EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

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\title Galois groups and connection matrices of $q$-difference equations
\author {\rm { Pavel I. Etingof}}
Department of Mathematics,
Harvard University,
Cambridge, MA 02138, USA.\newline
\email etingof\@math.harvard.edu \endemail
\subjclass 12H10;39A10 \endsubjclass
\date April 6, 1995 \enddate
\communicatedby{David Kazhdan}


We study the Galois group of a matrix $q$-difference equation
with rational coefficients which is regular at $0$ and $\infty$,
in the sense of (difference) Picard-Vessiot theory, and
show that it coincides
with the algebraic group generated by matrices $C(u)C(w)^{-1}$ $u,w\in\C^*$,
where $C(z)$ is the Birkhoff connection matrix of the equation.

%\vskip .1in
%\centerline{April 3, 1995}
%\vskip .05in
%\centerline{\bf Abstract}
%\vskip .1in
%We study the Galois group of a matrix $q$-difference equation
%with rational coefficients which is regular at $0$ and $\infty$,
%in the sense of (difference) Picard-Vessiot theory, and
%show that it coincides
%with the algebraic group generated by matrices $C(z)C(w)^{-1}$ $z,w\in\C^*$,
%where $C(z)$ is the Birkhoff connection matrix of the equation.
%\vskip .1in

\centerline{\bf 1. Differential algebra}
\vskip .1in

The notion of the Galois group of a linear ordinary differential equation
or a holonomic system of such equations is well known.
For example, consider the differential equation
\frac{df(z)}{dz}=a(z)f(z),\tag 1.1
where $a$ is an $N\times N$-matrix valued function
and $f$ is an unknown $\C^N$-valued function of one complex variable $z$
(both are assumed holomorphic in $z$ in a certain region).
To define the Galois group, one fixes a field of functions $F$
containing the coefficients of the equation
and invariant under $d/dz$. Let $L$ be the field
generated over $F$ by all solutions of the system. This field is 
invariant under $d/dz$. 
The Galois group $G$ of the system is, by definition,
the group of all automorphisms $g$
of the field $L$
such that $g$ fixes all elements of $F$ and $[g,d/dz]=0$.

The group $G$ is naturally isomorphic
to a linear algebraic group over $\C$.
Indeed, let $f_1,...,f_N\in L$ be a basis of the space of solutions of the system.
Then for any $g\in G$, $g(f_i)$ are also solutions of the system, so
$g(f_i)=\sum_j g_{ij}f_j$. Moreover, $g$ is uniquely determined by the matrix $g_{ij}$.
Thus we get an embedding of $G$ into $GL_N(\C)$, given by $g\to \{g_{ij}\}^t$
(superscript $t$ means transposition).
It can be shown that the image of this map is a closed subgroup.
If we replace the basis $f_1,...,f_N$
by another basis, this embedding will be conjugated by the corresponding transformation

The properties of
group $G$ are the subject of Picard-Vessiot (or differential Galois) theory,
and are described in \cite{Ko,R,Ka}. This theory allows to prove that certain
linear differential equations of second and higher order cannot
be reduced to a sequence of first order equations, by showing that their
Galois groups are not solvable.

Computation of the Galois group of a given system of differential equations
is, in general, a nontrivial problem, like in the usual Galois theory.
However, in some cases, this group is easily computable. One of the most interesting
cases is the following.

Let $X$ be a smooth
projective algebraic variety over $\C$. Let $S$ be a hypersurface on $X$.
Let $D$ be an $N$-dimensional
 holonomic system of linear differential equations over $X$ with rational coefficients
which are regular outside $S$. Assume that the system has regular singularities at $S$.
Then the Galois group of $D$ over the field $F=\C(X)$ coincides with the Zariski closure
$\bar \Gamma$ of the monodromy group $\Gamma$ of $D$ in $GL_N(\C)$.

This well known theorem is proved, roughly,
as follows. Assume for simplicity that $X$ is a curve, $S$ is a finite set.
We assume that we fixed a base point $x\in X\setminus S$ and a basis
of solutions of the system near this point. Let $G$ be the Galois group.
It is clear that $\Gamma\subset G$, since analytic continuation along a nontrivial
loop starting and ending at $x$
clearly defines an automorphism of the solution field $F$ fixing rational functions and
$d/dz$. Therefore, $\bar \Gamma\subset G$. Now assume that $\bar\Gamma$ is a proper closed
subgroup in $G$. Then, by the main theorem of the differential Galois theory (\cite{Ka},
Theorem 5.9)
$\C(X)=L^G\subset L^{\bar\Gamma}$ is a proper subfield, where $L^H$ is the field of
$H$-invariants in $L$. But it is easy to see that $L^{\bar\Gamma}=\C(X)$. It is even
true that
$L^\Gamma=\C(X)$. Indeed, let $f\in L^\Gamma$.
Let $\tilde X$ be the universal cover of $X\setminus S$. Then $f$
is meromorphic on $\tilde X$. Because $f$ is invariant under the monodromy group,
it is meromorphic on $X\setminus S$. Since our equation has regular singularities, $f$ has
to have polynomial growth near $S$. Thus, $f$ is meromorphic at $S$ too, so $f\in\C(X)$.

Among other things, this theorem implies the following.
Assume that our variety $X$, hypersurface $S$, and system $D$ are defined
over the field $\bar\Bbb Q$ of algebraic numbers. Then the group $\bar\Gamma$ is
defined over algebraic numbers; that is, we can find a basis of solutions
of $D$ in which $\bar\Gamma=H(\C)$, where $H$ is an affine algebraic
group over $\bar\Bbb Q$. Indeed, the definition of the Galois group
is purely algebraic, so it is defined over $\bar\Bbb Q$ automatically.
Therefore, so is $\bar\Gamma$. More generally, one can see
that if the data $X,S,D$
are defined over any subfield $K\subset\C$, the group $\bar\Gamma$ is defined over some
finite extension $M$ of $K$ ($\bar\Gamma=H(\C)$, $H$ is defined over $M$; however, note that
$H$ is not unique). This is true because $G$ has the same property (this is also
discussed in \cite{De}). In contrast to this,
note that $\Gamma$ itself usually contains
transcendental points of $\bar\Gamma$, i.e. points
which are not defined over $\bar\Bbb Q$.

In the next sections we describe difference counterparts of these statements.
For simplicity we restrict ourselves to the case of equations with one independent variable.

\bf 2. Difference algebra.

The following notions of difference algebra can be found in \cite{Co},\cite{Fr}, and
references therein.\nobreak

Let $F$ be a field of characteristic $0$. We say that $F$ is an (inversive) difference
field if it is equipped with an automorphism $T:F\to F$. For example, if $M$ is
the field of meromorphic functions in $\C^*$, and $Tf(z)=f(qz)$, where
$q\in\C^*$ is a fixed number, then $(M,T)$ is a difference field.
Also, any $T$-invariant subfield of $M$ is a difference field with the same $T$.

Let $(F,T)$ be a difference field. An element $f\in F$ is called a constant if $Tf=f$.
Constants in $F$ form a field which we denote by $C_F$.

We say that $L$ is a difference field extension of $F$ if $L$ is a difference field
and $F$ is a difference subfield of $L$ (with the same $T$).

Let $F$ be a difference field, and $L$ be a difference field extension of $F$.
Consider a system of difference equations
Tf=af,\quad a\in GL_N(F)\tag 2.1
with respect to $f\in GL_N(L)$.
We say that $L$ is a solution field of (2.1) over $F$ if
there exists a solution $f\in GL_N(L)$ of (2.1), and $L=F(f)$ (by $F(f)$ we mean
the field obtained by adding all entries of $f$ to $F$; it is clearly closed under $T$
because of (2.1)).
If $f'\in GL_N(L)$ is any other solution of (2.1) then $f'=fR$, $R\in GL_N(C_L)$.
Thus solutions of (2.1) in $GL_N(L)$ form a principal homogeneous space of $GL_N(C_L)$.

We say that a solution field $L$ is a Picard-Vessiot extension of $F$ if
$C_F$ is algebraically closed, and $C_L=C_F$.

Note that if $F\subset K\subset L$ is any intermediate field and $L$ is a Picard-Vessiot
extension of $F$, it is also a Picard-Vessiot extension of $K$.

\remark{Remark} Strictly speaking, our definition of a Picard-Vessiot extension
is a little more general than in \cite{Co},\cite{Fr},
where the authors considered extensions generated by scalar, higher order difference equations,
of the form $T^nf+a_1T^{n-1}f+...+a_nf=0$, $a_i\in F$, $f\in L$. However, the theory
of matrix equations (2.1) is completely analogous to the theory of scalar equations
of higher order, so we make no distinction between them.

Let $F\subset L$ be a Picard-Vessiot extension associated to (2.1).
The Galois group of $L$ over $F$,
$G=\text{Gal}(L/F)$ is, by definition, the group of all automorphisms of $L$
fixing all elements of $F$ and commuting with $T$.

Let $f$ be a solution of (2.1) in $GL_N(L)$. For any $g\in G$, $g(f)$ is another solution.
So there exists a unique matrix $R_g\in GL_N(C_F)$ such that $g(f)=fR_g$. Thus the assignment
$g\to R_g$ defines an embedding $G\to GL_N(C_F)$. The image of this embedding
is a closed subgroup, because it can be described as follows.

Let $I$ be the set of all polynomials $P$ of $N^2$ variables over $F$
such that $P(f)=0$. It is clear that $I$ is a radical ideal, and $F[f]$ is isomorphic
to $F[X]/I$, $X=\{x_{ij}\}$, $i,j=1,...,N$, via the map
$f\to X$. Let $Y$ be the spectrum of the ring $F[X]/I$. It is an affine algebraic variety
defined over $F$. We call this variety the variety of relations for $L$ and call the ideal $I$
the ideal of relations. We have $L=F(f)=F(Y)$. It is clear that the image of $G$ in
$GL_N(C_F)$ coincides with the set of those $R_g\in GL_N(C_F)$ whose right action maps $Y$
to itself. So it is a Zariski closed matrix group.

For Picard-Vessiot extensions, one has the following main theorem of Galois theory.

\proclaim{Theorem 2.1}{\rm(}\cite{Fr}{\rm)} If $F\subset L$ is a Picard-Vessiot extension then there exists a 1-1 correspondence between
intermediate difference fields $K$, $F\subset K\subset L$, which are 
relatively algebraically closed in $L$,
and connected closed subgroups $K'$ in $\text{Gal}(L/F)$.
This correspondence is given by the formulas $K=L^{K'}$, $K'=\text{Gal}(L/K)$. 
The transcendency degree of $L$ over $K$ equals the dimension of $K'$. 

\bf 3. $q$-Difference equations

 From now on we will assume that the field $F$ is the field $\C(z)$ of rational functions 
in one variable, and $T$ acts by $Tf(z)=f(qz)$, where $q\in\C^*$ is a fixed number 
with $|q|<1$. Let $M$ be the field of all meromorphic functions in $\C^*$. This is a difference field,
with the same $T$. We call a Picard-Vessiot extension of $\C(z)$ meromorphic if it can be 
embedded into $M$ consistently with $T$. Since $\C(z)$ is algebraically closed in $M$
(a single-valued algebraic function is rational), 
the Galois group of any meromorphic Picard-Vessiot extension is connected. 
Also, it is known \cite{Fr} that 
if two Picard-Vessiot extensions of the same difference field can be placed inside of a 
common difference field, then they are isomorphic to each other
as difference field extensions.  This shows that if there exists 
at least one meromorphic Picard-Vessiot extension of $\C(z)$ corresponding to a given 
system of equations, then the Galois group depends only on the system and not on 
the choice of such extension.

We will also assume that the element $a$ used in (2.1) 
(which is now a rational matrix function) is such that $a(0)=a(\infty)=1$. Then
it is easy to construct two meromorphic Picard-Vessiot extensions of $\C(z)$ corresponding
to (2.1) as follows. 

1. Let 
f_0(z)=\prod_{j=0}^\infty a(q^jz)^{-1}=a(z)^{-1}a(qz)^{-1}...\tag 3.1
$f_0$ is a solution of (2.1), as a meromorphic function. Let $L_0=\C(z,f_0)$.

2. Let 
f_\infty(z)=\prod_{j=1}^\infty a(q^{-j}z)=a(q^{-1}z)a(q^{-2}z)...\tag 3.2
$f_\infty$ is a solution of (2.1). Let $L_\infty=\C(z,f_\infty)$. 
Then $L_0,L_\infty$ are Picard-Vessiot extensions of $\C(z)$ (the constant field is $\C$).
This follows from the fact that all elements of $L_0$ ($L_\infty$) are meromorphic
at $0$ ($\infty$), so if we have an element that is periodic: $h(z)=h(qz)$, in either of them,
this element has to be constant.

We know that these extensions should be identical. But how to identify them? This is done
as follows.

Consider the Birkhoff connection matrix of (2.1):
C(z)=f_0(z)^{-1}f_\infty(z)=\prod_{j=-\infty}^\infty a(q^{-j}z)=...a(qz)a(z)a(q^{-1}z)... \tag 
This is an elliptic function with values in $N\times N$ matrices: it is meromorphic in $\C^*$ and

\proclaim{Theorem 3.1} Let $w\in\C^*$ be such that $C(w)$ is finite and invertible.
Then the map $f_\infty\to f_0 C(w)$ defines an isomorphism of Picard-Vessiot extensions
$\tau_w:L_\infty\to L_0$.

{\it Proof.} The proof relies on the following simple but crucial analytic lemma.

\proclaim{Lemma} Assume that we have an identity
\sum_{j=1}^n m_j(z)p_j(z)=0,\tag 3.4
where $m_j$ are meromorphic functions in $\C$, 
and $p_j$ are periodic meromorphic 
functions in $\C^*$:
$p_j(qz)=p_j(z)$. Then
\sum_{j=1}^n m_j(z)p_j(w)=0,\tag 3.5
for generic $z,w$.

{\it Proof of the Lemma.} Fix $w\in\C^*$ such that $p_j$ are nonsingular at $w$.
Using (3.4) and periodicity of $p_j$, for sufficiently large $k$ 
(so that $m_j(q^kw)$ are defined) we get
\sum_{j=1}^n m_j(q^kw)p_j(w)=0.\tag 3.6
Consider the function $h(z)=\sum_{j=1}^nm_j(z)p_j(w)$. We have: $h(q^kw)=0$ for big $k$.
Also, $h$ is meromorphic at $0$. Therefore, $h=0$, as desired. $\square$

{\it Proof of the theorem.} Let $I_0$, $I_\infty$ be the ideals of relations of $L_0$, $L_\infty$, 
and let $Y_0$, $Y_\infty$ be the corresponding varieties of relations. Let 
$P\in I_\infty$.
Then we have $P(z,f_\infty(z))=0$. We can rewrite it as $P(z,f_0(z)C(z))=0$.
This relation is exactly of the type (3.4), because $f_0,z$ are meromorphic at $0$, and 
$C$ is periodic. Therefore, by the Lemma $P(z,f_0(z)C(w))=0$ for generic $z,w$.

Now consider the map $\tau_w: \C(z)[X]\to\C(z)[X]$ ($X=\{x_{ij}\}$) given by $\tau_w(X)=XC(w)$.
I claim that this homomorphism maps $I_\infty$ to $I_0$. 
Indeed, we just showed that if $P\in I_\infty$
then $\tau_w(P)\in I_0$, since $\tau_w(P)(z,f_0(z))=P(z,f_0(z)C(w))=0$. Therefore,
$\tau_w$ descends to a morphism of algebraic varieties $Y_0\to Y_\infty$. This morphism is an
isomorphism -- the inverse to $\tau_w$ is constructed analogously to $\tau_w$. 
In particular, we get an isomorphism of the fields of rational functions on these 
varieties, i.e. a difference field isomorphism $L_\infty\to L_0$, which is identity
on $\C(z)$, as desired. $\square$
\vskip .05in

Now choose $u,w\in\C^*$ 
and consider the composition $\tau_w^{-1}\tau_u: L_\infty\to L_\infty$.
This composition acts by $f_\infty\to f_\infty C(u)C(w)^{-1}$ and defines an element
of the Galois group $\text{Gal}(L_\infty/F)$. Thus, we proved

\proclaim{Corollary} The matrix $C(u)C(w)^{-1}$, for all values of $u,w$
for which it is defined and nondegenerate, belongs to  
$G=\text{Gal}(L_\infty/F)$ (where $G$ is regarded as a closed subgroup in $GL_N(\C)$).

Let $\Gamma$ be the group generated by the matrices $C(u)C(w)^{-1}$. 

\proclaim{Proposition 3.2} $\Gamma$ is a closed connected subgroup of 

\demo{Proof} Connectedness is obvious, because the set of matrices $C(u)C(w)^{-1}$ is  
connected. Let us show that $\Gamma$ is closed.

Let $E$ be the elliptic curve $\C^*/q^\Z$. Let $E_0\subset E$ be the open set consisting
of all points $w$ such that $C(w)$ is defined and invertible.
We have a regular map of algebraic varieties: $\mu_1: E_0^2\to GL_N(\C)$ defined by
$\mu_1(u,w)=C(u)C(w)^{-1}$. Consider the regular map
$\mu_k:E_0^{2k}\to GL_N(\C)$ defined by 
\mu_1(u_1,w_1)...\mu_1(u_k,w_k).\tag 3.7
Let $X_k$ be the image of $\mu_k$, and let $\bar X_k$ be the closure of $X_k$.
It is clear that $\bar X_k$ are irreducible (because of irreducibility   
of $E_0^{2k}$), and $\bar X_k\subset \bar X_{k+1}$. This means that they must stabilize:
there exists $m$ such that $\bar X_m=\cup_{k\ge 1}\bar X_k$. Observe that 
$\Gamma=\cup_{k\ge 0}X_k$. Therefore, $\bar \Gamma=\bar X_m$. But
a standard theorem of algebraic geometry claims that if $f:X \to Y$ is a morphism of 
quasiprojective varieties, and $f(X)$ is dense in $Y$ then $f(X)$ contains an open subset
of $Y$. Applying this to $\mu_m:E_0^{2m}\to \bar X_m$,
we can let $X_m^0\subset X_m\subset\Gamma$ be an open subset of 
$\bar X_m=\bar\Gamma$. Thus, we have a connected affine algebraic group $\bar\Gamma$ and 
a subgroup $\Gamma$ containing an open subset of $\bar\Gamma$. This implies that 

\proclaim{Theorem 3.3} $\text{Gal}(L_\infty/F)=\Gamma$. \endproclaim

\demo{Proof} Let $h\in L_\infty$, and assume that $h$ is
fixed by all matrices $C(u)C(w)^{-1}$. 
We have $h(z)=P(z,f_\infty(z))$, where $P$ is some rational function. 
In particular, $h$ is meromorphic at infinity. By the invariance property, we get
$h(z)=P(z,f_\infty(z)C(u)C(w)^{-1})$ for such $u,w$ that $C(u),C(w)$ are nondegenerate. 
In particular, for $u=z$
we get $h(z)=P(z,f_0(z)C(w)^{-1})$. This shows that $h$ is meromorphic at the origin. 
Thus, $h$ is meromorphic on the whole Riemann sphere, i.e. rational. This proves:
L_\infty^\Gamma=F=\C(z).\tag 3.8
So, by Theorem 2.1, it follows that $\Gamma$ is the whole Galois group, as desired. $\square$

\remark{Remark} Informally speaking, the matrices $C(u)C(w)^{-1}$ play the role of  
monodromy matrices in the difference case, and the group $\Gamma$ plays 
the role of the monodromy
group. Taking the product $C(u)C(w)^{-1}$, $u\ne w$ corresponds (in the differential setting)
to moving from $\infty$ to $0$
along one path, and returning along another, to obtain a nontrivial loop from 
$\pi_1(\C P^1\setminus S,\infty)$ (assuming $0,\infty$ are regular points). The regularity
of the points $0,\infty$ is expressed in the difference case by the identities 

\proclaim{Corollary} Let $K$ be any subfield of $\C$.
Assume that $a\in K(z)$, $q\in K^*$. Then
the group $\Gamma$ generated by the matrices $C(u)C(w)^{-1}$
is of the form $H(\C)$, where $H$ is an affine algebraic group
defined over $K$.

\remark{Remarks} 1. Note that even if the equations have rational coefficients,
the monodromy matrices
$C(u)C(w)^{-1}$ will be generally transcendental, since they are given by infinite products.
Still, the group generated by them in this case will be defined over rational numbers.

2. Note that the claim is that the two groups (the span of $C(u)C(w)^{-1}$ and $H(\C)$)
 are {\it literally} the same, 
not only up to conjugation.

\demo{Proof} By Theorem 3.3, it is enough to prove this for the Galois group.
To do this, it is enough to prove that the variety of relations $Y_\infty$
of the field $L_\infty$ over $F$ is defined over $K(z)$. Let us write  
the function $f_\infty$ as a series in powers of $1/z$ near $z=\infty$.
It is easy to see from (3.2), that the coefficients of this series are 
rational functions in $q$ with coefficients in $K$. For example, the coefficient
to $1/z$ is $a_{-1}q/(1-q)$, where $a=1+a_{-1}/z+...$. Thus, if $q\in K^*$, 
these coefficients are automatically in $K$. Any relation $P(z,f_\infty)=0$ can therefore
be regarded as a relation in $\C((1/z))$. Therefore, one can find a basis of
the ideal $I_\infty$ which is defined over $K(z)$, i.e. 
$Y_\infty$ is defined over $K(z)$,
as desired. $\square$

Theorem 3.3 allows to obtain an easy solution of the inverse problem of Galois 
theory for difference equations over $\C(z)$:
\proclaim{Proposition 3.4} Let $G$ be any connected affine algebraic group over $\C$. 
Then there exists a Picard-Vessiot extension of $\C(z)$ (in the sense of Section 2)
whose Galois group is $G$. 

\demo{Sketch of Proof} Pick $a\in G(\C(z))$ in such a way that $a(0)=a(\infty)=1$, and 
the elements $a(u)a(w)^{-1}$, taken for all $u,w$ where $a$ is regular,
generate $G$. This will be so ``for most elements'' of $G(\C()$. 
Consider now equation (2.1) with $q$ very close to $0$. Since $C(z)$ is approximately
equal to $a(z)$, $C(u)C(w)^{-1}$ generate $G$. Therefore, by Theorem 3.3, $G$  is the 
Galois group of the Picard-Vessiot extension associated to (2.1). $\square$ 

\remark{Remark} This proof shows that if the group $G$ is defined over a field $K\subset \C$, then 
it is possible to find a system of the form (2.1) with $a\in GL_N(K(z))$ and $q\in K^*$
and a meromorphic Picard-Vessiot extension of $\C(z)$ associated to it, whose Galois group 
is $G$.

Finally, let us discuss solvability of difference equations.
First, let us consider some simple difference equations. 

1. Suppose we have an equation 
f(qz)=a(z)f(z),\tag 3.9
$a(z)$ is a meromorphic scalar-valued function in $\C^*$. 

Let $r\in \R^+$ be such that $a$ has no 
zeros or poles on $|z|=r$. Then $a$ can be uniquely written as a product
$a=a_+a_-a_0$, where $a_0=Cz^m$, $a_+$ is holomorphic in $|z|\le r$, $a_-$ is holomorphic
in $|z|\ge r$, and $a_+(0)=a_-(\infty)=1$.
This is done with the help of Cauchy integral. Now we set 
f_+(z) & = \prod_{m=0}^\infty a_+(q^mz)^{-1},\cr
f_-(z) & = \prod_{m=1}^\infty a_-(q^{-m}z),\cr 
f_0(z) & = (-1)^m\Theta(Cz)^{-1}\Theta(z)^{-m+1},}\tag 3.10
\Theta(z)=\prod_{m\ge 0}(1-q^{m+1})(1-q^mz)(1-q^{m+1}z^{-1})\tag 3.11
is the theta-function.  

It is easy to check that $f_j(qz)=a_j(z)f_j(z)$, $j=+,-,0$. Therefore, $f=f_+f_-f_0$ is
a meromorphic solution of (3.9). Also, any other meromorphic solution of (3.9)  
has the form $f(z)\e(z)$, where $\e(z)$ is an elliptic function, i.e. a meromorphic function in $\C^*$
such that $\e(qz)=\e(z)$.

2. Suppose we have a system of the form  
f_1(qz)=f_1(z)+a(z)f_2(z),\quad f_2(qz)=f_2(z), \tag 3.12
where $a$ is a scalar meromorphic function.
Setting $f_2=1$, we reduce it to
the equation $f(qz)=f(z)+a(z)$. Let $r\in \R^+$ be such that $a$ has no 
zeros or poles on $|z|=r$. Then $a$ can be uniquely written as a sum
$a=a_++a_-+a_0$, where $a_0=C$, $a_+$ is holomorphic in $|z|\le r$, $a_-$ is holomorphic
in $|z|\ge r$, and $a_+(0)=a_-(\infty)=0$. 
This is done by using the Cauchy integral. Now we set 
 f_+(z) & = -\sum_{m=0}^\infty a_+(q^mz),\cr
 f_-(z) & = \sum_{m=1}^\infty a_-(q^{-m}z),\cr 
 f_0(z) & = -Cz\Theta'(z)/\Theta(z).}\tag 3.13
It is easy to check that $f_j(qz)=f_j(z)+a_j(z)$, $j=+,-,0$. Therefore, if $h=f_++f_-+f_0$ then
the matrix $f=\biggl(\matrix 1&h\\0&1\endmatrix\biggr)$ is
a meromorphic solution of (3.12). Moreover, any nondegenerate meromorphic matrix solution
of (3.12) is of the form $f(z)\e(z)$, where $\e$ is a nondegenerate 
$2\times 2$ matrix whose entries are elliptic functions.  

Let $K$ be a difference field extension of $F=\C(z)$, $F\subset K\subset M$.
After \cite{Fr}, we call $K$ a Liouville extension if there exists a tower
of difference fields of the form $F=F_0\subset F_1\subset...\subset F_n=K$ such that
for each $j$, $F_{j+1}=F_j(f)$, where $f$ is a nondegenerate solution of (3.9) or (3.12), 
with $a\in F_j$. We call a difference field extension $K_0$ a generalized Liouville
extension if $K_0\subset K$, where $K$ is Liouville. 

So, the property of an extension $K$ to be generalized Liouville is equivalent to the possibility
of computing any element of $K$ via infinite sums and products, starting with rational functions
(note that we don't allow nonabelian products of the form (3.1), (3.2) -- since any
equation we are considering admits a solution of such form). Therefore, if a meromorphic
Picard-Vessiot extension of $\C(z)$ associated to an equation of the form (2.1)
is generalized Liouville, we will say that this equation can be solved in q-quadratures. 

The following theorem is a special case of a more general statement which can be found
in \cite{Fr}:

\proclaim{Theorem 3.5} A meromorphic Picard-Vessiot extension of $\C(z)$ is   
generalized Liouville if and only if its Galois group is solvable. 

This implies:

\proclaim{Proposition 3.6} An equation of the form (2.1) with $a(0)=a(\infty)=1$ can be solved 
in q-quadratures if and only if the group generated by the matrices $C(u)C(w)^{\hskip-2pt -\hskip-2pt 1}$ (where $C(z)$ is the Birkhoff connection matrix (3.3)), is solvable.

\remark{Final remark}We have been considering the case of difference equations 
in one variable regular at $0,\infty$ ($a(0)=a(\infty)=1)$. 
This was done for the sake of brevity. 
The above theory carries over, with minor modifications, 
to a larger class of equations --
such that $a(0),a(\infty)$ are arbitrary invertible 
matrices, and to analogous systems 
of difference equations with several variables. 
In particular, it can be applied to $q$-hypergeometric systems and 
their generalizations. 

\vskip .1in
\centerline {\bf Acknowledgements.} 
\vskip .1in

I would like to thank Maxim Kontsevich and
Misha Verbitsky for inspiring me to write this note, and David
Kazhdan for useful discussions.


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\publaddr New York\yr 1965\endref

\ref\key De\by Deligne P.\paper Cat\'egories Tannakiennes\jour Grothendieck 
Festschrift\vol 2\publ Birkh\"auser\publaddr Bos\-ton\yr 1991\pages 111-195

\ref\key Fr\by Franke, C.H.\paper Picard-Vessiot theory of linear 
homogeneous difference equations\jour Trans. of AMS\vol 108\pages 491-515
\yr 1963\endref 

\ref\key Ka\by Kaplansky, I.\book
An introduction to differential algebra\publ Hermann\publaddr Paris\yr 1957

\ref\key Ko\by Kolchin, E.R.\book 
Differential algebra and algebraic groups\publ Acad. Press\publaddr New York
\yr 1973\endref

\ref\key R\by Ritt, J.F.\book Differential algebra\publ Colloquium publ. of AMS
\vol XXXIII\yr 1950\endref