formulate our results we need to introduce a ring slightly larger than $K,$
namely the ring

$S=K[z,z^{-1},\zeta(z,\Lambda)]\subset F$

generated over $K$ by $z,$$z^{-1}$ and the Weierstrass zeta function
$\zeta(z,\Lambda).$ Recall that the latter is a primitive of $-\wp(z,\Lambda)$
and satisfies, for $\omega\in\Lambda$,

$\zeta(z+\omega,\Lambda)-\zeta(z,\Lambda)=\eta(\omega,\Lambda),$

where the additive homomorphism $\eta(\cdot,\Lambda):\Lambda\to\mathbb{C}$ is
Legendre’s eta function. It is easy to see that the ring $S$ does not depend
on which $\Lambda\subset\Lambda_{0}$ we use: once we adjoin one
$\zeta(z,\Lambda),$ they are all in $S$. It is also easy to see that $\phi$
and $\psi$ induce automorphisms of $S$.

####  1.2.3. Previous results in the case (2Ell)

In [dS2], the following analogue of Theorem 1 was proved:

######  Theorem 3.

Assume that $2\leq p,q$ and $(p,q)=1$. Then in the case (2Ell) we have

$S_{\phi}(F/K)\cap S_{\psi}(F/K)=S.$

######  Remark.

(i) The reader should note the assumption on $p$ and $q$ being relatively
prime integers $\geq 2.$ This is stronger than assuming $p$ and $q$ to be only
multiplicatively independent. This stronger assumption was needed in only one
lemma of [dS2], but so far could not be avoided.

(ii) The case (2Ell) brings up two completely new issues, absent from the
rational cases discussed so far. One is the issue of _periodicity_. The method
of [Sch-Si] starts with a formal analysis of the solutions to our $\phi$\- and
$\psi$-difference equations at common fixed points of $\phi$ and $\psi.$ Using
estimates on coefficients in Taylor expansions one shows that certain formal
power series converge in some open disks around these fixed points. Using the
difference equations one writes down a functional equation for these
functions, that allows to continue them meromorphically all the way up to a
“natural boundary”. While each of the three cases (2S), (2Q) and (2M) has its
own peculiarities, and is technically different, the upshot in all three cases
is that a certain matrix with meromorphic entries is proved to be _globally_
meromorphic on $\mathbb{P}^{1}(\mathbb{C}),$ hence a matrix with entries in
$K=\mathbb{C}(x).$ This matrix is used to descend a certain difference module
attached to our system of equations from $K$ to $\mathbb{C},$ and this leads
to a proof of Theorem 1.

In the case (2Ell) the analysis of the situation starts along the same lines.
However, the matrix of globally meromorphic functions on $\mathbb{C}$ thus
produced bears, a priori, no relation to the lattices $\Lambda.$ It starts its
life as a matrix of formal power series, convergent in some disk
$|z|<\varepsilon,$ and is then continued meromorphically using a functional
equation with respect to $z\mapsto qz,$ losing the connection to the lattices.
In fact, examples show that this matrix _need not be_ a matrix of elliptic
functions.

The Periodicity Theorem of [dS1], and its vast generalization in [dS2], show
that just enough of the periodicity can be salvaged to push this approach to
an end. A certain generalization of the “baby case” of this theorem,
considered in [dS1], will be instrumental in the present work, when we deal
with equations of the first order.

(iii) The second new issue in the case (2Ell) has to do with the emergence of
certain _vector bundles_ over the elliptic curve $E_{\Lambda}$, that we
associate to our system of difference equations. Luckily, vector bundles over
elliptic curves have been fully analyzed in Atiyah’s work [At]. Their
classification allows us to understand the ($\phi,\psi)$-difference modules
associated to an $f\in S_{\phi}(F/K)\cap S_{\psi}(F/K)$. The ensuing structure
theorem for elliptic $(\phi,\psi)$-difference modules is the main theorem of
[dS2], and Theorem 3 is a corollary of it. The need to include
$\zeta(z,\Lambda)$ in $S$ reflects the non-triviality of these vector bundles.
Over the field $\mathbb{C}(z)$ none of this shows up, essentially because
every vector bundle over $\mathbb{G}_{a}$ or $\mathbb{G}_{m}$ is trivial.

###  1.3. The main results

Our main result is an elliptic analogue of Theorem 2 (Theorem 1.3 of [A-D-
H-W]). In fact, both our result and Theorem 2 admit a mild generalization. Let
$AS_{\psi}(F/K)$ be the collection of all $u\in F$ for which there exists an
$n\geq 0$ such that

$\psi^{n}(u)\in K(u,\psi(u),\dots,\psi^{n-1}(u)).$

Clearly $S_{\psi}(F/K)\subset AS_{\psi}(F/K).$

######  Theorem 4.

Let $(K,F,\phi,\psi)$ be as in case (2Ell) and assume that $2\leq p,q$ and
$(p,q)=1.$ Let $f\in S_{\phi}(F/K)$ and $g\in AS_{\psi}(F/K)$. If $f,g\notin
S,$ then $f$ and $g$ are algebraically independent over $K$.

The proof follows the strategy of [A-D-H-W]. Theorem 4 will be deduced from
the following analogue of Theorem 4.1 there, which concerns a _single_ power
series _ $f\in F.$_

######  Theorem 5.

Let $(K,F,\phi,\psi)$ be as in case (2Ell) and assume that $2\leq p,q$ and
$(p,q)=1.$ Let $f\in S_{\phi}(F/K)$ and assume that $f\notin S.$ Then
$\\{f,\psi(f),\psi^{2}(f),\dots\\}$ are algebraically independent over $K$.

To explain the input from our earlier work, we have to formally introduce the
notion of a difference module, to which we already alluded several times. A
_$\phi$ -difference module _$(M,\Phi)$ over $K$ (called, in short, a
$\phi$-module) is a finite dimensional $K$-vector space $M$ equipped with a
$\phi$-linear bijective endomorphism $\Phi$. Its _rank_ $\mathrm{rk}(M)$ is
the dimension of $M$ as a $K$-vector space. The set of $\Phi$-fixed points
$M^{\Phi}$ is a $C$-subspace of dimension $\leq\mathrm{rk}(M).$

Since $\psi$ commutes with $\phi,$ the module

$M^{(\psi)}=(K\otimes_{\psi,K}M,1\otimes\Phi)$

is another $\phi$-module. Our $M$ is called $\psi$_-isomonodromic_(or
$\psi$-integrable) if $M\simeq M^{(\psi)}$.

To any $\phi$-difference equation (1.1) one can attach a $\phi$-module $M$ of
rank $n$ whose fixed points $M^{\Phi}$ correspond to the solutions of the
equation in $K$. This is classical, and explained in section 2.2 below. For
this reason we shall refer to $M^{\Phi}$ also as the space of “solutions” of
$M.$

To any $\phi$-module $M$ of rank $n$ over $K$ one can associate a _difference
Galois group_ $\mathcal{G}$, which is a Zariski closed subgroup of
$GL_{n,\mathbb{C}}$, uniquely determined up to conjugation (and reviewed in
section 2.5 below). This linear algebraic group measures the algebraic
relations that exist between the solutions of $M$, not over $K$ itself (where
there might be none, or too few solutions), but after we have base-changed to
a suitable universal extension - the Picard-Vessiot extension - in which a
full set of solutions can be found. The larger $\mathcal{G}$ is, the fewer
such relations exist. The analogy with classical Galois theory, in which the
Galois group measures the algebraic relations between the roots of a
polynomial in a splitting field, is obvious.

The input, deduced from the main theorem of [dS2], needed in the proof of
Theorem 5, is the following. We continue to assume that $2\leq p,q$ and
$(p,q)=1.$

######  Theorem 6.

Assume that $M$ is $\psi$-isomonodromic. Then its difference Galois group
$\mathcal{G}$ is solvable.

In addition, we shall need a generalization of the “baby” Periodicity Theorem
of [dS1], explained in section 4.2.

Except for these two results, the rest of the proof of Theorems 4 and 5
imitates [A-D-H-W]. As this depends on results scattered through many
references [A-D-H, A-D-H-W, D-H-R, DV-H-W1, DV-H-W2, O-W], we shall make an
effort to collect all the prerequisites in a way that facilitates the reading.

###  1.4. Outline of the paper

Section 2 will be devoted to generalities on difference equations, difference
modules, Picard-Vessiot extensions and the difference Galois group. The
standard reference here is [S-vdP], although our language will sometimes be
different.

Section 3 will be devoted to the more recent theory of _parametrized_ Picard-
Vessiot theory and the _parametrized_ difference Galois group, to be found in
the references cited above.

In Section 4 we shall prove the two results that we need as input in the proof
of Theorem 5, relying on [dS1, dS2].

Section 5 will start by explaining how to deduce Theorem 4 from Theorem 5. We
shall then carry out the proof of Theorem 5, following the program of [A-D-
H-W].

##  2\. Review of classical Picard-Vessiot theory

###  2.1. The ground field

Let $K$ be defined as in section 1.2, in the case (2Ell). We shall need the
following facts about it.

######  Proposition 7.

(i) $K$ is a $C^{1}$ field (any homogenous polynomial of degree $d$ in $n>d$
variables has a nontrivial zero in $K$).

(ii) If $G$ is a connected linear algebraic group over $K$ then any $G$-torsor
over $K$ is trivial.

(iii) $K$ does not have any non-trivial finite extension $L/K$ to which $\phi$
(or $\psi$) extends as an automorphism.

###### Proof.

(i) It is enough to prove the claim for every $K_{\Lambda}$, where this is
Tsen’s theorem: the function field of any curve over an algebraically closed
field of characteristic 0 is a $C^{1}$-field.

(ii) This is Springer’s theorem: a $C^{1}$-field of characteristic 0 is of
cohomological dimension $\leq 1$. By Steinberg’s theorem this implies that
every torsor of a connected linear algebraic group over $K$ is trivial. See
[Se] ch. III.2.

(iii) (Compare [D-R], Proposition 6). Suppose $L$ is a finite extension of $K$
to which $\phi$ extends as an automorphism. Then, for $\Lambda$ small enough,
$L=L_{\Lambda}K$ where $L_{\Lambda}$ is an extension of $K_{\Lambda}$,
$[L:K]=[L_{\Lambda}:K_{\Lambda}]$. Let $\Lambda^{\prime}\subset\Lambda$ and
$L_{\Lambda^{\prime}}=L_{\Lambda}K_{\Lambda^{\prime}}.$ Then for
$\Lambda^{\prime}$ sufficiently small $\psi(L_{\Lambda})\subset
L_{\Lambda^{\prime}}$. Replacing $\Lambda$ by $\Lambda^{\prime}$ we may
therefore assume that $\psi(L_{\Lambda})\subset L_{\Lambda}.$ Thus $\psi$
extends to an endomorphism of $L_{\Lambda}.$ Let $\pi:Y\to E_{\Lambda}$ be the
covering of complete nonsingular curves corresponding to $L_{\Lambda}\supset
K_{\Lambda}$ and $\alpha:Y\to Y$ the morphism inducing $\psi$ on
$L_{\Lambda}.$ Since $\pi\circ\alpha=[q]\circ\pi$ we get that
$\deg(\alpha)=q^{2}.$ By the Riemann-Hurwitz formula

$2g_{Y}-2=(2g_{Y}-2)q^{2}+\sum_{x\in Ram(\alpha)}(e_{x}-1)$

where $g_{Y}\geq 1$ is the genus of $Y$ and $Ram(\alpha)$ the ramification
locus of $\alpha$, $e_{x}$ being the ramification index. This equation can
only hold if $g_{Y}=1$ (and $\alpha$ is everywhere unramified). In particular,
$\pi$ is an isogeny of elliptic curves, hence $L_{\Lambda}\subset K$ and
$L=K.$ ∎

Define

$S=K[z,z^{-1},\zeta(z,\Lambda)]\subset F.$

If $\Lambda^{\prime}\subset\Lambda$ is another lattice then

$\wp(z,\Lambda)-\sum_{\omega\in\Lambda/\Lambda^{\prime}}\wp(z+\omega,\Lambda^{\prime})$

is a meromorphic $\Lambda$-periodic function. Its poles are contained in
$\Lambda$, but at 0 the poles of $\wp(z,\Lambda)$ and of
$\wp(z,\Lambda^{\prime})$ cancel each other, while the other terms have no
pole. It follows that this $\Lambda$-periodic function has no poles, hence is
a constant. Integrating, we find that

$\zeta(z,\Lambda)-\sum_{\omega\in\Lambda/\Lambda^{\prime}}\zeta(z+\omega,\Lambda^{\prime})=az+b$

for some $a,b\in\mathbb{C}.$ On the other hand
$\zeta(z+\omega,\Lambda^{\prime})-\zeta(z,\Lambda^{\prime})\in
K_{\Lambda^{\prime}}\subset K.$ It follows that

$\zeta(z,\Lambda)-[\Lambda:\Lambda^{\prime}]\zeta(z,\Lambda^{\prime})\in
K[z,z^{-1}].$

This shows that the definition of $S$ does not depend on which
$\Lambda\subset\Lambda_{0}$ we use. Since for any rational number $r=m/n$
$\zeta(rz,\Lambda)-r\zeta(z,\Lambda)\in K_{n\Lambda}\subset K,$ $\phi$ and
$\psi$ induce automorphisms of $S$.

######  Problem.

Does the field of fractions of $S$ satisfy Proposition 7?

###  2.2. Difference equations, difference systems and difference modules

In this subsection and the next ones, the $\phi$-field $(K,\phi)$ can be
arbitrary. The standard reference is [S-vdP]. As usual, to the difference
equation (1.1) we associate the companion matrix

$A=\left(\begin{array}[]{ccccc}0&1\\\ &0&1\\\ &&\ddots\\\ &&&0&1\\\
-a_{n}/a_{0}&&\cdots&&-a_{1}/a_{0}\end{array}\right),$

and the first order linear system of equations

(2.1) $\phi(Y)=AY$

for which we seek solutions $Y=\,^{t}(u_{1},\dots,u_{n})$ in $\phi$-ring
extensions $L$ of $K$. Notice that if $u$ is a solution of (1.1) then
${}^{t}(u,\phi(u),\dots,\phi^{n-1}(u))$ is a solution of (2.1).

From now on we concentrate on first order systems of equations of the form
(2.1) with $A\in GL_{n}(K)$ _arbitrarily given_.

With the system (2.1) we associate the $\phi$-difference module
$M=(K^{n},\Phi)$ where

$\Phi(v)=A^{-1}\phi(v).$

Notice that a solution $v\in K^{n}$ to (2.1) is nothing but an element of
$M^{\Phi},$ the fixed points of $\Phi$ in $M$. This is a
$C=K^{\phi}$-subspace, and the well-known Wronskian Lemma shows that

$\dim_{C}M^{\Phi}\leq\mathrm{rk}M.$

By abuse of language, we shall refer to $M^{\Phi}$ also as the space of
“solutions” of $M$.

An equality $\dim_{C}M^{\Phi}=\mathrm{rk}M$ holds if and only if a full set of
solutions of (2.1) exists in $K$, if and only if $M$ is isomorphic to the
trivial module $(K^{n},\phi).$ In such a case a matrix $U\in GL_{n}(K)$
satisfying

$\phi(U)=AU$

is called a _fundamental matrix_ for (2.1). Its columns form a basis of
$M^{\Phi}$ over $C$.

Given any $\phi$-module, a choice of a basis of $M$ over $K$ shows that it is
of the above form. A choice of another basis results in a _gauge
transformation_ , replacing $A$ with

$A^{\prime}=\phi(P)^{-1}AP$

for some $P\in GL_{n}(K).$ Conversely, the systems of equations defined by $A$
and by $A^{\prime}$ are equivalent if and only if $A$ and $A^{\prime}$ are
gauge equivalent. The transition from a system of equations to a $\phi$-module
can therefore be reversed. Thanks to Birkhoff’s cyclicity lemma, the
transition from a single linear equation of order $n$ to a system of equations
of order one can also be reversed. The three notions are therefore equivalent,
and which language one chooses to work with is very much a matter of taste.

###  2.3. Isomonodromy and $(\Phi,\Psi)$-difference modules

Assume now that the $\phi$-field $K$ is endowed with a second automorphism
$\psi$, _commuting_ with $\phi$. If $(M,\Phi)$ is a $\phi$-module over $K$,
then

$M^{(\psi)}=(K\otimes_{\psi,K}M,1\otimes\Phi)$

is another $\phi$-module, called the $\psi$_-transform_ of $M$. If
$M=(K^{n},\Phi)$ with $\Phi(v)=A^{-1}\phi(v)$ then $M^{(\psi)}$ is likewise
given by the matrix $\psi(A).$

The notion of a $(\phi,\psi)$-difference module is naturally defined. It is a
finite dimensional vector space $M$ over $K$ equipped with bijective
$\phi$-linear (resp. $\psi$-linear) endomorphisms $\Phi$ (resp. $\Psi$)
commuting with each other: $\Phi\circ\Psi=\Psi\circ\Phi$.

######  Lemma 8.

For a $\phi$-module $M$ over $K$ of rank $n$, the following are equivalent:

(i) $M^{(\psi)}\simeq M$ as a $\phi$-module.

(ii) $M$ admits a structure of a $(\phi,\psi)$-module extending the given
$\phi$-module structure.

(iii) If $A=A_{\phi}$ is the matrix associated to $M$ in some basis, there
exists a matrix $A_{\psi}\in GL_{n}(K)$ satisfying the compatibility condition

$\phi(A_{\psi})A_{\phi}=\psi(A_{\phi})A_{\psi}.$

The proof is left as an easy exercise. A $\phi$-module satisfying the above
conditions is called $\psi$-_isomonodromic_ (or $\psi$-integrable). Property
(iii) shows that the definition is symmetric: If $\left(M,\Phi\right)$ is
$\psi$-isomonodromic and $\Psi$ is the $\psi$-linear operator as in (ii), then
$(M,\Psi)$ is $\phi$-isomonodromic as a $\psi$-module. The terminology is
derived from the differential set-up, of which the theory of difference
equations is a discrete analogue.

###  2.4. Picard-Vessiot theory

####  2.4.1. Picard-Vessiot rings and extensions

It is natural to look for an extension of $K$ in which (2.1) attains a full
set of solutions, or, equivalently, over which the associated module $M$ is
trivialized, after base change. Easy examples show that such an extension
might have to have zero divisors. The best we can do is encapsulated in the
following definition.

######  Definition 9.

(i) A $\phi$_-ring_ is a commutative unital ring $R$ equipped with an
_endomorphism_ $\phi$. It is called $\phi$-_simple_ if it does not have any
non-zero ideals $I$ invariant under $\phi$, i.e. satisfying $\phi(I)\subset
I.$

(ii) A _Picard-Vessiot (PV) ring_ for the $\phi$-module $M$ (associated to
$A\in GL_{n}(K)$ as above) is a simple $\phi$-ring extension $(R,\phi)$ of
$(K,\phi)$ over which $M_{R}=(R\otimes_{K}M,\phi\otimes\Phi)$ is trivialized
(i.e. becomes isomorphic to $(R^{n},\phi)$), and such that
$R=K[u_{ij},\det(U)^{-1}]$ if $U=(u_{ij})\in GL_{n}(R)$ is a fundamental
matrix of (2.1).

Here are the main properties of PV rings.

  * •

PV rings exist, are noetherian and (like any $\phi$-simple ring) reduced.
Furthermore, a PV ring $R$ is a finite product $R_{1}\times\cdots\times R_{t}$
of integral domains, permuted cyclically by $\phi.$

  * •

Since $\phi$ was assumed to be an automorphism of $K$ and $A$ is invertible, a
PV ring $R$ happens to be _inversive: $\phi$ _is an automorphism of $R$.

  * •

The field of constants $C_{R}=R^{\phi}$ is an algebraic extension of
$C=K^{\phi}.$ If $C$ is algebraically closed, $C=C_{R}.$

  * •

The fundamental matrix $U\in GL_{n}(R)$ is unique up to $U\mapsto UV$ with
$V\in GL_{n}(C_{R}).$

  * •

If $C$ is algebraically closed, any two PV rings for $M$ are (noncanonically)
isomorphic.

  * •

Let $L=Quot(R)$ be the total ring of fractions of $R$ (the localization of $R$
in the $\phi$-invariant multiplicative set of non-zero divisors of $R$). Thus
$L=L_{1}\times\cdots\times L_{t}$ is a finite product of fields, which are
permuted cyclically by $\phi$. We have $L^{\phi}=C_{R}.$ A $\phi$-ring $L$ of
this type is a called a $\phi$_-pseudofield_.

Assume from now on that $C$ is algebraically closed.

######  Lemma 10.

Let $L$ be a $\phi$-pseudofield extension of $K$ which trivializes $M$ and is
generated over $K$ (as a pseudofield) by the entries $u_{ij}$ of a fundamental
matrix $U.$ Suppose that $L^{\phi}=C.$ Then $R=K[u_{ij},\det(U)^{-1}]\subset
L$ is $\phi$-simple, hence it is a PV ring for $M$, and $L$ is its total ring
of fractions.

The last lemma is of great practical value, because it is often much easier to
check that $L^{\phi}=C$ then to verify directly that $R$ is $\phi$-simple. The
$\phi$-pseudofield $L$ is called the PV _extension_ associated with $M$.

  * •

Notation as above, $L_{1}$ is a $\phi^{t}$-PV extension for $(M,\Phi^{t})$
over $(K,\phi^{t}).$ Note that the matrix associated to $(M,\Phi^{t})$ is

$A_{[t]}=\phi^{t-1}(A)\cdots\phi(A)A.$

Thus, at the expense of replacing $\phi$ by a suitable power, we may assume
that $L$ is a field and $R$ a domain. In the current paper, this will turn out
to be always possible.

A PV ring $R$ for (2.1) is constructed as follows. Let $X=(X_{ij})$ be an
$n\times n$ matrix of indeterminates. Let $\phi$ act on the ring
$\widetilde{R}=K[X_{ij},\det(X)^{-1}]$ via its given action on $K$ and the
formula

$\phi(X)=AX,$

i.e. $\phi(X_{ij})=\sum_{\nu=1}^{n}a_{i\nu}X_{\nu j}$. Let $I$ be a maximal
$\phi$-invariant ideal in $\widetilde{R}$. Then

$R=\widetilde{R}/I$

is a PV ring for (2.1), and $U=X\mod I$ is a fundamental matrix in
$GL_{n}(R).$ We remark that since $\widetilde{R}$ is noetherian, any
$\phi$-invariant ideal $I$ satisfies $\phi(I)=I.$

The reduced $K$-scheme $W=Spec(R)$ is called the PV _scheme_ associated with
$R.$ Since the choice of a fundamental matrix $U$ amounts to a presentation
$R=\widetilde{R}/I$ as above, the choice of $U$ determines a closed embedding

$W\hookrightarrow GL_{n,K}.$

In general, the $K$-scheme $W$ might not have any $K$-points. We shall see
soon (Proposition 11) that if $K$ satisfies the conclusions of Proposition 7,
$W(K)\neq\emptyset$. This will be an important observation in our context.

####  2.4.2. The map $\tau$

If $h\in GL_{n}(K)=\mathrm{Hom}_{K}(K[X_{ij},\det(X)^{-1}],K)$ we let

$\tau(h)=\phi\circ h\circ\phi^{-1}\in GL_{n}(K).$

If $X_{h}=h(X)$ is the matrix in $GL_{n}(K)$ representing the $K$-point $h$,
then since $\phi^{-1}(X)=\phi^{-1}(A)^{-1}X$ we have

$X_{\tau(h)}=\tau(h)(X)=A^{-1}\phi(X_{h}).$

If $h\in W(K),$ i.e. $h$ factors through $I$, then since $\phi^{-1}(I)=I,$ so
does $\tau(h).$ Regarded as a subset of $GL_{n}(K),$ if $P\in W(K)$ then

$\tau(P)=A^{-1}\phi(P)\in W(K).$

The set of $K$-points of the Picard-Vessiot scheme is therefore, if not empty,
invariant under $\tau.$

###  2.5. The difference Galois group of $(M,\Phi)$

We continue to assume that $C=K^{\phi}$ is algebraically closed. Let
$(M,\Phi)$ be a $\phi$-module, $A$ the matrix associated to it in some basis,
$R$ a PV ring and $L=Quot(R)$ the associated PV extension.

Let $B$ be a $C$-algebra, with a trivial $\phi$-action. Writing
$-_{B}=B\otimes_{C}-$ we let

$\mathcal{G}(B)=\mathrm{Aut}_{\phi}(R_{B}/K_{B})=\mathrm{Aut}_{\phi}(L_{B}/K_{B})$

be the group of automorphisms of $R_{B}$ that fix $K_{B}$ pointwise and
commute with $\phi$. This yields a functor

$\mathcal{G}:\mathrm{Alg}_{C}\rightsquigarrow\mathrm{Groups}.$

Then:

  * •

$\mathcal{G}$ is representable by a closed subgroup scheme of $GL_{n,C}.$ If
$\sigma\in\mathcal{G}(B)$

$\sigma(U)=U\cdot V(\sigma)$

with $V(\sigma)\in GL_{n}(B)$ and $\sigma\mapsto V(\sigma)$ embeds
$\mathcal{G}$ in $GL_{n,C}.$ If $char(C)=0$ then $\mathcal{G}$ is reduced, but
in positive characteristic we must include the possibility of non-reduced
$\mathcal{G}.$

  * •

If $A$ is replaced by $\phi(P)^{-1}AP$ (change of basis of $M$, $P\in
GL_{n}(K)$) and $U$ is replaced by $P^{-1}U$ then, since $\sigma(P)=P$, we get
_the same embedding_ $\mathcal{G}\hookrightarrow GL_{n,C}.$ If $U$ is replaced
by another fundamental matrix for (2.1), necessarily of the form $UT$ with
$T\in GL_{n}(C),$ then $V(\sigma)$ is replaced by $T^{-1}V(\sigma)T.$ Thus
$\mathcal{G}$ _is uniquely determined up to conjugation in_ $GL_{n}(C).$

  * •

The coordinate ring of $\mathcal{G}$ is given by
$C[\mathcal{G}]=(R\otimes_{K}R)^{\phi}$. Let

$Z=(U^{-1}\otimes 1)\cdot(1\otimes U)\in GL_{n}(R\otimes_{K}R),$

i.e. $Z_{ij}=\sum_{\nu=1}^{n}(U^{-1})_{i\nu}\otimes U{}_{\nu j}$. Then

$\phi(Z)=(U^{-1}\otimes 1)\cdot(A^{-1}\otimes 1)\cdot(1\otimes
A)\cdot(1\otimes U)=Z$

so $Z_{ij}\in C[\mathcal{G}]$ and $C[\mathcal{G}]=C[Z_{ij},\det Z^{-1}].$ We
have

$\sigma\in\mathcal{G}(B)=\mathrm{\mathrm{Hom}}(C[\mathcal{G}],B)\leftrightsquigarrow(Z\mapsto
V(\sigma))$

and $\mathcal{G}\hookrightarrow GL_{n,C}$ implies the comultiplication

$m^{*}(Z)=Z\otimes Z,$

i.e. $m^{*}(Z_{ij})=\sum_{\nu=1}^{n}Z_{i\nu}\otimes Z_{\nu j}$.

  * •

Inside $R\otimes_{K}R$ we have the canonical isomorphism

$R\otimes_{K}K[\mathcal{G}]=R\otimes_{C}C[\mathcal{G}]\simeq R\otimes_{K}R$

(since $(U\otimes 1)\cdot Z=1\otimes U$), which means that $W=Spec(R)$ is a
_torsor_ of $\mathcal{G}_{K}.$ We conclude that $W(K)\neq\emptyset$ is a
necessary and sufficient condition for $W$ to be the trivial torsor, i.e. to
be (noncanonically) isomorphic to $\mathcal{G}_{K}.$

  * •

If $L$ is a field, $\mathrm{tr.deg.}L/K=\dim\mathcal{G}.$

######  Proposition 11.

Assume that $char(C)=0$ and that $K$ satisfies the conclusions of Proposition
7 for every power of $\phi$. Then $W(K)\neq\emptyset$.

###### Proof.

If $\mathcal{G}$ is connected, this follows from part (ii) of Proposition 7.
Following Proposition 1.20 in [S-vdP] we explain how part (iii) of the same
Proposition allows us to get rid of the assumption that $\mathcal{G}$ is
connected. Let

$R=R_{1}\times\cdots\times R_{t}$

be the decomposition of $R$ into a product of integral domains, permuted
cyclically by $\phi$. Since $K$ does not have any finite extension to which
$\phi^{t}$ extends, it is algebraically closed in the field
$L_{i}=Quot(R_{i})$. This means that $W_{i}=Spec(R_{i})$ remains irreducible
over the algebraic closure $\overline{K}$ of $K$. It follows that one of the
$W_{i}$, say $W_{1},$ is a torsor of $\mathcal{G}_{K}^{0}.$ Since
$\mathcal{G}_{K}^{0}$ is connected, $W_{1}(K)\neq\emptyset$, hence
$W(K)\neq\emptyset.$ ∎

We continue to assume that $C$ is algebraically closed of characteristic 0.

######  Theorem 12.

([S-vdP] Theorem I.1.21) (i) Let $H\subset GL_{n,C}$ be a closed subgroup. If
in some basis $A\in H(K),$ then we can choose $U\in H(R)$ and
$\mathcal{G}\subset H$. For a general fundamental matrix $U\in GL_{n}(R),$
some conjugate of $\mathcal{G}$ by an element of $GL_{n}(C)$ will be contained
in $H$.

(ii) Assume that the conclusions of Proposition 7 hold. Then conversely, there
exists a basis of $M$ with respect to which $A\in\mathcal{G}(K).$
Equivalently, for the original $A$ there exists a $P\in GL_{n}(K)$ such that
$\phi(P)^{-1}AP\in\mathcal{G}(K).$

(iii) Under the assumptions of (ii) $\mathcal{G}$ is characterized (up to
conjugation by $GL_{n}(C)$) as a _minimal_ element of the set

$\mathscr{H}=\\{H\subset GL_{n,C}|\,H\,\mathrm{closed,}\,\exists P\in
GL_{n}(K)\,\mathrm{s.t.}\,\,\,\phi(P)^{-1}AP\in H(K)\\}.$

Every other element of $\mathscr{H}$ therefore contains a conjugate of
$\mathcal{G}$.

(iv) $\mathcal{G}/\mathcal{G}^{0}$ is cyclic.

###### Proof.

(i) Assume $A\in H(K)$, and $H$ is given explicitly as
$Spec(C[X_{ij},\det(X)^{-1}]/N)$ where $N$ is a Hopf ideal. Let $\phi$ act on
$K[X_{ij},\det(X)^{-1}]$ via the given action on $K$ and via $\phi(X)=AX$.
Then $N_{K}$ is a $\phi$-ideal because if $f\in N$ then

$\phi(f)=(\alpha\times 1)\circ m^{*}(f)$

where $m^{*}$ is the comultiplication and $\alpha$ the homomorphism
$C[X_{ij},\det(X)^{-1}]\to K$ substituting $A$ for $X$. But

$m^{*}(f)\in
N\otimes_{C}C[X_{ij},\det(X)^{-1}]+C[X_{ij},\det(X)^{-1}]\otimes_{C}N$

and $\alpha(N)=0$ since $A\in H(K).$ Thus $\phi(f)\in K\otimes_{C}N=N_{K}$.

Let $I$ be a maximal $\phi$-ideal in $K[X_{ij},\det(X)^{-1}]$ _containing_
$N_{K}$ and $U=X\mod I$. Then

$W=Spec(R)=Spec(K[X_{ij},\det(X)^{-1}]/I)\subset H_{K}$

is the PV scheme, and $U\in H(R)$ (corresponding to the canonical homomorphism

$C[X_{ij},\det(X)^{-1}]/N\to K[X_{ij},\det(X)^{-1}]/I.)$

It follows that for $\sigma\in\mathrm{Aut_{\phi}}(R_{B}/K_{B})$ we have
$\sigma(U)\in H(R_{B})$, hence

$V(\sigma)=U^{-1}\sigma(U)\in H(R_{B})^{\phi}=H(B).$

Any other fundamental matrix is of the form $UT$ with $T\in GL_{n}(C)$, so
$T\mathcal{G}T^{-1}\subset H.$

(ii) Under our assumptions, $W(K)$ is non-empty. Any $P\in W(K)\subset
GL_{n}(K)$ satisfies

$W=P\mathcal{G}_{K}.$

Since $\tau(P)\in W(K)$ as well (see 2.4.2),
$\tau(P)^{-1}P=\phi(P)^{-1}AP\in\mathcal{G}(K)$ and there exists a basis of
$M$ for which $A\in\mathcal{G}(K).$

(iii) By (i) every member of $\mathscr{H}$ contains $\mathcal{G}$ up to
conjugacy. By (ii) every $\mathcal{G}$ (uniquely determined up to conjugacy)
belongs to $\mathscr{H}.$ Thus $\mathcal{G}$ is the unique minimal member of
$\mathscr{H},$ up to conjugacy. Note that it is not a-priori clear that all
the minimal members of $\mathscr{H}$ are conjugate, but this follows from the
proof.

(iv) See [S-vdP]. ∎

###  2.6. The Galois correspondence

Let $R$ be a PV ring for $M$ and $L$ its total ring of fractions. Let
$\mathcal{G}$ be the difference Galois group of $M.$

We quote the following basic theorem. We say that $a\in L$ is fixed by
$\mathcal{G}$ and write $a\in L^{\mathcal{G}}$ if for every
$B\in\mathrm{Alg}_{C}$ the element $1\otimes a\in L_{B}$ is fixed by
$\mathcal{G}(B).$ The set $L^{\mathcal{G}}$ is a $\phi$-sub-pseudofield of
$L$.

######  Theorem 13.

(i) For any closed subgroup $\mathcal{G}^{\prime}\subset\mathcal{G}$ let

$\text{$\mathscr{F}(\mathcal{G}^{\prime})=K^{\prime}=$}L^{\mathcal{G}^{\prime}},$

a $\phi$-sub-pseudofield of $L$ containing $K$. For any $\phi$-pseudofield
$K\subset K^{\prime}\subset L$ let
$\mathscr{G}(K^{\prime})=\mathcal{G}^{\prime}$ be the closed subgroup of
$\mathcal{G}$ whose $B$-points, for $B\in\mathrm{Alg}_{C}$ are

$\mathscr{G}(K^{\prime})(B)=\mathcal{G}^{\prime}(B)=\mathrm{Aut_{\phi}}(L_{B}/K^{\prime}_{B}).$

Then $\mathcal{G}^{\prime}\mapsto
K^{\prime}=\mathscr{F}(\mathcal{G}^{\prime})$ and
$K^{\prime}\mapsto\mathcal{G}^{\prime}=\mathscr{G}(K^{\prime})$ is a 1-1
correspondence between closed subgroups of $\mathcal{G}$ and $\phi$-sub-
pseudofields of $L$ containing $K$. In particular $L^{\mathcal{G}}=K.$

(ii) $\mathcal{G}^{\prime}$ is normal in $\mathcal{G}$ if and only if
$K^{\prime}$ is a PV extension of some difference $\phi$-module $M^{\prime}.$
In this case the difference Galois group of $K^{\prime}/K$ is
$\mathcal{G}/\mathcal{G}^{\prime}$.

(iii) $\mathcal{G}^{0}$ corresponds to the algebraic closure of $K$ in $L$. If
we assume (replacing $\phi$ by some $\phi^{r}$) that $L$ is a field, then
since $L$ is a finitely generated field extension of $K$, the algebraic
closure of $K$ in $L$ is a finite extension. Under Proposition 7(iii) we
conclude that $K$ must be algebraically closed in $L$, hence $\mathcal{G}$
must be connected.

###  2.7. An example

Let $K$ and $\phi$ be as in case (Ell). Let $A\in GL_{2}(K)$ be the matrix

$A=\left(\begin{array}[]{cc}q&g_{q}(z,\Lambda)\\\ 0&1\end{array}\right)$

where $g_{q}(z,\Lambda)=\zeta(qz,\Lambda)-q\zeta(z,\Lambda)\in K.$ The field

$E=K(z,\zeta(z,\Lambda))$

is a PV extension for the system $\phi(Y)=AY,$ and

$U=\left(\begin{array}[]{cc}z&\zeta(z,\Lambda)\\\ 0&1\end{array}\right)$

is a fundamental matrix. Note that $\phi$ induces an automorphism of $E$. The
difference Galois group is

$\mathcal{G}=\left\\{\left(\begin{array}[]{cc}\alpha&\beta\\\
0&1\end{array}\right)|\,\alpha,\beta\in\mathbb{C}\right\\}$

and the unipotent subgroup ($\alpha=1$) corresponds, in the Galois
correspondence, to $K(z).$ The field $K(\zeta(z,\Lambda))$ is also a
$\phi$-subfield corresponding to the torus ($\beta=0)$, but is not a normal
extension of $K.$ Note that this field, unlike $E$, depends on the lattice
$\Lambda$.

##  3\. Review of parametrized Picard-Vessiot theory

###  3.1. $\psi$-linear algebraic groups

####  3.1.1. Generalities

In this section $C$ will be an algebraically closed field of characteristic 0,
equipped with an __ automorphism $\psi$ (denoted $\sigma$ in most of the
references). For example, if $K$ is a $(\phi,\psi)$-field and $C=K^{\phi}$
then $C$ inherits an action of $\psi,$ although this action might well be
trivial, as it is in the case (2Ell). We let $\mathrm{Alg}_{C}^{\psi}$ denote
the category of $C$-$\psi$-algebras __1 11 _It is important, when developing
the general formalism, to abandon the requirement that $\psi$ be invertible on
a general $C$-$\psi$-algebra. Thus while we maintain the assumption that
$\psi$ is an automorphism of $C$, hence $(C,\psi)$ is “inversive”, we must
allow rings in which $\psi$ is only an endomorphism, perhaps not even
injective, in the category $\mathrm{Alg}_{C}^{\psi}.$_. All our schemes and
$\psi$-schemes will be affine. If $R\in\mathrm{Alg}_{C}^{\psi}$ we denote by
$Spec^{\psi}(R)$ the functor

$Spec^{\psi}(R):\mathrm{Alg}_{C}^{\psi}\to\mathrm{Sets}$

defined by $Spec^{\psi}(R)(S)=\mathrm{Hom}_{C}^{\psi}(R,S)$ (homomorphisms of
$C$-$\psi$-algebras). Note that if $h\in Spec^{\psi}(R)(S)$ then
$\psi(h)=\psi\circ h=h\circ\psi\in Spec^{\psi}(R)(S)$ as well, so the functor
factors through the category of $\psi$-sets.

Let $L$ be a $\psi$-field extension of $C$. A subset $\\{a_{1},\dots,a_{n}\\}$
of $L$ is called $\psi$_-algebraically independent_ over $C$ if the collection
{$\psi^{i}a_{j}|\,0\leq i,\,\,1\leq j\leq n\\}$ is algebraically independent
over $C$. The $\psi$-transcendence degree of $L$ over $C$, denoted
$\psi\mathrm{tr.deg.}(L/C)$, is the cardinality of a maximal
$\psi$-algebraically independent set in $L.$ This notion is well defined (any
two such maximal sets have the same cardinality).

We refer to appendix A of [DV-H-W1] for an introduction to $\psi$-schemes and
$\psi$-group schemes. A $\psi$-_algebraic group_ $G$ over $C$ is a
$\psi$-group scheme that is $\psi$_-algebraic_ over $C$. This means that its
coordinate ring $C\\{G\\}$ is finitely $\psi$-generated over $C:$ it contains
a finite set $\\{u_{1},\dots,u_{n}\\}$ such that $\psi^{i}(u_{j})$ for $0\leq
i$ and $1\leq j\leq n$ generate $C\\{G\\}$ as a $C$-algebra. It is called
$\psi$-_reduced_ if $\psi$ is injective on $C\\{G\\},$ _perfectly
$\psi$-reduced _if the equation
$u^{e_{0}}\psi(u)^{e_{1}}\cdots\psi^{m}(u)^{e_{m}}=0$ ($e_{i}\geq 0$) forces
$u=0$ in $C\\{G\\},$ and $\psi$_-integral_ if $C\\{G\\}$ is an integral domain
and $\psi$ is injective.

If $G$ is a $\psi$-algebraic group over $C$ its $\psi$-dimension is a non-
negative integer, defined in Definition $A.25$ of [DV-H-W1]. If $G$ is
$\psi$-integral then

$\psi\dim(G)=\psi\mathrm{tr.deg.}(Quot(C\\{G\\})/C).$

If $\mathcal{G}$ is a (classical) algebraic group over $C$ then the functor
$B\mapsto\mathcal{G}(B^{\flat})$ from $\mathrm{Alg}_{C}^{\psi}$ to
$\mathrm{Groups},$ where $B^{\flat}$ is the $C$-algebra $B$ with the
$\psi$-structure forgotten, is representable by a $\psi$-algebraic group that
we denote $[\psi]\mathcal{G}.$ Suppose $\mathcal{G}$ is exhibited as a closed
subgroup of $GL_{n,C},$ so that

(3.1) $\mathcal{G}=Spec(C[X_{ij},\det(X)^{-1}]/I)$

where $1\leq i,j\leq n$ and $I$ is a Hopf ideal. Then
$G=[\psi]\mathcal{G}=Spec^{\psi}C\\{G\\}$ where
$C\\{G\\}=C[\psi^{k}X_{ij},\det\psi^{k}(X)^{-1}]/I_{\psi}$, $0\leq k<\infty,$
the $\psi^{k}X_{ij}$ are symbols treated as independent variables, the
$\psi$-action is the obvious one, and $I_{\psi}$ is the Hopf $\psi$-ideal
generated by all $\psi^{k}(h)$ for $h\in I$. As might be expected,

$\psi\dim([\psi]\mathcal{G})=\dim(\mathcal{G}).$

As a non-trivial example of a $\psi$-algebraic group, consider the
$\psi$-closed subgroup of $[\psi]\mathbb{G}_{m}$ given by the equation

(3.2) $X^{e_{0}}\psi(X)^{e_{1}}\cdots\psi^{m}(X)^{e_{m}}=1$

for some $e_{i}\in\mathbb{Z}$ (here $\mathbb{G}_{m}=Spec(C[X,X^{-1}])$.

By Lemma A.40 of [DV-H-W1] any closed $\psi$-subgroup of
$[\psi]\mathbb{G}_{m}$ is defined by a (possibly infinite) collection of
$\psi$-monomials of the form (3.2). All that we shall need is the following
weaker result.

######  Lemma 14.

If $G\subsetneq[\psi]\mathbb{G}_{m}$ is a proper closed $\psi$-subgroup of the
multiplicative group, then there exists a non-trivial $\psi$-monomial of the
form (3.2) satisfied by $G$, and $\psi\dim(G)=0.$

####  3.1.2. (Classical) Zariski closure

Let $\mathcal{G}$ be a (classical) linear algebraic group over $C$ and
$G\subseteq[\psi]\mathcal{G}$ a $\psi$-closed subgroup. We say that $G$ is
_Zariski dense_ in $\mathcal{G}$ if for any proper subvariety (or subgroup,
since $G$ is a group functor, this will turn out to be the same)
$\mathcal{H}\subset\mathcal{G}$, $G\nsubseteq[\psi]\mathcal{H}.$ If
$\mathcal{G}$ is a subgroup of $GL_{n,C}$ given by the Hopf algebra (3.1), and

$G=Spec^{\psi}(C[\psi^{k}X_{ij},\det\psi^{k}(X)^{-1}]/J)$

for a Hopf $\psi$-ideal $J$, a necessary and sufficient condition for $G$ to
be Zariski dense in $\mathcal{G}$ is that $J\cap C[X_{ij},\det(X)^{-1}]=I,$
i.e. the “ordinary” equations, not involving $\psi$, in the ideal defining
$G$, are just those defining $\mathcal{G}$. In such a case $I_{\psi}\subset
J$, because $J$ is a $\psi$-ideal and $I_{\psi}$ is the smallest $\psi$-ideal
containing $I$, but this inclusion might be strict if
$G\subsetneq[\psi]\mathcal{G}.$

Conversely, starting with a $\psi$-algebraic group $G$ presented in the above
form, and defining

$J\cap C[X_{ij},\det(X)^{-1}]=:I$

one sees that $I$ is Hopf ideal in $C[X_{ij},\det(X)^{-1}]$ and the algebraic
group $\mathcal{G}$ that it defines is the (classical) _Zariski closure_ of
$G$.

####  3.1.3. The structure of $\psi$-linear algebraic groups whose Zariski
closure is simple

We shall need the following result, which, for simplicity, we only state in
the case: $C=\mathbb{C}$ and $\psi$ is the identity. It shows that if
$\mathcal{G}$ is _simple_ , proper $\psi$-subgroups of $[\psi]\mathcal{G},$
which are nevertheless Zariski dense in it, are (under a mild technical
condition of $\psi$-reducedness) of a very special form. Alternatively, one
may start with an arbitrary $\psi$-reduced $\psi$-linear group $G$ and ask (i)
that it be _properly contained_ in its (classical) Zariski closure, i.e. $G$
“is not classical”, and (ii) that this Zariski closure be simple222In fact,
allowing $m=0$ and $\alpha=1$ in the Proposition, one may drop (i) and assume
only that the Zariski closure of $G$ is simple..

######  Proposition 15 ([DV-H-W2], Proposition A.19 and Theorem A.20).

Let $\mathcal{G}$ be a _simple_ linear algebraic group over $\mathbb{C}$ and
$G\subsetneq[\psi]\mathcal{G}$ a _proper_ , $\psi$-closed, $\psi$-reduced
subgroup of $[\psi]\mathcal{G}.$ Assume that $G$ is Zariski dense in
$\mathcal{G}.$ Then there exists an automorphism
$\alpha\in\mathrm{Aut}(\mathcal{G})$ and an integer $m\geq 1$ such that

$G(B)=\\{g\in\mathcal{G}(B)|\,\psi^{m}(g)=\alpha(g)\\}$

for every $B\in\mathrm{Alg}_{\mathbb{C}}^{\psi}.$ Furthermore, replacing $m$
by a multiple of it we find that there exists an $h\in\mathcal{G}(\mathbb{C})$
such that

$G(B)\subset\\{g\in\mathcal{G}(B)|\,\psi^{m}(g)=hgh^{-1}\\}.$

###  3.2. Parametrized Picard-Vessiot extensions

Let $(K,\phi$) be an inversive $\phi$-field, and assume that it is endowed
with another automorphism $\psi$, commuting with $\phi.$ Assume that the field
of $\phi$-constants $C=K^{\phi}$ is algebraically closed of characteristic 0,
and note that it inherits a structure of a $\psi$-field. Let $(M$,$\Phi)$ be a
$\phi$-module of rank $n$ over $K$, and let $A\in GL_{n}(K)$ be the matrix
associated with it in some basis.

######  Definition 16.

A $\psi$_-Picard-Vessiot extension,_ called also a _parametrized Picard-
Vessiot (PPV) extension_ for $(M,\Phi)$ (or the system $\phi(Y)=AY$), is a
$\phi$-pseudofield $L_{\psi}$ containing $K$ which:

(i) carries, in addition, a structure of a (not necessarily inversive)
$\psi$-field, commuting with $\phi$,

(ii) trivializes $M$ after base-change, and if $U\in GL_{n}(L_{\psi})$ is a
fundamental matrix for $\phi(Y)=AY,$

$L_{\psi}=K(u_{ij})_{\psi}=K(\psi^{k}(u_{ij}))$

is generated as a total ring (a ring in which every non-zero divisor is
invertible) by the elements $\psi^{k}(u_{ij})$ for $1\leq i,j\leq n$ and
$0\leq k<\infty.$ We shall express this property by saying that $L_{\psi}$ is
the $\psi$_-hull_ (as a total ring) of $K(u_{ij}).$

(iii) $L_{\psi}^{\phi}=K^{\phi}=C.$

Here are the main facts about PPV extensions.

  * •

A PPV extension $L_{\psi}$ as above exists ([O-W], Theorem 2.28 and Corollary
2.29). This is tricky! One is inclined to construct inductively (classical) PV
extensions for the $\phi$-modules

$M_{d}=M\oplus M^{(\psi)}\oplus\cdots\oplus M^{(\psi^{d})}$

and go to the limit when $d\to\infty.$ The difficulty is in showing that we
can get $L_{\psi}$ to be a _finite_ product of fields. One should keep track
of the number of connected components in this inductive procedure, and prove
that it stays bounded.

  * •

Let

$R_{\psi}=K[u_{ij},\det(U)^{-1}]_{\psi}=K[\psi^{k}(u_{ij}),\psi^{k}(\det
U)^{-1}]$

be the _ring_ $\psi$_-hull_ of $K[u_{ij},\det(U)^{-1}]$ inside $L_{\psi}.$
Then $R_{\psi}$ is $\phi$-simple and $L_{\psi}$ is its total ring of
fractions. One calls $R_{\psi}$ the PPV _ring_ of $M$. Since $U$ is uniquely
determined up to right multiplication by $V\in GL_{n}(C),$ $R_{\psi}$ is
uniquly determined as a subring of $L_{\psi}.$

  * •

Let $L=K(u_{ij})$ and $R=K[u_{ij},\det(U)^{-1}]$ (inside $L_{\psi}$). Then $L$
is a (classical) PV extension and $R$ a PV ring for $M$.

###  3.3. The parametrized difference Galois group

####  3.3.1. General facts and definitions

Assumptions and notation as above, fix a PPV extension $L_{\psi}$ and the PPV
ring $R_{\psi}\subset L_{\psi}.$ Consider the functor
$G:\mathrm{Alg}_{C}^{\psi}\rightsquigarrow\mathrm{Groups}$ given by

$G(B)=\mathrm{Aut}_{\phi,\psi}((R_{\psi})_{B}/K_{B})=\mathrm{Aut}_{\phi,\psi}((L_{\psi})_{B}/K_{B}),$

the automorphisms of $(R_{\psi})_{B}=B\otimes_{C}R_{\psi}$ that fix
$B\otimes_{C}K$ pointwise and commute with both $\phi$ and $\psi$. Here $B$ is
given the trivial $\phi$-action. If $\sigma\in G(B)$ then

$\sigma(U)=UV(\sigma)$

with $V(\sigma)\in GL_{n}(B)$. Moreover, since $\sigma$ commutes with $\psi,$
we have for every $i\geq 0$

(3.3) $\sigma(\psi^{i}(U))=\psi^{i}(U)\psi^{i}(V(\sigma)).$

Thus the choice of $U$ determines an embedding
$G\hookrightarrow[\psi]GL_{n,C}.$

The main facts about $G$, mirroring the facts listed for the classical
difference Galois group $\mathcal{G}$, are the following (see [O-W], section
2.7).

  * •

$G$ is representable by a $\psi$-linear algebraic group.

  * •

The Hopf $\psi$-algebra $C\\{G\\}$ of $G$ is
$(R_{\psi}\otimes_{K}R_{\psi})^{\phi}.$

  * •

The natural map

$R_{\psi}\otimes_{C}C\\{G\\}=R_{\psi}\otimes_{K}K\\{G\\}\simeq
R_{\psi}\otimes_{K}R_{\psi}$

sending $r\otimes h$ ($r\in R_{\psi},\,\,h\in C\\{G\\}$) to $r\otimes 1\cdot
h$ is an isomorphism of $K$-$(\phi,\psi)$-algebras. This means that
$W_{\psi}=Spec^{\psi}(R_{\psi})$ is a $\psi$-$G_{K}$-torsor.

  * •

If $L_{\psi}$ is a field, $\psi\dim(G)=\psi\mathrm{tr.deg.}(L_{\psi}/K)$.

  * •

The fixed field of a PPV extension under the parametrized Galois group being
defined in the same way as the fixed field of a PV extension under the
classical Galois group, we have

$L_{\psi}^{G}=K.$

  * •

More generally, there is a 1-1 correspondence between $\psi$-algebraic
subgroups of $G$ and intermediate $\phi$-pseudofields of $L_{\psi}$ stable
under $\psi$. Normal $\psi$-subgroups correspond to PPV extensions (of some
other $\phi$-modules $M^{\prime})$.

####  3.3.2. Relation between $G$ and $\mathcal{G}$

Let $L=K(u_{ij})\subset L_{\psi}$ be the classical PV extension inside the PPV
extension, and $R\subset R_{\psi}$ the PPV ring. Let $\mathcal{G}$ be the
classical Galois group. The realization of $\sigma\in G(B)$ as $V(\sigma)\in
GL_{n}(B)$ via its action on $U,$ namely

$\sigma:U\mapsto UV(\sigma)$

shows that $\sigma$ restricts to an automorphism of $L_{B}$ over $K_{B},$
hence a map of functors

$G\hookrightarrow[\psi]\mathcal{G},$

which is evidently injective. In general, it need not be an isomorphism, as
$\sigma\in[\psi]\mathcal{G}(B)=\mathcal{G}(B^{\flat})$ “does not know” about
the extra automorphism $\psi$, and may not extend to $L_{\psi}$ so that the
extra compatibilities (3.3) are satisfied. However, since

$C[\mathcal{G}]=(R\otimes_{K}R)^{\phi}\hookrightarrow(R_{\psi}\otimes_{K}R_{\psi})^{\phi}=C\\{G\\}$

is injective, any function from $C[\mathcal{G}]$ that vanishes on $G$ is 0. It
follows that there does not exist a proper (classical) subgroup
$\mathcal{H}\subset\mathcal{G}$ with $G\subset[\psi]\mathcal{H},$ hence

  * •

$G$ is Zariski dense in $\mathcal{G}$ ([A-D-H-W], Proposition 2.1.2).

####  3.3.3. A Galoisian criterion for being $\psi$-isomonodromic

The $\psi$-Galois group $G$ of a difference $\phi$-module $M$ enables us to
state a criterion for $M$ to be $\psi$-isomonodromic, i.e. for $M\simeq
M^{(\psi)}.$

######  Proposition 17 (Theorem 2.55 of [O-W]).

The $\phi$-module $M$ is $\psi$-isomonodromic if and only if there exists an
$h\in GL_{n}(C)$ such that

$\psi(X)=hXh^{-1}$

holds in $G$ (i.e. for any $B\in\mathrm{Alg}_{C}^{\psi}$ and any $\sigma\in
G(B)\subset\mathcal{G}(B)\subset GL_{n}(B),$ $\psi(\sigma)=h\sigma h^{-1}$).

###### Proof.

Assume that $M$ is $\psi$-isomonodromic. Then there exists a matrix
$A_{\psi}\in GL_{n}(K)$, such that with $A_{\phi}=A$, we have the
compatibility relation

$\phi(A_{\psi})A_{\phi}=\psi(A_{\phi})A_{\psi}.$

Using this relation and the relation $1=\phi(U)^{-1}A_{\phi}U$ we see that

$h=\psi(U)^{-1}A_{\psi}U$

is fixed under $\phi,$ hence belongs to $GL_{n}(L_{\psi}^{\phi})=GL_{n}(C)$.
Let $\sigma\in G(B)$ and compute $\sigma(\psi(U))$ in two ways. On the one
hand

$\sigma(\psi(U))=\sigma(A_{\psi}Uh^{-1})=A_{\psi}UV(\sigma)h^{-1}.$

On the other hand

$\sigma(\psi(U))=\psi(\sigma(U))=\psi(UV(\sigma))=A_{\psi}Uh^{-1}\psi(V(\sigma)).$

Comparing the two expressions we get $\psi(V(\sigma))=hV(\sigma)h^{-1}.$ This
string of identities can be reversed. Starting with $h$ as above and defining
$A_{\psi}=\psi(U)hU^{-1}$, we see that $A_{\psi}$ is fixed by every $\sigma$
in the Galois group, hence lies in $GL_{n}(K)$, and we get the desired
compatibility relation between $A_{\psi}$ and $A_{\phi}.$ ∎

######  Remark 18.

The last proof can be given a matrix-free version. If $h:M\simeq M^{(\psi)}$
is an isomorphism of $\phi$-modules, then $h$ can be base-changed to
$L_{\psi}$ and then, since it commutes with $\Phi$, induces an isomorphism
between the modules of solutions. If $\sigma\in G$ (and not only in
$\mathcal{G})$ then $\sigma$ induces a commutative diagram

$\begin{array}[]{ccc}M_{L_{\psi}}^{\Phi}&\overset{h}{\to}&(M_{L_{\psi}}^{(\psi)})^{\Phi}\\\
\sigma\downarrow&&\downarrow\psi(\sigma)\\\
M_{L_{\psi}}^{\Phi}&\overset{h}{\to}&(M_{L_{\psi}}^{(\psi)})^{\Phi}\end{array}$

yielding the relation $\psi(\sigma)=h\sigma h^{-1}.$ If we identify, as usual,
the spaces of solutions $M_{L_{\psi}}^{\Phi}$ and
$(M_{L_{\psi}}^{(\psi)})^{\Phi}$ with $\mathbb{C}^{n}$, in the bases given by
the columns of $U$ and $\psi(U),$ then $h$ becomes a matrix in
$GL_{n}(\mathbb{C})$ as in the Proposition. Conversely, a descent argument
shows that given such a diagram relating the spaces of solutions (_after_ base
change to $L_{\psi}$) yields an isomorphism $M\simeq M^{(\psi)}$ (_before_ the
base-change). In fact, this “conceptual proof” is not any different than the
“matrix proof” by Ovchinnikov and Wibmer. Unwinding the arguments, one sees
that the two proofs are one and the same.

###  3.4. Example 2.7 continued

Suppose we add, in Example 2.7, a second difference operator $\psi
f(z)=f(pz),$ as in the case (2Ell). Then the $\phi$-module corresponding to
the system $\phi(Y)=AY$ is $\psi$-isomonodromic, and the corresponding system
$\psi(Y)=BY$ is given by

$B=\left(\begin{array}[]{cc}p&g_{p}(z,\Lambda)\\\ 0&1\end{array}\right).$

The compatability relation

$\phi(B)A=\psi(A)B$

is satisfied. The field $E$ is also a PPV extension, being $\psi$-stable. The
$\psi$-Galois group $G$ is “classical”, i.e.

$G=[\psi]\mathcal{G}.$

##  4\. Some preliminary results

###  4.1. Isomonodromy and solvability

Let $(K,F,\phi,\psi)$ be as in the case (2Ell) and assume that $2\leq p,q$ and
$(p,q)=1.$ Let $M$ be a $\phi$-module over $K$, $A$ the associated matrix (in
a fixed basis), $R$ a PV ring for $M$, $L=Quot(R)$ the corresponding PV
extension, $U\in GL_{n}(R)$ a fundamental matrix, and $\mathcal{G}\subset
GL_{n,\mathbb{C}}$ the difference Galois group, its embedding in
$GL_{n,\mathbb{C}}$ determined by the choice of $U$. The following theorem
will be used in the proof of Theorem 5, but has independent interest.

######  Theorem (= Theorem 6).

Assume that $M$ is $\psi$-isomonodromic. Then $\mathcal{G}$ is solvable.

###### Proof.

By Theorem 12(i), it is enough to show that with respect to a suitable basis
of $M$ the matrix $A$ is upper triangular. Indeed, if this is the case, take
$H$ to be the Borel subgroup of upper triangular matrices. Since (a conjugate
of) $\mathcal{G}\subset H$ and $H$ is solvable, $\mathcal{G}$ is solvable too.

Endow $M=(M,\Phi,\Psi)$ with a $(\phi,\psi)$-module structure. Call $M$
solvable if there exists a sequence

$0\subset M_{1}\subset\cdots\subset M_{n}=M$

of $(\phi,\psi)$-submodules $M_{i},$ $\mathrm{rk}(M_{i})=i$. This would
clearly imply that $A=A_{\phi}$ is gauge-equivalent to a matrix in upper
triangular form.

We show that $M$ is solvable. By induction, it is enough to show that $M$
contains a rank-one ($\phi,\psi)$-submodule. We apply Theorem 35 of [dS2].
Using the notation there, let $(r_{1},\dots,r_{k})$ be the type of $M,$
$r_{1}\leq r_{2}\leq\cdots\leq r_{k},$ $\sum_{i=1}^{k}r_{i}=n.$ Let
$e_{1},\dots,e_{n}$ be the basis of $M$ in which $A$ has the form prescribed
by that theorem. Recall that if we write $A=(A_{ij})$ in block-form,
$A_{ij}\in M_{r_{i}\times r_{j}}(K)$, then

$A_{ij}(z)=U_{r_{i}}(z/p)T_{ij}(z)U_{r_{j}}(z)^{-1}.$

The marix $U_{r}(z)$ is unipotent upper-triangular. The matrix

$T_{ij}=\left(\begin{array}[]{cc}0&T_{ij}^{*}\\\ 0&0\end{array}\right)$

where $T_{ij}^{*}$ is a square upper-triangular $s\times s$ marix for
$s\leq\min(r_{i},r_{j})$, with _constant_ (i.e. $\mathbb{C}$-valued) entries.
An analogous description, with a constant matrix $S$ replacing $T$, gives the
matrix $B,$ associated with $\Psi$ in the same basis.

Let

$i_{1}=1,\,\,i_{2}=r_{1}+1,\dots,i_{k}=r_{1}+\cdots+r_{k-1}+1$

be the first indices in each of the $k$ blocks. Let
$M^{\prime}=Span_{K}\\{e_{i_{1}},\dots,e_{i_{k}}\\}.$ Then $M^{\prime}$ is a
rank $k$ $(\phi,\psi)$-submodule of $M$. Moreover, $\Phi|_{M^{\prime}}$ and
$\Psi|_{M^{\prime}}$ are given in this basis by _constant_ matrices. In other
words, $M^{\prime}$ descends to $\mathbb{C},$
$M^{\prime}=M^{\prime}_{0}\otimes_{\mathbb{C}}K,$ where $M^{\prime}_{0}$ is a
$\mathbb{C}$-representation of
$\Gamma=\left\langle\phi,\psi\right\rangle\simeq\mathbb{Z}^{2}$, and $\Phi$
and $\Psi$ are extended semilinearly from $M^{\prime}_{0}$ to $M^{\prime}$.
Since any two commuting endomorphisms of a $\mathbb{C}$-vector space have a
common eigenvector, $M^{\prime}$ has a rank-one $(\phi,\psi)$-submodule
$M_{1}\subset M^{\prime}\subset M$, which concludes the proof. ∎

Incidentally, note that we have given an affirmative answer to Problem 36 in
[dS2].

###  4.2. A periodicity theorem

In this subsection we generalize Theorem 1.1 of [dS1]. Let $\mathscr{D}$ be
the $\mathbb{Q}$-vector space of discretely supported functions
$f:\mathbb{C}\to\mathbb{Q},$ i.e. functions for which
$\mathrm{supp}(f)=\\{z|\,f(z)\neq 0\\}$ has no accumulation point in
$\mathbb{C}.$ For any lattice $\Lambda\subset\mathbb{C}$ let
$\mathscr{D}_{\Lambda}$ be the subspace of $f\in\mathscr{D}$ which are
$\Lambda$-periodic. We may identify

$\mathscr{D}_{\Lambda}=\mathrm{Div}(E_{\Lambda})_{\mathbb{Q}}$

with the group of $\mathbb{Q}$-divisors on the elliptic curve $E_{\Lambda}.$

Given two functions $f,\widetilde{f}\in\mathscr{D}$ we say that
$\widetilde{f}$ is a _modification at 0_ of $f$ if $\widetilde{f}(z)=f(z)$ for
every $z\neq 0.$

Let $2\leq p,q\in\mathbb{N}$ be relatively prime integers: $(p,q)=1.$ Consider
the operators

$\phi f(z)=f(qz),\,\,\,\psi f(z)=f(pz)$

on $\mathscr{D}.$ These operators preserve every $\mathscr{D}_{\Lambda}.$

######  Proposition 19.

Let $f\in\mathscr{D}$. Assume that for some $\Lambda$-periodic
$f_{p},f_{q}\in\mathscr{D}_{\Lambda}$ the relations

$f_{q}(z)=f(qz)-f(z)$ $f_{p}(z)=\sum_{i=0}^{m}e_{m-i}f(p^{1-i}z)$

($e_{i}\in\mathbb{Q},$ $e_{m}=1,$ $e_{0}\neq 0$) hold for all $z\neq 0$. Then,
after replacing $\Lambda$ by a sublattice, a suitable modification
$\widetilde{f}$ of $f$ at $0$ is $\Lambda$-periodic.

Theorem 1.1 of [dS1] concerned the case $f_{p}(z)=f(pz)-f(z).$ In this case,
or more generally if $\sum_{i=0}^{m}e_{m-i}=0$, we can not forgo the need to
modify $f$ at 0 because if two $f$’s agree outside 0, they yield the same
$f_{p}$ and $f_{q}$.

We shall now show how to modify the proof to treat the more general case given
here.

Observe first that for some $r_{\nu}\in\mathbb{Q}$, $r_{1}=1,$ we have for
every $z\neq 0$

(4.1)
$f(z)=\sum_{\nu=1}^{\infty}f_{q}(\frac{z}{q^{\nu}})=\sum_{\nu=1}^{\infty}r_{\nu}f_{p}(\frac{z}{p^{\nu}}).$

Formally, this is clear for the first sum, and in the second sum one solves
recursively for the $r_{\nu}$. Since all our functions are discretely
supported, for any given $z$ the infinite sums are actually finite, and the
formal identity becomes an equality.

Let $S_{p}\subset\mathbb{C}/\Lambda$ and $S_{q}\subset\mathbb{C}/\Lambda$ be
the supports of $f_{p}$ and $f_{q}$ (modulo $\Lambda$). Let
$\pi_{\Lambda}:\mathbb{C}\to\mathbb{C}/\Lambda$ be the projection and
$\widetilde{S}_{p}=\pi_{\Lambda}^{-1}(S_{p}),\,\,\,\widetilde{S}_{q}=\pi_{\Lambda}^{-1}(S_{q}).$
Let $\widetilde{S}$ be the support of $f$, and
$S=\pi_{\Lambda}(\widetilde{S}).$ By (4.1) we have

(4.2)
$\widetilde{S}-\\{0\\}\subset\bigcup_{\nu=1}^{\infty}p^{\nu}\widetilde{S}_{p}\cap\bigcup_{\nu=1}^{\infty}q^{\nu}\widetilde{S}_{q}.$

######  Lemma 20.

The set $S$ is finite.

###### Proof.

See Lemma 2.3 of [dS1]. It is enough to assume here that $p$ and $q$ are
multiplicatively independent. ∎

######  Lemma 21.

Let $z\in\widetilde{S}_{q},$ $z\notin\mathbb{Q}\Lambda,$ and let $n_{q}(z)$ be
the largest $n\geq 0$ such that $q^{n}z\in\widetilde{S}_{q}$ (it exists since
$S_{q}$ is finite and the points $q^{n}z$ have distinct images modulo
$\Lambda$). Note that $n_{q}(z)=n_{q}(z+\lambda)$ for $\lambda\in\Lambda$ so
that

$n_{q}=1+\max_{z\in\widetilde{S}_{q},\,\,z\notin\mathbb{Q}\Lambda}n_{q}(z)$

exists. Then

$f(z+q^{2n_{q}}\lambda)=f(z)$

for every $z\notin\mathbb{Q}\Lambda$ and $\lambda\in\Lambda$.

###### Proof.

The proof preceding Proposition 2.4 in [dS1] holds, word for word, except that
the $P$ there is our $q$ here. Thus, away from torsion points of the lattice,
$f$ is $q^{2n_{q}}\Lambda$-periodic. The proof of this Lemma, which relies on
the previous one, still assumes only the multiplicative independence of $p$
and $q$. ∎

We now treat torsion points $z\in\mathbb{Q}\Lambda$, for which we have to
assume $(p,q)=1.$ We may assume, without loss of generality, that $f,$ hence
$f_{p}$ and $f_{q}$, are supported on $\mathbb{Q}\Lambda$, because away from
$\mathbb{Q}\Lambda$ we have already proved periodicity, so we may subtract the
part of $f$ supported on non-torsion points from the original $f$ without
affecting the hypotheses.

Since $S$ is finite we may rescale $\Lambda$ and assume that $f$ is supported
on $pq$$\Lambda.$ Then $f_{p}$ is supported on $q\Lambda$ and $f_{q}$ on
$p\Lambda$, as becomes evident from (4.1).

######  Lemma 22.

If both $f_{p}$ and $f_{q}$ are $N\Lambda$-periodic, then so is a suitable
modification of $f$ at 0.

###### Proof.

The proof of Proposition 2.2 in [dS1] works the same, substituting the
relations (4.1) for the relations used there. ∎

This concludes the proof of Proposition 19.

##  5\. Proof of the main theorems

###  5.1. Deduction of Theorem 4 from Theorem 5

From now on we let $(K,F,\phi,\psi)$ be as in the case (2Ell), assuming that
$2\leq p,q,\,\,\,(p,q)=1.$ Assume that Theorem 5 is proven, and let $f$ and
$g$ be as in Theorem 4. Let $n$ be the first integer such that

$\psi^{n}(g)\in K(g,\psi(g),\dots,\psi^{n-1}(g)).$

Clearly all the $\psi^{i}(g),$ $i\geq n,$ also belong there.

If $g$ were algebraic over $K$, so would be all the $\psi^{i}(g),$ and the
field

$K(g)_{\psi}=K(g,\psi(g),\dots,\psi^{n-1}(g))$

would be a finite extension of $K$ to which $\psi$ extends as an endomorphism.
In fact, since $\psi$ is an automorphism of $K$ and $[K(g)_{\psi}:K]<\infty,$
it would be an _automorphism_ of $K(g)_{\psi}$. By Proposition 7(iii) this is
impossible. Hence $g$ is transcendental over $K$.

Suppose $f$ and $g$ were algebraically dependent over $K.$ Then this
dependence must involve $f$, hence $f$ is algebraic over $K(g)_{\psi},$ and so
would be all the $\psi^{i}(f).$ It follows that

$\mathrm{tr.deg.}(K(f,g)_{\psi}/K)=\mathrm{tr.deg.}(K(g)_{\psi}/K)\leq
n<\infty.$

A fortiori,

$\mathrm{tr.deg.}(K(f)_{\psi}/K)<\infty,$

contradicting the conclusion of Theorem 5.

###  5.2. First order equations

Consider the difference equation

(5.1) $\phi(y)=ay$

with $a\in K^{\times}.$ The associated $\phi$-module is $M=Ke$ with
$\Phi(e)=a^{-1}e.$ Let $L_{\psi}$ be a PPV extension for $M$, and $u\in
L_{\psi}$ a solution: $\phi(u)=au.$ Replacing $\phi$ and $\psi$ by some powers
$\phi^{r}$ and $\psi^{s}$ we may assume that $L_{\psi}$ is a field. Indeed,
let

$L_{\psi}=L_{1}\times\cdots\times L_{r}$

be the decomposition of the $\phi$-pseudofield $L_{\psi}$ into a product of
fields. Then $\phi^{r}$ is an endomorphism of $L_{1},$ and some power
$\psi^{s}$ of $\psi$ must also preserve it, and induces an endomorphism of
$L_{1}$. The subfield of $L_{1}$ generated by (the projection of) $u$ and all
its $\psi^{s}$-transforms is a PPV extension for $M$ as a $\phi^{r}$-module,
which is stable by $\psi^{s}.$

Assume therefore that $L_{\psi}$ is a field.

######  Proposition 23.

The following are equivalent:

(i) $u$ is $\psi$-algebraic over $K,$ i.e.
$\mathrm{tr.deg.}(L_{\psi}/K)<\infty.$

(ii) $u$ satisfies an equation $\psi(u)=\widetilde{a}u$ for some
$\widetilde{a}\in K^{\times}.$

(iii) The $\phi$-module $M$ descends to $\mathbb{C}:$ there exist $b\in
K^{\times}$and $c\in\mathbb{C}^{\times}$ such that

$a=c\frac{\phi(b)}{b}.$

######  Corollary 24.

If $ord_{0}(a)\neq 0$ then $u$ is $\psi$-transcendental over $K$.

###### Proof.

In this case, (iii) can not hold, so (i) can not hold either. ∎

In [dS1] we proved $(ii)\Rightarrow(iii),$ but we shall not need this step
here. Clearly $(ii)\Rightarrow(i)$, and $(iii)\Rightarrow(ii)$ because if
(iii) holds we may assume, replacing $u$ by $u/b$, that
$a\in\mathbb{C}^{\times}.$ Then

$(u^{\psi-1})^{\phi-1}=(u^{\phi-1})^{\psi-1}=a^{\psi-1}=1,$

so $\widetilde{a}=u^{\psi-1}\in L_{\psi}^{\phi}=\mathbb{C}$, and (ii) holds.
(If we did not assume $a\in\mathbb{C}^{\times}$ we would only get
$\widetilde{a}\in K^{\times}$.) We shall now prove $(i)\Rightarrow(iii)$.

###### Proof.

Let $G$ be the $\psi$-Galois group of (5.1). It is a $\psi$-closed subgroup of
$[\psi]\mathbb{G}_{m}.$ If $R_{\psi}=K[u,u^{-1}]_{\psi}\subset L_{\psi}$ is
the PPV ring then for any $B\in\mathrm{Alg}_{\mathbb{C}}^{\psi}$ and
$\sigma\in G(B)$ we embed $\sigma\mapsto v_{\sigma}\in B^{\times}$ where
$\sigma(u)=uv_{\sigma}.$

Assume that $u$ is $\psi$-algebraic. Then

$\psi\dim(G)=\psi\mathrm{tr.deg.}(L_{\psi}/K)=0<1=\psi\dim([\psi]\mathbb{G}_{m}),$

so $G$ is a proper closed $\psi$-subgroup of $[\psi]\mathbb{G}_{m}.$ It
follows from Lemma 14 that there is a $\psi$-monomial relation

$\mu(v_{\sigma})=v_{\sigma}^{e_{0}}\psi(v_{\sigma})^{e_{1}}\cdots\psi^{m}(v_{\sigma})^{e_{m}}=1$

($e_{i}\in\mathbb{Z},$ $e_{m}\neq 0$) that holds for all $\sigma\in G$.

Let $B\in\mathrm{Alg}_{\mathbb{C}}^{\psi}$ and $\sigma\in G(B).$ Then

$\sigma(\mu(u))=\mu(\sigma(u))=\mu(uv_{\sigma})=\mu(u)$

so $b^{\prime}=\mu(u)\in L_{\psi}^{G}=K.$ We conclude that

(5.2) $\mu(a)=\mu(u^{\phi-1})=\mu(u)^{\phi-1}=\phi(b^{\prime})/b^{\prime}.$

Let $\alpha(z)=ord_{z}(a)$ and $\beta(z)=ord_{z}(b^{\prime}).$ These are the
divisors of the elliptic functions $a$ and $b^{\prime}$, so for some
$\Lambda\subset\Lambda_{0}$ we have $\alpha,\beta\in\mathscr{D}_{\Lambda}.$
Furthermore, taking the divisor of the relation (5.2) gives

$\sum_{i=0}^{m}e_{i}\psi^{i}(\alpha)=\phi(\beta)-\beta.$

Define

$\delta(z)=\sum_{\nu=1}^{\infty}\alpha(\frac{z}{q^{\nu}})\,\,\,(z\neq
0);\,\,\,\,\delta(0)=0.$

Then $\delta\in\mathscr{D}$ and for $z\neq 0$

$\begin{cases}\begin{array}[]{c}\alpha(z)=\delta(qz)-\delta(z)\\\
\beta(z)=\sum_{i=0}^{m}e_{i}\delta(p^{i}z).\end{array}\end{cases}$

Applying the Periodicity Theorem 19 to $f_{p}(z)=\beta(pz),$
$f_{q}(z)=\alpha(p^{m}z)$ and $f(z)=\delta(p^{m}z)$ we conclude:

  * •

After replacing $\Lambda$ by a sublattice, a suitable modification
$\widetilde{\delta}$ of $\delta$ at 0 is $\Lambda$-periodic.

  * •

We must have $\alpha(0)=0.$

The first assertion is the Periodicity Theorem. For the second, if $z\neq 0$
we have

$\alpha(z)=\delta(qz)-\delta(z)=\widetilde{\delta}(qz)-\widetilde{\delta}(z).$

Let $\Lambda$ be a periodicity lattice for both $\alpha$ and
$\widetilde{\delta}.$ Take $0\neq\lambda\in\Lambda$. Then
$\alpha(\lambda)=\widetilde{\delta}(q\lambda)-\widetilde{\delta}(\lambda)=0,$
hence $\alpha(0)=0.$

We may therefore assume that $\delta(z)$ is already periodic and
$\alpha(z)=\delta(qz)-\delta(z)$ holds everywhere, including at 0. Observe,
however, that in the process of modifying $\delta$ at 0, we might no longer
have $\delta(0)=0.$

Let $\Pi$ be a fundamental parllelopiped for $\mathbb{C}/\Lambda$ where
$\Lambda$ is a periodicity lattice for $\alpha,\beta$ and $\delta$. Then

$0=\sum_{z\in\Pi}\alpha(z)=\sum_{z\in\Pi}\delta(qz)-\sum_{z\in\Pi}\delta(z)=(q^{2}-1)\sum_{z\in\Pi}\delta(z),$

so $\delta\in\mathrm{Div}^{0}(\mathbb{C}/\Lambda)$. We also have

$\sum_{z\in\Pi}z\alpha(z)=q^{-1}\sum_{z\in\Pi}qz\delta(qz)-\sum_{z\in\Pi}z\delta(z)$

$=q^{-1}\sum_{z\in
q\Pi}z\delta(z)-\sum_{z\in\Pi}z\delta(z)=(q-1)\sum_{z\in\Pi}z\delta(z).$

In the last step we used the fact that $q\Pi$ is the union of $q^{2}$
translates $\Pi+\lambda$ and that

$\sum_{z\in\Pi+\lambda}z\delta(z)=\sum_{z\in\Pi}z\delta(z)$

because $\delta$ is $\Lambda$-periodic and $\sum_{z\in\Pi}\delta(z)=0.$

Let $\Lambda^{\prime}=(q-1)\Lambda$. Let $\Pi^{\prime}$ be a fundamental
parllelopiped for $\Lambda^{\prime}$. Then by the same argument as above

$\sum_{z\in\Pi^{\prime}}z\delta(z)=(q-1)^{2}\sum_{z\in\Pi}z\delta(z)=(q-1)\sum_{z\in\Pi}z\alpha(z)\in\Lambda^{\prime}$

because by Abel-Jacobi $\sum_{z\in\Pi}z\alpha(z)\in\Lambda$.

Replacing $\Lambda$ by $\Lambda^{\prime}$ we conclude, again by Abel-Jacobi,
that $\delta$ is the divisor of some $b\in K_{\Lambda}.$ Let
$c=a/(\phi(b)/b).$ Then $c$ is $\Lambda$-elliptic and its divisor is

$\alpha-(\phi(\delta)-\delta)=0.$

Thus $c$ is constant, and the proof is complete. ∎

######  Corollary 25 (First order case of Theorem 5).

Let $f\in F$ satisfy the first order, linear homogenous equation

$\phi(f)=af$

with $a\in K^{\times}$. Then either $f\in S$ or
$\\{f,\psi(f),\psi^{2}(f),\dots\\}$ are algebraically independent over $K$.

###### Proof.

We may embed $K(f)_{\psi}$ in the PPV extension $L_{\psi}.$ If
$\\{f,\psi(f),\psi^{2}(f),\dots\\}$ are not algebraically independent over $K$
then, according to the last Proposition $f$ satisfies also a linear homogenous
$\psi$-difference equation over $K.$ By Theorem 3 we must have $f\in S$. ∎

######  Remark.

According to [dS1], in the order one case, if $f$ is $\psi$-algebraic over
$K,$ we can even infer that for some $m\in\mathbb{Z},$ $z^{m}f\in K$.

Generalizing from first order homogenous equations to first order inhomogenous
equations is done exactly as in [A-D-H-W], Proposition 4.5. We do not repeat
the proof, as it can be duplicated word for word, and will not be needed in
the sequel, see the remark below. The only difference is that at the last step
in [A-D-H-W], assuming $f$ was $\psi$-algebraic over $K$, Theorem 1.1 of that
paper was invoked to deduce that $f\in K.$ Here we should invoke Theorem 3
instead, so we can only deduce $f\in S$. We arrive at the following
Proposition.

######  Proposition 26.

Let $f\in F$ satisfy the inhomogenous difference equation

$\phi(f)=af+b$

with $a,b\in K$. Then either $f\in S$ or $\\{f,\psi(f),\psi^{2}(f),\dots\\}$
are algebraically independent over $K$.

######  Remark.

We shall not need this Proposition. In the last stage of our proof of Theorem
5 we shall deal with the same type of inhomogenous equation, but where $b\in
S.$ We shall give full details there.

###  5.3. The case of a simple $\mathcal{G}$

####  5.3.1. Recall of notation and assumptions

Let

$M=(K^{n},\Phi),\,\,\Phi(v)=A^{-1}\phi(v)$

be the rank-$n$ $\phi$-module over $K$ associated with the system
$\phi(Y)=AY.$

Let $L\subset L_{\psi}$ be PV and PPV extensions for $M$, $\mathcal{G}$ the
(classical) difference Galois group and $G$ the (parametrized) $\psi$-Galois
group.

Fixing a fundamental matrix $U$ with entries in $L$ we get embeddings

$G\subset[\psi]\mathcal{G}\subset[\psi]GL_{n,\mathbb{C}}.$

When we base change $M$ to $L$ we get the full, $n$-dimensional, complex
vector space of “solutions”

(5.3) $\mathcal{V}=M_{L}^{\Phi}=U\mathbb{C}^{n}\subset M_{L}=L^{n}.$

If instead of $L$ we use the even larger PPV extension $L_{\psi}$ we get the
same complex vector space $\mathcal{V}$, as all the solutions already lie in
$L^{n}$. However, over $L_{\psi}$ we get also the solution spaces
$\psi^{i}(\mathcal{V})$ of all the $\psi$-transforms $M^{(\psi^{i})},$ $i\geq
0.$

The difference Galois group $\mathcal{G}$ acts on $\mathcal{V}$. If
$\sigma\in\mathcal{G}(\mathbb{C})$ then for $v\in\mathbb{C}^{n}$

$\sigma(Uv)=UV(\sigma)v,$

so $\sigma\mapsto V(\sigma)$ is the matrix representation of $\mathcal{G}$ on
$\mathcal{V}$ in the basis consisting of the columns of $U$.

####  5.3.2. The simple case

######  Lemma 27 ([A-D-H-W], Lemma 3.9).

Assume that $L_{\psi}$ is a field. Then $\mathcal{G}$ is connected and $G$ is
$\psi$-integral, hence in particular $\psi$-reduced.

The proof relies on Proposition 7. Recall that being $\psi$-integral means
that the coordinate ring $\mathbb{C}\\{G\\}$ is a domain and $\psi$ is
injective on it.

######  Proposition 28 ([A-D-H-W], Proposition 4.11).

Assume that $L_{\psi}$ is a field. If $\mathcal{G}$ is simple, then
$G=[\psi]\mathcal{G}.$ In particular,

$\psi\mathrm{tr.deg.}(L_{\psi}/K)=\psi\dim(G)=\dim(\mathcal{G})>0.$

###### Proof.

We repeat the arguments of [A-D-H-W]. Note first that $G$ is Zariski dense in
$\mathcal{G}$ (always true) and $\psi$-reduced (by the previous lemma). By
Proposition 15, if $G\subsetneq[\psi]\mathcal{G}$, there exists an $h\in
GL_{n}(\mathbb{C})$ and an $m\geq 1$ such that

$\psi^{m}(X)=hXh^{-1}$

holds in $G.$

By Proposition 17 $M$ is then $\psi^{m}$-isomonodromic.

By Theorem 6, $\mathcal{G}$ must be solvable, contradicting the assumption
that it was simple. This contradiction shows that we must have had
$G=[\psi]\mathcal{G}.$ ∎

######  Proposition 29 ([A-D-H-W], Proposition 4.12).

Assume that $L_{\psi}$ is a field. Then either $\mathcal{G}$ is connected and
solvable or $\psi\mathrm{tr.deg.}(L_{\psi}/K)>0.$

###### Proof.

Same as in [A-D-H-W]. Connectendness follows from Lemma 27. If $\mathcal{G}$
is not solvable then it has a simple quotient $\mathcal{G}/\mathcal{N}.$ The
Galois correspondence theorem is used to obtain a PV extension
$L^{\prime}=L^{\mathcal{N}}$ for a $\phi$-module $M^{\prime}$, whose
difference Galois group is $\mathcal{G}/\mathcal{N}.$ The PPV extension of the
same $M^{\prime}$ is a subfield $L^{\prime}_{\psi}\subset L_{\psi},$ and by
the previous Proposition $\psi\mathrm{tr.deg.}(L_{\psi}^{\prime}/K)>0.$ A
fortiori, $\psi\mathrm{tr.deg.}(L_{\psi}/K)>0.$ ∎

###  5.4. Conclusion of the proof: Galois acrobatics

We prove the following claim, which clearly implies Theorem 5, by induction on
$n$. The assumptions on $p$ and $q$ are maintained.

######  Claim 30.

Let $n\geq 1,$ $A\in GL_{n}(K)$ and $u=\,^{t}(u_{1},\dots,u_{n})\in F^{n}$ a
solution of $\phi(Y)=AY.$ Assume that all the coordinates $u_{i}$ are
$\psi$-algebraic, i.e. $\mathrm{tr.deg.}(K(u)_{\psi}/K)<\infty$ where
$K(u)_{\psi}$ is the field $\psi$-hull of $K(u)$ in $F$. Then $u\in S^{n}.$

The case $n=1$ follows from Corollary 25. The following Lemma will be used to
reduce the general case to certain inhomogenous first order equations, albeit
with coefficients outside $K$.

######  Lemma 31.

Without loss of generality, we may assume:

  1. (1)

The PPV extension $L_{\psi}$ is a field, and all its elements are
$\psi$-algebraic over $K$. Equivalently, $\psi\dim(G)=0.$

  2. (2)

$\mathcal{G}$ is solvable, and the matrices $A$ and $U$ are upper triangular.

  3. (3)

The diagonal elements of $A$ are in $\mathbb{C}^{\times}.$

  4. (4)

$K(u)_{\psi}\subset L_{\psi}$ and the vector $u$ is the last column of $U$.

###### Proof.

As before, replacing $\phi$ and $\psi$ by some powers $\phi^{r}$ and
$\psi^{s}$, and $A$ by $A_{[r]}$, we may assume, _not changing $u$,_ that the
PPV extension $L_{\psi}$ of $M$ is a field, $K(u)_{\psi}$ (a priori a subfield
of $F$) is embedded as a $(\phi,\psi)$-subfield of $L_{\psi}$, and that the
(classical) difference Galois group $\mathcal{G}$ is connected. Let $U$ be a
fundamental matrix, with entries in $L_{\psi}.$ The given vector $u$ is some
linear combination of its columns.

Let $L_{\psi}^{a}$ be the subfield of $L_{\psi}$ consisting of elements that
are $\psi$-algebraic. It is stable under $\phi$ and $\psi$, and also under
$G.$

Let $\mathcal{V}=M_{L_{\psi}}^{\Phi}=U\mathbb{C}^{n}\subset L_{\psi}^{n}$ be
the space of solutions, and
$\mathcal{V}^{a}=\mathcal{V}\cap(L_{\psi}^{a})^{n}.$ Since $\mathcal{V}^{a}$
is fixed by $G$ and $G$ is Zariski dense in $\mathcal{G}$, the (classical)
Galois group $\mathcal{G}$ fixes $\mathcal{V}^{a}$ as well. Let $H$ be the
maximal parabolic subgroup of $GL_{n,\mathbb{C}}$ which is the stabilizer of
$\mathcal{V}^{a}$. Since $\mathcal{G}\subset H$ we may assume that $A$ and $U$
lie in $H.$ After a $\mathbb{C}$-linear change of coordinates we may assume
that

$A=\left(\begin{array}[]{cc}A_{11}&A_{12}\\\
0&A_{22}\end{array}\right),\,\,\,U=\left(\begin{array}[]{cc}U_{11}&U_{12}\\\
0&U_{22}\end{array}\right)$

in block form ($A_{11}$ and $U_{11}$ of size $n_{1}\times n_{1}),$ and the
first $n_{1}$ columns of $U$ span $\mathcal{V}^{a}\simeq\mathbb{C}^{n_{1}}.$
Thus $u$ is a $\mathbb{C}$-linear combination of the first $n_{1}$ columns of
$U$. Replacing our system of equations by the system $\phi(Y)=A_{11}Y$ we may
assume that $n=n_{1},$ $A=A_{11},$ $\mathcal{V}=\mathcal{V}^{a}$ etc., hence
$L_{\psi}=L_{\psi}^{a}.$ This proves (1).

By Proposition (29) the Galois group $\mathcal{G}$ is solvable. By the Lie-
Kolchin theorem we may assume that $\mathcal{G}$ is contained in the upper-
triangular Borel subgroup of $GL_{n,\mathbb{C}}.$ By Theorem 12, so are $U$
and $A$. This proves (2).

Recall that the vector $u$ is linear combination of the columns of $U$. If the
last non-zero entry of $u$ is $u_{r}$ then we can assume, by a further change
of coordinates, not affecting the upper triangular form, that $u$ is the $r$th
column of $U$. Replacing the system of equations by the one correspoding to
the upper-left $r\times r$ block we may assume that $r=n,$ and $u$ is the last
column of $U$. This is (4).

By induction, we may assume that (3) had been proved for all the diagonal
elements of $A,$ but the first one. Since $u_{11}$ is a solution of
$\phi(y)=a_{11}y$, and is $\psi$-algebraic, Proposition 23 shows that there
exists a $b\in K^{\times}$ such that
$a_{11}/(\phi(b)/b)\in\mathbb{C}^{\times}.$ Replacing $A$ by the gauge-
equivalent $\phi(P)^{-1}AP,$ where $P=\mathrm{diag}.[b,1,\dots,1],$ we get
(3). ∎

From now on the proof can be concluded either by the methods of [dS2], or by
the Galois theoretic “acrobatics” adapted from [A-D-H-W]. We chose to use the
second approach.

Assume therefore that we are in the situation of the Lemma. In particular
$u_{i}=u_{in}$ is the $i$th entry in the last column of the fundamental matrix
$U$. The $u_{i}$ lie in $F$, but also in $L_{\psi}$, by our assumption (4)
that $K(u)_{\psi}\subset L_{\psi}$.

We may also assume that $E=Quot(S)$ is contained in $L_{\psi}.$ If this is not
the case, augment the matrix $A$ by adding to it along the diagonal a $2\times
2$ block as in example 2.7. The fundamental matrix gets augmented by the block

$\left(\begin{array}[]{cc}z&\zeta(z,\Lambda)\\\ 0&1\end{array}\right),$

hence the PPV extension for the augmented system contains $E$. If we prove the
main theorem for the augmented system, we clearly have proved it also for the
original one.

By induction we may assume that $u_{2},\dots,u_{n}\in S$. We have to show that
$u^{\prime}=u_{1}\in S.$

This $u^{\prime}\in F$ satisfies the equation

$\phi(u^{\prime})=au^{\prime}+b$

where $a=a_{11}\in\mathbb{C}^{\times},$ and
$b=a_{12}u_{2n}+\cdots+a_{1n}u_{nn}\in S$ (by our induction hypothesis). Let
$v=u_{11}$, so that $\phi(v)=av.$

Since $(\phi-a)(u^{\prime})\in S\subset S_{\phi}(F/K)$ clearly $u^{\prime}\in
S_{\phi}(F/K).$ To conclude the proof we shall show, following the ideas of
the proof of Proposition 4.5 in [A-D-H-W] that

$u^{\prime}\in S_{\psi}(F/K)$

as well. Theorem 3 will show then that $u^{\prime}\in S$, as desired.

Consider the matrix

$U^{\prime}=\left(\begin{array}[]{cc}v&u^{\prime}\\\ 0&1\end{array}\right)\in
GL_{2}(L_{\psi}).$

This is a fundamental matrix for the system

$\phi(Y)=\left(\begin{array}[]{cc}a&b\\\ 0&1\end{array}\right)Y,$

regarded as a system of difference equations over $E$. The field

$L^{\prime}_{\psi}=E(v,u^{\prime})_{\psi}\subset L_{\psi}$

is its PPV extension. Furthermore, the $\psi$-Galois group of the last system
is

$G^{\prime}=Gal^{\psi}(L^{\prime}_{\psi}/E)\subset\left\\{\left(\begin{array}[]{cc}\alpha&\beta\\\
0&1\end{array}\right)\in GL_{2}\right\\}$

and its intersection with the unipotent radical (where $\alpha=1$), denoted
$G^{\prime}_{u},$ corresponds via the Galois correspondence to $E(v)_{\psi}:$

$G^{\prime}_{u}=Gal^{\psi}(L^{\prime}_{\psi}/E(v)_{\psi}).$

This is a $\psi$-subgroup of $[\psi]\mathbb{G}_{a}$. By our assumption that
$u^{\prime}$ is $\psi$-algebraic over $K$, hence clearly over $E$,

$\psi\dim
G^{\prime}_{u}=\psi\mathrm{tr.deg.}(L^{\prime}_{\psi}/E(v)_{\psi})=0.$

It follows from Corollary A.3 of [DV-H-W1] that there exists an
$0\neq\mathcal{L}_{1}\in\mathbb{C}[\psi]$ such that for any
$B\in\mathrm{Alg}_{\mathbb{C}}^{\psi}$ we have

$G^{\prime}_{u}(B)\subset\left\\{\left(\begin{array}[]{cc}1&\beta\\\
0&1\end{array}\right)\in GL_{2}(B)|\,\,\mathcal{L}_{1}(\beta)=0\right\\}.$

Observe that

$\phi(\frac{u^{\prime}}{v})=\frac{u^{\prime}}{v}+\frac{b}{av},$

and $b/av\in E(v)_{\psi},$ so $L^{\prime}_{\psi}/E(v)_{\psi}$ is a PPV
extension for $\phi(y)=y+(b/av),$ equivalently for the system

$\phi(Y)=\left(\begin{array}[]{cc}1&b/av\\\ &1\end{array}\right)Y$

over $E(v)_{\psi}.$ The action of $\tau\in G^{\prime}_{u}(B)$ is given by

$\tau(\frac{u^{\prime}}{v})=\frac{u^{\prime}}{v}+\beta_{\tau}$

where $\beta_{\tau}\in B$ corresponds to the above realization of $\tau$ as a
unipotent $2\times 2$ matrix. Indeed, if

$\tau(U^{\prime})=U^{\prime}\left(\begin{array}[]{cc}1&\beta_{\tau}\\\
0&1\end{array}\right)$

then $\tau(u^{\prime})=v\beta_{\tau}+u^{\prime}$ and $\tau(v)=v$.

It follows that for any $\tau\in G^{\prime}_{u}(B)$

$\tau\left(\mathcal{L}_{1}(\frac{u^{\prime}}{v})\otimes
1\right)=\mathcal{L}_{1}(\tau(\frac{u^{\prime}}{v}))\otimes
1=\mathcal{L}_{1}(\frac{u^{\prime}}{v}+\beta_{\tau})\otimes
1=\mathcal{L}_{1}(\frac{u^{\prime}}{v})\otimes 1,$

hence $\mathcal{L}_{1}(u^{\prime}/v)\in E(v)_{\psi}.$ But

$(v^{\psi-1})^{\phi-1}=(v^{\phi-1})^{\psi-1}=a^{\psi-1}=1,$

so $d=v^{\psi-1}\in\mathbb{C}$ and $\psi(v)=dv,$ $E(v)_{\psi}=E(v).$ It
follows that there exists a second operator
$\mathcal{L}_{2}\in\mathbb{C}[\psi],$ easily derived from $\mathcal{L}_{1},$
such that

$\mathcal{L}_{2}(u^{\prime})\in E(v).$

This $\mathcal{L}_{2}(u^{\prime})$ satisfies the equation

$\phi(y)=ay+\mathcal{L}_{2}(b)$

where we have used the fact that $a$ was constant, and where
$\mathcal{L}_{2}(b)\in S.$ By Lemma 4.7 of [A-D-H-W], with $E$ serving as the
base field _and_ the intermediate field, there exists a $g\in E$ with

$\phi(g)=ag+\mathcal{L}_{2}(b).$

We are indebted to Charlotte Hardouin for pointing out the following lemma.

######  Lemma 32.

In fact, $g\in S$.

###### Proof.

Let $I=\\{s\in S|\,sg\in S\\}$ be the ideal of denominators of $g.$ If $s\in
I$ then $\phi(s)\in I$ because

$\phi(s)g=a^{-1}(\phi(sg)-\phi(s)\mathcal{L}_{2}(b))\in S.$

Since $S$ is a simple $\phi$-ring (it is the localization at $z$ of the PV
ring $K[z,\zeta(z,\Lambda)]$ associated to the system considered in Example
2.7), we must have $1\in I,$ so $g\in S.$ ∎

It follows that
$\phi(\mathcal{L}_{2}(u^{\prime})-g)=a(\mathcal{L}_{2}(u^{\prime})-g).$ Since
also $\phi(v)=av,$ the element $d^{\prime}=(\mathcal{L}_{2}(u^{\prime})-g)/v$
is fixed by $\phi,$ hence lies in $\mathbb{C}$, and

$\mathcal{L}_{2}(u^{\prime})=d^{\prime}v+g.$

Since $(\psi-d)d^{\prime}v=d^{\prime}(\psi-d)v=0$

$(\psi-d)\circ\mathcal{L}_{2}(u^{\prime})=(\psi-d)(g)\in S.$

As ($\psi-d)\circ\mathcal{L}_{2}\in\mathbb{C}[\psi]$ and any element of $S$ is
annihilated by a non-trivial operator from $K[\psi],$ we deduce that
$u^{\prime}\in S_{\psi}(F/K),$ and the proof is complete.

_Acknowledgements._ The author would like to thank Charlotte Hardouin and
Thomas Dreyfus for fruitful discussions regarding this work. He would also
like to thank Zev Rosengarten and the participants of the Kazhdan Seminar at
the Hebrew University for carefully and critically attending his lectures on
the subject.

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