# Solutions of the ${\mathfrak{sl}_{2}}$ qKZ equations modulo an integer

Evgeny Mukhin and Alexander Varchenko

###### Abstract.

We study the qKZ difference equations with values in the $n$-th tensor power
of the vector ${\mathfrak{sl}_{2}}$ representation $V$, variables
$z_{1},\dots,z_{n}$ and integer step $\kappa$. For any integer $N$ relatively
prime to the step $\kappa$, we construct a family of polynomials $f_{r}(z)$ in
variables $z_{1},\dots,z_{n}$ with values in $V^{\otimes n}$ such that the
coordinates of these polynomials with respect to the standard basis of
$V^{\otimes n}$ are polynomials with integer coefficients. We show that the
polynomials $f_{r}(z)$ satisfy the qKZ equations modulo $N$.

Polynomials $f_{r}(z)$ are modulo $N$ analogs of the hypergeometric solutions
of the qKZ equations given in the form of multidimensional Barnes integrals.

$\kern-3.99994pt^{\circ}\mskip-0.99998mu$Department of Mathematical Sciences,
Indiana University –Purdue University Indianapolis
402 North Blackford St, Indianapolis, IN 46202-3216, USA

⋆ Department of Mathematics, University of North Carolina at Chapel Hill
Chapel Hill, NC 27599-3250, USA

Key words: qKZ equations; master polynomials, weight functions, difference
differentials.

2020 Mathematics Subject Classification: 11D79 (12H25, 32G34, 33C05, 33E30)

††footnotetext: ${}^{\circ}\mskip-0.99998mu$E-mail<EMAIL_ADDRESS>supported in part by Simons Foundation grants 353831, 709444
${}^{\star}\mskip-0.99998mu$E-mail<EMAIL_ADDRESS>supported in part by
NSF grant DMS-1954266

##  1\. Introduction

The Knizhnik-Zamolodchikov (KZ) differential equations are a system of linear
differential equations, satisfied by conformal blocks on the sphere in the WZW
model of conformal field theory. The quantum Knizhnik-Zamolodchikov (qKZ)
equations are a difference version of the KZ equations which naturally appear
in the representation theory of Yangians (rational case) and quantum affine
algebras (trigonometric case). As a rule one considers the KZ and qKZ
equations over the field of complex numbers. Then these differential and
difference equations are solved in multidimensional hypergeometric integrals.

In [SV] the KZ differential equations were considered modulo an integer $N$.
It turned out that modulo $N$ the KZ equations have a family of polynomial
solutions. The construction of these solutions was analogous to the
construction of the multidimensional hypergeometric solutions, and these
polynomial solutions were called the $N$-hypergeometric solutions.

In this paper we consider modulo $N$ the rational ${\mathfrak{sl}_{2}}$ qKZ
equations with values in the $n$-th tensor power of the vector representation
of $V$ and with an integer step $\kappa$. The qKZ equations for a function
$f(z_{1},\dots,z_{n})$ with values in $V^{\otimes n}$ have the form

$\displaystyle
f(z_{1},\dots,z_{a}-\kappa,\dots,z_{n})\,=\,K_{a}(z\>;\kappa)\,f(z),\qquad
a=1,\dots,n,$

where linear operators $K_{a}(z\>;\kappa)$ are given in terms of the rational
${\mathfrak{sl}_{2}}$ $R$-matrix, see (2.3). The operators $K_{a}(z\>;\kappa)$
commute with the diagonal action of ${\mathfrak{sl}_{2}}$, and, therefore, it
is sufficient to solve the qKZ equations only with values in the space of
singular vectors of a given weight.

We fix an integer $N$ relatively prime to $\kappa$ and construct $V^{\otimes
n}$-valued polynomials in $z_{1},\dots,z_{n}$ such that their coordinates in
the standard basis of $V^{\otimes n}$ are polynomials with integer
coefficients. Then we show that these $V^{\otimes n}$-valued polynomials
satisfy the qKZ equations modulo $N$.

The idea of the construction is as follows. The hypergeometric complex
$V^{\otimes n}$-valued solutions of the qKZ are given in [TV, MV]. The
solutions with values in the subspace of singular vectors of weight $n-2l$ are
written as $l$-dimensional integrals of the form

(1.1)
$f(z)=\int_{{\mathbbm{i}}{\mathbb{R}}^{l}}\Phi(t,z)\,w(t,z)\,W(t,z)dt\,.$

Here $t=(t_{1},\dots,t_{l})$, $\Phi(t,z)$ is a scalar master function given as
a ratio of products of Euler gamma functions, $w(t,z)$ is the vector-valued
rational weight function, and $W(t,z)$ is a scalar $\kappa$-periodic function
with respect to all variables $t,z$; cf. Section A.3. To show that $f(z)$ is a
solution, one observes that the difference

(1.2)
$\Phi(t,z_{1},\dots,z_{a}-\kappa,\dots,z_{n})\,w(t,z_{1},\dots,z_{a}-\kappa,\dots,z_{n})\,-\,K_{a}(z\>,\kappa)\,\Phi(t,z)\,w(t,z)$

is a discrete differential with respect to variables $t_{1},\dots,t_{l}$, that
is, it can be written in the form
$\sum_{s=1}^{l}(G_{a,s}(t)-G_{a,s}(t_{1},\dots,t_{s}-\kappa,\dots,t_{l}))$ for
appropriate functions $G_{a,s}$.

Then using Stokes’ theorem one shows that the integral of a discrete
difference equals zero, and hence $f(z)$ is a solution of the qKZ equations.
The hypergeometric solution $f(z)$ depends on the choice of the
$\kappa$-periodic function $W(t,z)$ from a suitable finite-dimensional vector
space. The functions $W(t,z)$ of that space provide the convergence of the
integral in (1.1) and applicability of Stokes’ theorem.

One may consider the qKZ difference equations with values in a tensor product
$\otimes_{i=1}^{n}V_{i}$ of finite-dimensional $\mathfrak{sl}_{k}$-modules and
construct the corresponding hypergeometric solutions like in formula (1.1).
The functions $\Phi(t,z)$ and $\vec{w}(t,z)$ used in (1.1) have meaning in
that more general situation, see [TV]. They depend on the Cartan matrix and
the highest weights of the modules. In our case the highest weight of the
vector representation is $\lambda=1$ and the Cartan matrix is $C=(2)$.

The main idea of the construction of polynomial solutions of the qKZ equations
modulo $N$, is to modify in formulas for $\Phi(t,z)$ and $w(t,z)$ the numbers
$1$ coming from the highest weight and the numbers $2$ coming from the Cartan
matrix in such a way, that they do not change modulo $N$ but become multiples
of the step $\kappa$. So, we change in the formulas:

$\displaystyle 1\to 1+Nm=-\kappa k,\qquad 2\to 2+Nm^{\prime}=\kappa
k^{\prime},$

where $m,m^{\prime},k,k^{\prime}$ are integers. This is possible if and only
if $N$ and $\kappa$ are relatively prime.

After that change, the ratio of products of gamma functions in the master
function $\Phi$ becomes a polynomial in $t,z$ due to the fundamental property
of the gamma function:

$\displaystyle(-\kappa)^{k}\,\frac{\Gamma\big{(}\frac{z-k\kappa}{-\kappa}\big{)}}{\Gamma\big{(}\frac{z}{-\kappa}\big{)}}=z(z-\kappa)\dots(z-(k-1)\kappa)=:[z]_{k}.$

This modified master function (now called the master polynomial) is a product
of linear factors organized into Pochhammer polynomials of the form $[x]_{k}$
and $[x]_{k^{\prime}}$ with appropriate $x$, see Figure 1. The product $\Phi
w$ also becomes a polynomial. Moreover, each of ${n\choose l}$ coordinates of
$\Phi w$ is a sum of $l!$ terms with each term being a product of Pochhammer
polynomials of the same kind, see Figure 2.

Then we check that modulo $N$ the difference (1.2) is still a discrete
differential.

It remains to find a way to integrate over $t$ which eliminates discrete
differentials of polynomials with integer coefficients. Such a procedure is
clearly impossible over complex numbers (since the discrete derivative map is
surjective), but is well-known modulo an integer $N$. Namely, let a polynomial
$G(t,z)$ be a discrete differential with respect to the $t$-variables, and let

$\displaystyle
G(t,z)=\sum_{r_{1},\dots,r_{l}}a_{r_{1},\dots,r_{l}}(z)[t_{1}]_{r_{1}}\dots[t_{l}]_{r_{l}},\qquad
a_{r_{1},\dots,r_{l}}(z)\in{\mathbb{Z}}[z_{1},\dots,z_{n}].$

Then a coefficient $a_{r_{1},\dots,r_{l}}(z)$ equals zero modulo $N$ if
$r_{i}\equiv-1\pmod{N}$ for $i=1,\dots,l$.

The assignment to a polynomial $G(t,z)$ of a coefficient
$a_{r_{1},\dots,r_{l}}(z)$ with $r_{i}\equiv-1\pmod{N}$ for $i=1,\dots,l$, may
be considered as a modulo $N$ analog of the integral over $t$ of the
polynomial $G(t,z)$. We call it a difference $r$-integral. There is a
generalization of the difference $r$-integral defined for all sequences of
non-negative integers $r=(r_{1},\dots,r_{l})$ which assigns to a polynomial
$G(t,z)$ the polynomial $N_{r}a_{r_{1},\dots,r_{l}}(z)$, where $N_{r}$ is the
smallest integer such that $N_{r}(r_{i}+1)\equiv 0\pmod{N}$. This assignment
also gives zero in ${\mathbb{Z}}/NZ$ , if $G(t,z)$ is a discrete differential.
See Section 3.2.

Thus, if

$\Phi(t,z)w(t,z)=\sum_{r_{1},\dots,r_{l}}c_{r_{1},\dots,r_{l}}(z)[t_{1}]_{r_{1}}\dots[t_{l}]_{r_{l}},$

and $(r_{1},\dots,r_{l})$ is a sequence of non-negative integers then
$f_{r}=N_{r}c_{r_{1},\dots,r_{l}}(z)$ is a solution of the qKZ equations
modulo $N$. Keeping in mind the language of solutions over complex numbers, we
call the solutions constructed in this way, the $N$-hypergeometric solutions.

Since we consider only the tensor products of ${\mathfrak{sl}_{2}}$ vector
representations, and since we have no gamma functions and no analytic issues,
the proofs are simpler than in [MV] or [TV] and are combinatorial in nature.
Moreover, we show that our solutions satisfy the (stronger) symmetric form of
the qKZ equations, see (2.3). This happens due to the symmetric nature of our
“integration”.

The $N$-hypergeometric solution are non-homogeneous polynomials in
$z_{1},\dots,z_{n}$. One can consider the set of solutions of the form
$\sum_{r}g_{r}(z)f_{r}(z)$, where $g_{r}(z)$ are scalar polynomials with
integer coefficients satisfying the congruences

$N_{r}g_{r}(z_{1},\dots,z_{i}-\kappa,\dots,z_{n})\equiv
N_{r}g_{r}(z_{1},\dots,z_{i},\dots,z_{n})\pmod{N}$

for all $i$. It is an interesting problem to describe this set as a module
over the ring of polynomials which are periodic modulo $N$. It is nontrivial
because this module is not free in general and the answer depends on how the
integers $\kappa,N,n,l$ are related to each other. We do not address this
question in this first joint paper on the subject, but we show that the top
degree terms of an $N$-hypergeometric solutions of the qKZ equations
constitute an $N$-hypergeometric solution of the KZ equations constructed in
[SV], see Proposition 6.2, for which some studies have been done, see [V1, V2,
V3].

In contrast to the complex-valued case, the $N$-hypergeometric solutions is
generically do not span the space of singular vectors. Therefore, a natural
question is if there are polynomial modulo $N$ solutions of the qKZ equations
which are not $N$-hypergeometric.

In this paper we consider the qKZ equations with values in a tensor power of
the vector representation of ${\mathfrak{sl}_{2}}$. In the same way we may
construct $N$-hypergeometric solutions of the qKZ equations with values in a
tensor product of finite-dimensional $\mathfrak{sl}_{k}$-modules.

The paper is organized as follows. In Section 2 we define the qKZ equations.
In Section 3 we discuss the generalities of discrete differentiation and
integration modulo $N$. In Section 4 we define the main ingredients of our
solutions: master polynomial and weight functions. In Section 5 we formulate
and prove our main theorem which provides $N$-hypergeometric solutions of the
qKZ equations. These are vectors of non-homogeneous polynomials in $z$ with
integer coefficients solving the qKZ equations modulo $N$. In Section 6 we
show that the sum of the top degree terms of the $N$-hypergeometric solutions
of the qKZ equations coincide with the $N$-hypergeomtric solutions of the KZ
equations. In Appendix A we remind the constructions of the hypergeometric
solutions of the KZ and qKZ equations in the special case in which the
corresponding hypergeometric integrals are one-dimensional.

The authors thank Bonn University and the Max-Planck Institute for Mathematics
in Bonn for hospitality in May-June of 2022 when this paper was conceived.

##  2\. Difference qKZ equations

###  2.1. Notations

Consider the Lie algebra ${\mathfrak{sl}_{2}}$ with basis $e,f,h$ and
relations $[e,f]=h$, $[h,e]=2e$, $[h,f]=-2f$. Let $V$ be the two-dimensional
${\mathfrak{sl}_{2}}$-module with basis $v_{1},v_{2}$ and the action
$ev_{1}=0,ev_{2}=v_{1}$, $fv_{1}=v_{2},fv_{2}=0$,
$hv_{1}=v_{1},hv_{2}=-v_{2}$.

Fix a positive integer $n>1$. The ${\mathfrak{sl}_{2}}$-module $V^{\otimes n}$
has weight decomposition

$\displaystyle V^{\otimes n}=\oplus_{l=0}^{n}V^{\otimes n}[n-2l],$

where $V^{\otimes n}[n-2l]$ is the eigenspace of $h$ with eigenvalue $n-2l$.
Let ${\operatorname{Sing}}V^{\otimes n}[n-2l]\subset V^{\otimes n}[n-2l]$ be
the subspace of singular vectors (the vectors annihilated by $e$).

Let $\mathcal{I}_{l}$ be the set of all $l$-element subsets of
$\\{1,\dots,n\\}$. Denote

$\displaystyle v_{I}=v_{i_{1}}\otimes\dots\otimes v_{i_{n}}\ \in\ V^{\otimes
n},$

where $i_{j}=2$ if $i_{j}\in I$ and $i_{j}=1$ if $i_{j}\notin I$. The set
$\\{v_{I}\ |\ I\in\mathcal{I}_{l}\\}$ is a basis of $V^{\otimes n}[n-2l]$.

###  2.2. The qKZ equations

Define the rational $\,R$-matrix acting on $\>V^{\otimes 2}$,

$R(u)\,=\,\frac{u-P}{u-1}\,,\ignorespaces$

where $P$ is the permutation of factors of $V^{\otimes 2}$. The $R$-matrix
satisfies the Yang-Baxter and unitarity equations,

(2.1) $\displaystyle
R^{(12)}(u-v)R^{(13)}(u)R^{(23)}(v)\,=\,R^{(23)}(v)R^{(13)}(u)R^{(12)}(u-v)\,,$
(2.2) $\displaystyle R^{(12)}(u)R^{(21)}(-u)\,=\,1\,.$

The first equation is an equation in $\operatorname{End}(V^{\otimes 3})$. The
superscript indicates the factors of $\>V^{\otimes 3}$ on which the
corresponding operators act.

Let $\,z=(z_{1},\allowbreak\dots,\allowbreak z_{n})$. Define the qKZ operators
$\,K_{1},\dots,K_{n}\mskip 2.00002mu$ acting on $V^{\otimes n}$:

(2.3) $\displaystyle
K_{a}(z;\kappa)=R^{(a,a-1)}(z_{a}-z_{a-1}-\kappa)\,\dots\,R^{(a,1)}(z_{a}-z_{1}-\kappa)$
$\displaystyle\hskip 120.0pt\times\
R^{(a,n)}(z_{a}-z_{n})\,\dots\,R^{(a,a+1)}(z_{a}\mskip-0.99998mu-z_{a+1})\,,$

where $\kappa$ is a parameter.

The qKZ operators preserve the weight decomposition of $V^{\otimes n}$,
commute with the ${\mathfrak{sl}_{2}}$-action, and form a discrete flat
connection with step $\kappa$,

$\displaystyle
K_{a}(z_{1},\dots,z_{b}-\kappa,\dots,z_{n};\kappa)\,K_{b}(z;\kappa)=K_{b}(z_{1},\dots,z_{a}-\kappa\dots,z_{n};\kappa)\,K_{a}(z;\kappa)$

for $a,b=1,\dots,n$, see [FR].

The system of difference equations with step $\kappa$,

(2.4)
$f(z_{1},\dots,z_{a}-\kappa,\dots,z_{n})\,=\,K_{a}(z\>;\kappa)\,f(z),\qquad
a=1,\dots,n,\ignorespaces$

for a $V^{\otimes n}$-valued function $f(z)$ is called the qKZ equations.

Since the qKZ operators commute with the action of ${\mathfrak{sl}_{2}}$ in
$V^{\otimes n}$, the qKZ operators preserve the subspace
${\operatorname{Sing}}V^{\otimes n}[n-2l]$ for any integer $l$.

###  2.3. The symmetric qKZ equations

Let $P^{(a,a+1)}$ be the operator swapping the $a$-th and $a+1$-st tensor
factors of $V^{\otimes n}$.

Let $\mu\in S_{n}$ be the cyclic permutation $(1,2,\dots,n)$. In terms of
simple transpositions we have

(2.5) $\displaystyle\mu=s_{1,2}s_{2,3}\dots s_{n-1,n}\,.$

Set

$\displaystyle P^{(\mu)}=P^{(1,2)}P^{(2,3)}\dots P^{(n-1,n)}.$

The following system of equations is called the symmetric qKZ equations:

$\displaystyle f(z_{1},\dots,z_{a+1},z_{a},\dots,z_{n})$ $\displaystyle=$
$\displaystyle P^{(a,a+1)}R^{(a,a+1)}(z_{a}-z_{a+1})\,f(z),\quad
a=1,\dots,n-1,$ (2.6) $\displaystyle f(z_{1}-\kappa,z_{2},\dots,z_{n})$
$\displaystyle=$ $\displaystyle P^{(\mu)}f(z_{2},\dots,z_{n},z_{1}).$

The symmetric qKZ equations imply the qKZ equations.

######  Lemma 2.1.

Let $f(z)$ satisfy (2.3). Then $f(z)$ satisfies (2.4).

###### Proof.

For (2.4) with $a=1$ we have

$\displaystyle f(z_{1}-\kappa,z_{2},\dots,z_{n})=P^{(1,2)}\dots
P^{(n-1,n)}f(z_{2},\dots,z_{n},z_{1})$ $\displaystyle=P^{(1,2)}\dots
P^{(n-2,n-1)}P^{(n-1,n)}P^{(n-1,n)}R^{(n-1,n)}(z_{1}-z_{n})f(z_{2},\dots,z_{n-1},z_{1},z_{n})$
$\displaystyle=P^{(1,2)}\dots
P^{(n-2,n-1)}R^{(n-1,n)}(z_{1}-z_{n})P^{(n-2,n-1)}R^{(n-2,n-1)}(z_{1}-z_{n-1})f(z_{2},\dots,z_{1},z_{n-1},z_{n})$
$\displaystyle=P^{(1,2)}\dots
P^{(n-3,n-2)}R^{(n-2,n)}(z_{1}-z_{n})R^{(n-2,n-1)}(z_{1}-z_{n-1})f(z_{2},\dots,z_{1},z_{n-1},z_{n})=\dots$
$\displaystyle=R^{(1,n)}(z_{1}-z_{n})\dots
R^{(1,2}(z_{1}-z_{2})f(z)=K_{1}(z\>,\kappa)f(z).$

Now we use it to show (2.4) with $a=2$,

$\displaystyle
f(z_{1},z_{2}-\kappa,\dots,z_{n})=P^{(12)}R^{(12)}(z_{2}-z_{1}-\kappa)f(z_{2}-\kappa,z_{1},\dots,z_{n})$
$\displaystyle=P^{(1,2)}R^{(1,2)}(z_{2}-z_{1}-\kappa)R^{(1,n)}(z_{2}-z_{n})\,\dots\,R^{(1,2)}(z_{2}-z_{1})f(z_{2},z_{1},\dots,z_{n})$
$\displaystyle=P^{(1,2)}R^{(1,2)}(z_{2}-z_{1}-\kappa)R^{(1,n)}(z_{2}-z_{n})\,\dots\,R^{(1,2)}(z_{2}-z_{1})P^{(12)}R^{(1,2)}(z_{2}-z_{1})f(z)$
$\displaystyle=\,K_{2}(z;\kappa)\,f(z).$

The proof of equations (2.4) with $a>2$ is similar. ∎

In this paper we fix relatively prime integers $\kappa$, $N$. We consider
equations (2.3) modulo $N$, and without loss of generality, we assume that
$0<\kappa<N$. We construct vectors in $V^{\otimes n}[n-2l]$ whose coefficients
in the basis $v_{I}$ are polynomials in variables $z$ with integer
coefficients,

(2.7) $f(z)=\sum_{I\in\mathcal{I}_{l}}f_{I}(z)\,v_{I},\qquad
f_{I}(z)\in{\mathbb{Z}}[z],$

such that equations (2.3) hold modulo $N$ and such that $ef(z)=0$ modulo $N$.

Difference equations modulo $N$ have trivial symmetries. Namely, let $f(z)$ be
a polynomial solution modulo $N$ of a difference equation with step $\kappa$.
Let $g(z)$ be a periodic polynomial,
$g(z_{1},\dots,z_{i}-\kappa,\dots,z_{n})\equiv g(z)\pmod{N}$ for all $i$. Let
$h(x)$ be any polynomial in variables $z$ with values in $V^{\otimes n}$ whose
coordinates have integer coefficients. Then $f(z)g(z)+Nh(x)$ is also a
solution of the difference equations modulo $N$. Moreover, if $N=dd^{\prime}$,
$d,d^{\prime}\in{\mathbb{Z}}$, and all coefficients of all coordinates
$f_{I}(z)$ of $f(z)$ are divisible by $d$, then $f(z)g(z)$ is a solution for
any $g(x)$ which is periodic modulo $d^{\prime}$.

##  3\. Difference differentials

Assume again that the integers $\kappa,N$ are relatively prime and
$0<\kappa<N$. Let $k,k^{\prime}$ be the integers such that $0<k<N$,
$0<k^{\prime}<N$,

(3.1) $\displaystyle\kappa k\,\equiv\,-1,\qquad\kappa
k^{\prime}\,\equiv\,2\pmod{N}.$

The integers $k,k^{\prime}$ exist and are unique.

###  3.1. Pochhammer polynomials

Let $m$ be a positive integer. Define the Pochhammer polynomial

$\displaystyle[t]_{m}=\prod_{i=1}^{m}(t-(i-1)\kappa).$

We have

$\displaystyle[t-\kappa]_{m}=[t]_{m}\,\frac{t-\kappa
m}{t},\qquad[t+\kappa]_{m}=[t]_{m}\,\frac{t+\kappa}{t-(m-1)\kappa}.$

We also call a Pochhammer polynomial $[t]_{m}$ a string of length $m$ which
starts at $t$ and ends at $t-(m-1)\kappa$.

The string $[t]_{m}$ should be thought of as a discretized analog of the
monomial $t^{m}$. For example, in parallel to the monomials, there is a Newton
binomial formula for strings as well:

(3.2) $[t+z]_{m}=\sum_{i=1}^{m}{m\choose i}[t]_{i}[z]_{m-i}.$

If $N=p$ is prime, and $\kappa$ is still relatively prime to $p$, then

$\displaystyle[t]_{p}\equiv t^{p}-t\pmod{p}$

by the Little Fermat theorem. In this case we also have

$\displaystyle[t+z]_{p}\equiv(t+z)^{p}-(t+z)\equiv
t^{p}-t+z^{p}-z\equiv[t]_{p}+[z]_{p}.$

For $M\in{\mathbb{Z}}$, we call a polynomial $f(t)\in{\mathbb{Z}}[t]$ an
$M$-constant if $f(t-\kappa)\equiv f(t)\pmod{M}$. We note that strings, whose
length is a multiple of $N$, are $N$-constants. More generally, we have

$\displaystyle b[t-\kappa]_{a}=b[t]_{a}-ab\kappa[t-\kappa]_{a-1}.$

Therefore, the polynomial $b[t]_{a}$ is an $N$-constant if and only if
$ab\equiv 0\pmod{N}$.

Let $A$ be a ${\mathbb{Z}}$-algebra. Our main example is
$A={\mathbb{Z}}[z_{1},\dots,z_{n}]={\mathbb{Z}}[z]$. The strings $\\{[t]_{m},\
m\geqslant 0\\}$ form an $A$-basis of $A[t]$ which should be thought of as a
discrete analog of the monomial basis $\\{t^{m},\ \ m\geqslant 0\\}$.

More generally, $\\{\prod_{i=1}^{l}[t_{i}]_{m_{i}},\
m_{i}\in{\mathbb{Z}}_{\geqslant 0}\\}$ form an $A$-basis of
$A[t_{1},\dots,t_{l}]$ which we will often use.

###  3.2. Difference $r$-integrals

Let $r=(r_{1},\dots,r_{l})$ be a sequence of non-negative integers. Let
$N_{r}$ be the least positive integer such that $N_{r}(r_{i}+1)\equiv
0\pmod{N}$ for all $i$. In terms of greatest common divisors (gcd) and least
common multiples (lcm) we have

$\displaystyle
N_{r}=\frac{N}{\operatorname{gcd}(r_{1}+1,\dots,r_{l}+1,N)}={\rm{lcm}}\left(\frac{N}{{\rm{gcd}}(N,r_{1}+1)},\dots,\frac{N}{{\rm{gcd}}(N,r_{l}+1)}\right).$

We also set

$\displaystyle
M_{r}=\frac{N}{N_{r}}=\operatorname{gcd}(r_{1}+1,\dots,r_{l}+1,N).$

For a polynomial
$f(t_{1},\dots,t_{l})=\sum_{m_{1},\dots,m_{l}}c_{m}\prod_{i=1}^{l}[t_{i}]_{m_{i}}\in
A[t_{1},\dots,t_{l}]$ and a sequence $r=(r_{1},\dots,r_{l})$ of non-negative
integers, we define the difference $r$-integral of $f$ by the formula:

$\displaystyle\\{f\\}^{t_{1},\dots,t_{l}}_{r}=N_{r}c_{r}.$

The difference $r$-integral is an $A$-linear map
$\\{\cdot\\}^{t_{1},\dots,t_{l}}_{r}:A[t_{1},\dots,t_{l}]\to A$. In the next
section we show that this map vanishes on discrete differentials modulo $N$.

If $r$ is such that $r_{i}\equiv-1\pmod{N}$, $i=1,\dots,l$, then $N_{r}=1$. In
this case we call the sequence $r$ a maximal sequence  and call the difference
$r$-integral a maximal difference integral.

If $r$ is such that for some $i$ the number $r_{i}+1$ is relatively prime to
$N$, then $N_{r}=N$ and the corresponding difference $r$-integral is
identically zero modulo $N$. We call such $r$ a trivial sequence and the
difference $r$-integral a trivial difference integral.

If $N=p$ is prime, then all non-trivial difference integrals are maximal.

###  3.3. Discrete differentials

Define the discrete $t$-derivative $D_{t}:\ A[t]\to A[t]$,

$\displaystyle D_{t}f(t)=f(t)-f(t-\kappa).$

Similar to usual differentiation, the discrete derivative is an $A$ linear map
which satisfy the Leibniz rule:

$\displaystyle D_{t}(f(t)g(t))=(D_{t}f(t))\,g(t)+f(t-\kappa)\,(D_{t}g(t)).$

We call polynomials in the image of the discrete $t$-derivative the discrete
differentials. We have

(3.3) $D_{t}[t]_{m}=m\kappa\,[t-\kappa]_{m-1},$

Modulo $N$, the discrete derivative $D_{t}$ has kernel given by $N$-constants,
generated by

$\displaystyle a[t]_{b},\qquad a,b\in{\mathbb{Z}}_{>0},\ ab\equiv 0\pmod{N}.$

For any $N$-constant $g(t)$ we have

$\displaystyle D_{t}(g(t)\,f(t))\equiv g(t)\,D_{t}f(t)\pmod{N},$

which explains the name “$N$-constant”.

The next proposition asserts that a difference integral computed on a discrete
differential is zero.

######  Proposition 3.1.

Let $r=(r_{1},\dots,r_{l})$, $t=(t_{1},\dots,t_{l})$, and $f\in
A[t_{1},\dots,t_{l}]$. Then for any $j=1,\dots,l$,

$\displaystyle\\{D_{t_{j}}f\\}_{r}^{t_{1},\dots,t_{l}}\equiv 0\pmod{N}.$

###### Proof.

Write $f$ in the $A$-basis $\prod_{i=1}^{l}[t_{i}]_{m_{i}}$ and apply
$D_{t_{j}}$ using (3.3). That gives $D_{t_{j}}f$ in the same basis, and the
coefficient in $D_{t_{j}}f$ of each $[t_{j}]_{r_{j}}$ is a multiple of
$r_{j}+1$. Therefore, it is zero modulo $N$ when multiplied by $N_{r}$. ∎

###  3.4. The module of $t$-periods

In this section we discuss the set of all difference $r$-integrals of a
polynomial $f(t,z)\in{\mathbb{Z}}[t,z]$. For simplicity, we consider the case
when $t$ and $z$ are single variables. The results can be extended to the case
when $t=(t_{1},\dots,t_{l})$, $z=(z_{1},\dots,z_{n})$. At the end of this
section we formulate a remark on an analog of Fubini’s theorem for the
difference $r$-integrals.

Given a polynomial $f(t,z)=\sum_{r=0}^{m}c_{r}(z)[t]_{r}\in{\mathbb{Z}}[t,z]$,
we consider the following set of polynomials in $z$ with integer coefficients.
The set is denoted by $\\{f(t,z)\\}^{t}$ and consists of all polynomials of
the form

$\displaystyle\sum_{r=1}^{m}g_{r}(z)N_{r}c_{r}(z)=\sum_{r=1}^{m}g_{r}(z)\\{f(t,z)\\}^{t}_{r}\,,$

where each $g_{r}(z)$ is an arbitrary polynomial such that
$N_{r}g_{r}(z-\kappa)\equiv N_{r}g_{r}(z)\pmod{N}$, in other words, each
$g_{r}(z)$ is an arbitrary $M_{r}$-constant. The set $\\{f(t,z)\\}^{t}$ is
called the module of $t$-periods of the polynomial $f(t,z)$ relative to the
variable $t$ and the integer $N$. The module is the object of study in this
section.

The reason for this concept is the following observation. If a polynomial
$f(t,z)$ satisfies a difference equation up to a $t$-derivative:

$\displaystyle f(t,z-\kappa)-A(z)f(t,z)\equiv D_{t}(h(t,z))\pmod{N},$

then all polynomials $u(z)\in\\{f(t,z)\\}^{t}$ are solutions of the equation

$\displaystyle u(z-\kappa)\equiv A(z)u(z)\pmod{N}.$

We have $\\{f(t,z)\\}^{t}\subset{\mathbb{Z}}[z]$, but we are really interested
in the projection of $\\{f(t,z)\\}^{t}$ to ${\mathbb{Z}}/N{\mathbb{Z}}[z]$.
The image of $\\{f(t,z)\\}^{t}\\}$ in ${\mathbb{Z}}/N{\mathbb{Z}}[z]$ is a
${\mathbb{Z}}/N{\mathbb{Z}}$-module.

The definition of the module of $t$-periods uses the $A$-basis $\\{[t]_{m},\
m\geqslant 0\\}$. The next proposition says that one can equivalently use the
$A$-basis $\\{[t+n_{1}z+n_{2}]_{m},\ m\geqslant 0\\}$ for any fixed
$n_{1},n_{2}\in{\mathbb{Z}}$ without changing the module of $t$-periods.

######  Proposition 3.2.

Let $n_{1},n_{2}\in{\mathbb{Z}}$ and

$\displaystyle\sum_{r=0}^{m}c_{r}(z)[t]_{r}\equiv\sum_{r=0}^{m}b_{r}(z)[t+n_{1}z+n_{2}]_{r}\pmod{N},$

where $c_{r}(z),b_{r}(z)\in{\mathbb{Z}}[z]$. Then the following two sets are
equal:

(3.4)
$\displaystyle\left\\{\sum_{r=0}^{m}g_{r}(z)N_{r}c_{r}(z)\mid\operatorname{each}\
g_{r}(z){\rm{\ is\ an\ arbitrary}}\ M_{r}\operatorname{-constant}\right\\}$
$\displaystyle\phantom{aaaa}=\left\\{\sum_{r=0}^{m}g_{r}(z)N_{r}b_{r}(z)\mid\operatorname{each}\
g_{r}(z){\rm{\ is\ an\ arbitrary\ }}M_{r}\operatorname{-constant}\right\\}.$

###### Proof.

We give the proof for the case of $n_{2}=0,n_{1}=1$. The case $n_{2}=0$ and
non-zero $n_{1}$ is the same. The case $n_{1}=0$ is similar (and simpler). The
general case if is the combination of the previous two statements.

By (3.2),

$\displaystyle b_{r}(z)[t+z]_{r}=b_{r}(z)\sum_{i=0}^{r}{r\choose
i}[t]_{i}[z]_{r-i}.$

We claim that if polynomial $g_{i}(z)\in{\mathbb{Z}}[z]$ is an
$M_{i}$-constant, then the polynomial

$\displaystyle\tilde{g}_{r}(z)={r\choose
i}\frac{N_{i}}{N_{r}}\,[z]_{r-i}g_{i}(z)$

has integer coefficients and, moreover, it is an $M_{r}$-constant.

It is enough to prove this for the case $N=p^{a}$ where $p$ is prime. Let
$r+1=p^{b}x$, $i+1=p^{c}y$, where $x,y$ are integers not divisible by $p$.

Let us consider $b\leqslant a$, $c\leqslant a$, the other cases are similar.
Then $N_{r}=p^{a-b}$, $N_{i}=p^{b-c}$. If $c\leqslant b$, then $r-i$ is
divisible by $p^{c}$ and $[z]_{r-i}g_{i}N_{i}/N_{r}$ is an $M_{r}$-constant
since $iN_{i}/N_{r}$ is divisible by $p^{b}$. If $b<c$ then we use ${r\choose
i}$. It is well known that a binomial coefficient is divisible by $p^{d}$
where $d$ is the number of moving a digit when the difference $r-i$ is
computed in the base $p$ system. It follows that ${r\choose i}$ is divisible
by $p^{c-b}$ and the claim follows.

Thus
$g_{i}(z)N_{i}c_{i}(z)=g_{i}(z)N_{i}b_{i}(z)+\sum_{r=i+1}^{n}b_{r}(z)N_{r}\tilde{g}_{r}$
where $\tilde{g}_{r}$ are $M_{r}$-constants. It follows that the left-hand
side set in (3.4) is inside of the right set. Doing the change $t\to t-z$ and
repeating the argument, we obtain the other inclusion. ∎

Thus, a difference $r$-integral of $f(t,z)$ with respect to $t$ may change, if
the $A$-basis is shifted by a linear polynomial in $z$, but the module of
$t$-periods does not.

Another statement in the same spirit is the following proposition.

######  Proposition 3.3.

For any integers $n_{1},n_{2},n_{3}$, with $n_{3}>0$, and any polynomial
$f(t,z)\in{\mathbb{Z}}[t,z]$, we have

$\displaystyle\\{f(t,z)[t+n_{1}z+n_{2}]_{n_{3}N}\\}^{t}\equiv\\{f(t,z)\\}^{t}\pmod{N}.$

###### Proof.

The proof is similar to Proposition 3.2. We treat the case $n_{1}=n_{3}=1$ and
$n_{2}=0$.

We have modulo $N$

$\displaystyle[t]_{r}[t+z]_{N}\equiv[t]_{r}[(t-r\kappa)+z]_{N}=[t]_{r}\sum_{i=0}^{N}{N\choose
i}[t-r\kappa]_{i}[z]_{N-i}=\sum_{i=0}^{N}{N\choose i}[t]_{r+i}[z]_{N-i}.$

The claim is that if $g_{r}(z)$ is an $M_{r}$-constant, then

$\displaystyle\tilde{g}(z)={N\choose i}\frac{N_{r+i}}{N_{r}}[z]_{N-i}g_{r}(x)$

is an $M_{r+i}$-constant. This is checked for $N=p^{a}$ where $p$ is prime, by
using the fact that if $i=p^{c}x$, where $x$ is an integer not divisible by
$p$ and $c<a$, then ${N\choose i}$ is divisible by $p^{a-c}$.

Write $f(t,z)=\sum_{r=0}^{m}c_{r}(z)[t]_{m}$ and
$f(t,z)[t+z]_{N}=\sum_{r=0}^{N+n}b_{r}(z)[t]_{r}$. It follows from the claim
that for any $M_{i}$-constant $g_{i}(z)$, we have
$b_{i}(z)g_{i}(z)N_{i}=\sum_{r=i-N}^{i}c_{r}(z)N_{r}\tilde{g}_{r}(z)$ where we
set $c_{r}(z)=0$ for $r<0$ and where $\tilde{g}_{r}(z)$ are $M_{r}$-constants.
Therefore, $\\{f(t,z)[t+z]_{N}\\}^{t}\subset\\{f(t,z)\\}^{t}$ modulo $N$.

Clearly $N_{i}=N_{i+N}$ and $g_{i}(z)$ is an $M_{i}$-constant if and only if
$g_{i}(z)$ is an $M_{i+N}$-constant. Therefore, if $i\leqslant m$, then
$b_{i+N}(z)g_{i}(z)N_{i+N}=c_{i}(z)N_{i}g_{i}(z)+\sum_{r=i+1}^{i+N}c_{r}(z)N_{r}\tilde{g}_{r}(z)$.
Thus we obtain $c_{i}(z)N_{i}g_{i}(z)\in\\{f(t,z)[t+z]_{N}\\}^{t}$ recursively
starting from $i=m$, then continuing to $i=m-1$, then to $m-2$, and so on. ∎

We use Propositions 3.2 and 3.3 in Section 5 to explain that the family of the
modulo $N$ solutions of the qKZ equations constructed in Theorem 5.1 does not
depend on various choices.

######  Remark.

Consider an example. Let $f(t_{1},t_{2},z)=p(z)[t_{1}]_{2}[t_{2}]_{2}$ and
$N=4$. Then

$\displaystyle\\{\\{f\\}_{2}^{t_{1}}\\}^{t_{2}}_{2}=\\{\\{f\\}_{2}^{t_{2}}\\}^{t_{1}}_{2}=4p(z)\equiv
0\pmod{4},\quad{\textrm{but}}\quad\\{f\\}_{2,2}^{t_{1},t_{2}}=2p(z).$

More generally, we have an $N$-analog of Fubini’s theorem:

(3.5)
$\displaystyle\\{\\{f(t_{1},t_{2},z)\\}^{t_{1}}_{r_{1}}\\}^{t_{2}}_{r_{2}}=\\{\\{f(t_{1},t_{2},z)\\}^{t_{2}}_{r_{2}}\\}_{r_{1}}^{t_{1}}$
$\displaystyle\phantom{aaaaaa}=\frac{N_{r_{1}}N_{r_{2}}}{\operatorname{gcd}(N_{r_{1}},N_{r_{2}})}\\{f(t_{1},t_{2},z)\\}^{t_{1},t_{2}}_{r_{1},r_{2}}=\frac{N_{r_{1}}N_{r_{2}}}{\operatorname{gcd}(N_{r_{1}},N_{r_{2}})}\\{f(t_{1},t_{2},z)\\}^{t_{2},t_{1}}_{r_{2},r_{1}}.$

Note that the integer $N_{r_{1},r_{2}}$ is a divisor of $N_{r_{1}}N_{r_{2}}$
and, in general, is a proper divisor. Thus, in general, the double integration
gives more difference $r$-integrals than the repeated integration.

##  4\. Master function and weight functions

We define the main ingredients of the construction of the solutions of the qKZ
equation modulo $N$: the master polynomial and weight function.

###  4.1. Master polynomial

Recall integers $N,\kappa,k,k^{\prime}$ defined in (3.1). We also have
integers $n,l$ corresponding to the choice of the space
${\operatorname{Sing}}V^{\otimes n}[n-2l]$.

Let again $z=(z_{1},\dots,z_{n}),t=(t_{1},\dots,t_{l})$. Define the master
polynomial

$\displaystyle\Phi(t,z)=\prod_{a=1}^{n}\prod_{i=1}^{l}[t_{i}-z_{a}]_{k}\prod_{1\leqslant
i<j\leqslant l}[t_{i}-t_{j}+1]_{k^{\prime}}\,.$

The zeroes of the master polynomial are shown on Figure 1 for $l=2$. The start
of each string is indicated in blue. The string itself consist of $k$ or
$k^{\prime}$ zeroes of the master polynomial. The distance between neighboring
zeroes is $-\kappa$ as shown in red in Figure 1.

Note that the master polynomial is symmetric in variables $z_{1},\dots,z_{n}$.

As explained in the introduction, the master polynomial is obtained from the
master function used in the case of complex-valued solutions. This master
function originally is a ratio of products of gamma functions, see Section
A.3, which reduces to a polynomial under our assumptions.

$-\kappa\ \
$k$\dots$21$t_{1}-z_{1}$k$\dots$21$t_{1}-z_{2}$$\Large{\dots}$k$\dots$21$t_{1}-z_{n}$${}_{k^{\prime}}$$\dots$21$t_{1}-t_{2}+1$k$\dots$21$t_{2}-z_{1}$k$\dots$21$t_{2}-z_{2}$$\Large{\dots}$k$\dots$21$t_{2}-z_{n}$
Figure 1. Zeroes of the master polynomial
$\Phi(t_{1},t_{2},z_{1},\dots,z_{n})$.

###  4.2. Symmetrization modulo $N$

In the case of complex-valued solutions, one uses an action of a symmetric
group $S_{l}$ on functions of $t_{1},\dots,t_{l}$ given by the formula

$\displaystyle(\tau\circ
f)(t_{1},\dots,t_{l})=f(t_{\tau(1)},\dots,t_{\tau(l)})\prod_{i<j,\,\tau(i)>\tau(j)}\frac{t_{i}-t_{j}-1}{t_{i}-t_{j}+1},$

for any $\tau\in S_{l}$. We modify it as follows.

For $\tau\in S_{l}$ define

$\displaystyle(\tau
f)(t_{1},\dots,t_{l})=f(t_{\tau(1)},\dots,t_{\tau(l)})\prod_{i<j,\,\tau(i)>\tau(j)}\frac{t_{i}-t_{j}+1-\kappa
k^{\prime}}{t_{i}-t_{j}+1}.$

Note that this formula defines an action of $S_{l}$ modulo $N$ since $1-\kappa
k^{\prime}\equiv-1\pmod{N}$.

We also define symmetrization modulo $N$:

$\displaystyle\operatorname{Sym}_{t}f(t)=\sum_{\tau\in S_{l}}(\tau f)(t)\,.$

###  4.3. Weight function.

In the case of complex-valued solutions, the coordinates of the weight
function are defined as symmetrizations of the functions

$\displaystyle\prod_{i=1}^{r}\frac{1}{t_{i}-z_{a_{i}}}\prod_{l=1}^{a_{i}-1}\frac{t_{i}-z_{l}+1}{t_{i}-z_{l}}.$

We modify it as follows.

For $I=\\{1\leqslant a_{1}<\dots<a_{l}\leqslant n\\}\in\mathcal{I}_{l}$,
define the coordinates of the weight function

$\displaystyle w_{I}(t,z)$ $\displaystyle=$
$\displaystyle\operatorname{Sym}_{t}\left(\prod_{i=1}^{l}\frac{1}{t_{i}-z_{a_{i}}}\prod_{j=1}^{a_{i}-1}\frac{t_{i}-z_{j}-\kappa
k}{t_{i}-z_{j}}\right).$

Since $k\kappa\equiv-1\pmod{N}$, and $\operatorname{Sym}_{t}$ coincides modulo
$N$ with the previous symmetrization, this formula coincides with the formula
used in the complex-valued case modulo $N$.

It is convenient to denote

$\displaystyle
w_{I}^{\tau}(t,z)=\tau\left(\prod_{i=1}^{l}\frac{1}{t_{i}-z_{a_{i}}}\prod_{j=1}^{a_{i}-1}\frac{t_{i}-z_{j}-\kappa
k}{t_{i}-z_{j}}\right).$

Then $w_{I}(t,z)=\sum_{\tau\in S_{l}}w_{I}^{\tau}(t,z)$.

We also denote

$\displaystyle U_{I}(t,z)=\Phi(t,z)w_{I}(t,z),\qquad
U_{I}^{\tau}(t,z)=\Phi(t,z)w_{I}^{\tau}(t,z).$

Then we have a lemma saying that the summands $U_{I}^{\tau}(t,z)$ are
polynomials organized in products of strings.

######  Lemma 4.1.

For $\tau\in S_{l}$, the function $U_{I}^{\tau}(t,z)$ is a polynomial which is
a product of $\binom{l}{2}$ strings of length $k^{\prime}$, of $l(n-1)$
strings of length $k$, and of $l$ strings of length $k-1$,

(4.1) $\displaystyle
U_{I}^{\tau}(t,z)=\prod_{i<j,\,\tau(i)<\tau(j)}[t_{i}-t_{j}+1]_{k^{\prime}}\prod_{i<j,\,\tau(i)>\tau(j)}[t_{i}-t_{j}+1-\kappa]_{k^{\prime}}$
$\displaystyle\times\prod_{i=1}^{l}\Big{(}\prod_{s=1}^{a_{i}-1}[t_{\tau(i)}-z_{s}-\kappa]_{k}\Big{)}[t_{\tau(i)}-z_{a_{i}}-\kappa]_{k-1}\Big{(}\prod_{s=a_{i}+1}^{n}[t_{\tau(i)}-z_{s}]_{k}\Big{)}.$

###### Proof.

It is easy to see that the poles of $w_{I}^{\tau}$ cancel with the starts of
the strings existing in the master polynomial, while the zeroes append those
strings.

Examples of $U_{I}^{\tau}(t,z)$ are shown on Figure 2. In that picture $l=2$
and $I=\\{1,2\\}$. Zeroes of $U_{I}^{{\operatorname{id}}}(t,z)$ are pictured
on the left and zeroes of $U_{I}^{(1,2)}(t,z)$ on the right. The cancelled
zeroes of the master polynomial are pictured as red circles. The new zeroes
are shown in red bullets. ∎

k+1k$\dots$21$t_{1}-z_{1}$k$\dots$21$t_{1}-z_{2}$$\dots$k$\dots$21$t_{1}-z_{n}$${}_{k^{\prime}+1}$${}_{k^{\prime}}$$\dots$21$t_{1}-t_{2}+1$k$\dots$21$t_{2}-z_{1}$k$\dots$21$t_{2}-z_{2}$$\dots$k$\dots$21$t_{2}-z_{n}$k$\dots$21$t_{1}-z_{1}$k$\dots$21$t_{1}-z_{2}$$\Large{\dots}$k$\dots$21$t_{1}-z_{n}$${}_{k^{\prime}}$$\dots$21$t_{1}-t_{2}+1$k+1k$\dots$21$t_{2}-z_{1}$k$\dots$21$t_{2}-z_{2}$$\Large{\dots}$k$\dots$21$t_{2}-z_{n}$
Figure 2. Zeroes of polynomials $U_{1,2}^{{\operatorname{id}}}(t,z)$ and
$U_{1,2}^{(1,2)}(t,z)$.

The polynomials $U_{I}(t,z)$ are coordinates of the integrand which will be
used to construct solutions of the $qKZ$ equations. Now we pass to the
integrand itself.

Define the polynomial $U(t,z)$ with values in $V^{\otimes n}[n-2l]$ by the
formula

(4.2)
$U(t,z)=\sum_{I\in\mathcal{I}_{l}}\,U_{I}(t,z)\,v_{I}=\sum_{I\in\mathcal{I}_{l}}\sum_{\tau\in
S_{l}}\,\Phi(t,z)w_{I}^{\tau}(t,z)\,v_{I}.$

In the next section we show that any difference $r$-integral of $U(t,z)$
solves the symmetric qKZ equations modulo $N$ and is a singular vector modulo
$N$.

##  5\. The main theorem

In this section we construct solutions modulo $N$ of the qKZ equations.

###  5.1. The statement of the theorem.

######  Theorem 5.1.

Let $U(t,z)$ be the polynomial in $t_{1},\dots,t_{l},z_{1},\dots,z_{n}$ with
values in $V^{\otimes n}[n-2l]$ defined in (4.2). Then for any
$r=(r_{1},\dots,r_{l})$ the difference $r$-integral
${f}_{r}(z)=\\{U(t,z)\\}^{t}_{r}$ is a polynomial in $z_{1},\dots,z_{n}$ which
is a solution of the symmetric qKZ equations (2.3) modulo $N$. The vector
${f}_{r}(z)$ is singular modulo $N$.

We prove the theorem in the next three sections.

If $g(z)$ is a scalar $N$-constant polynomial and ${f}(z)$ is a solution of
the qKZ modulo $N$, then $g(z){f}(z)$ is a solution too. Moreover, we have a
stronger statement. Namely, let
$f(z)=\sum_{I\in\mathcal{I}_{l}}f_{I}(z)\,v_{I}$,
$f_{I}(z)\in{\mathbb{Z}}[z]$, be a solution of the qKZ equations modulo $N$,
and let $a\in{\mathbb{Z}}$ be such that all polynomials $f_{I}(z)/a$ have
integer coefficients. Then for any scalar
$(N/\operatorname{gcd}(a,N))$-constant $g(z)$, the product $g(z){f}(z)$ is
also a solution of the qKZ equations modulo $N$.

For example, if $g_{r}(z)\in{\mathbb{Z}}[z]$ is an $M_{r}$-constant then
$g_{r}(z){f}_{r}(z)$ is also a solution. Thus, by Theorem 5.1 all functions in
the module of $t$-periods $\\{U(t,z)\\}^{t}$ are solutions modulo $N$ of the
qKZ equations. We call all such solutions $N$-hypergeometric solutions. Thus,
the $N$-hypergeometric solutions are polynomials of the form
$\sum_{r}g_{r}(z){f}_{r}(z)$ where ${f}_{r}(z)\in N_{r}{\mathbb{Z}}[z]$ are
given in Theorem 5.1, and $g_{r}(z)\in{\mathbb{Z}}[z]$ are $M_{r}$-constants.
We give examples of $N$-hypergeometric solutions in Section 5.5.

Seemingly, there are several other ways to obtain solutions modulo $N$. The
first way is to choose integers $k$ and $k^{\prime}$ as any positive integer
solutions of congruences (3.1) and not as the least positive integer
solutions. Such a choice will change the master and weight functions. It is
clear that Theorem 5.1 still holds for the modified integrand $U(t,z)$.
However the choice of new $k$ and $k^{\prime}$ does not produce new solutions
by Proposition 3.3.

The second way is to construct solutions by integrating with respect to a
different $A$-basis, e.g.
$\\{\prod[t_{i}-z_{i}]_{m_{i}},m_{i}\in{\mathbb{Z}}_{\geqslant 0}\\}$. Then an
analog of Theorem 5.1 still holds. However, a different choice of an $A$-basis
produces no new solutions as well by Proposition 3.2.

The third way is to consider a divisor $\bar{N}$ of $N$ and a solution
$\bar{f}(z)$ of the qKZ equations modulo $\bar{N}$. Then $N\bar{f}(z)/\bar{N}$
is a solution of the qKZ equations modulo $N$. The solutions obtained in this
way also are not new.

###  5.2. The first $n-1$ equations.

In this section we check the first $n-1$ equations of (2.3). Moreover, we show
that these equations are already satisfied before taking the difference
$r$-integral.

Namely, we check modulo $N$

(5.1) ${g}(z_{1},\dots,z_{a+1},z_{a},\dots,z_{n})\equiv
P^{(a,a+1)}R^{(a,a+1)}(z_{a}-z_{a+1})\,{g}(z)$

for ${g}(z)=w(t,z)=\sum_{I\in\mathcal{I}_{l}}w_{I}v_{I}$.

This is sufficient because multiplication by the scalar master polynomial
$\Phi(z,t)$ preserves relation (5.1) as $\Phi$ is symmetric in
$z_{1},\dots,z_{n}$. The integration preserves this relation as well since
(5.1) does not depend on integration variables $t_{1},\dots,t_{l}$.

We have three cases.

First, let $I$ be such that $a,a+1\not\in I$. Then $w_{I}$ is symmetric in
$z_{a},z_{a+1}$. The vector $v_{I}$ has $v_{1}\otimes v_{1}$ in positions
$a,a+1$. For any $x$, we have $P\,R(x)v_{1}\otimes v_{1}=v_{1}\otimes v_{1}$.
Therefore, ${g}=w_{I}v_{I}$ satisfies (5.1) for each $I\in\mathcal{I}_{l}$.

Second, let $I$ be such that $a,a+1\in I$. The vector $v_{I}$ has
$v_{2}\otimes v_{2}$ in positions $a,a+1$. Again, for any $x$, we have
$P\,R(x)v_{2}\otimes v_{2}=v_{2}\otimes v_{2}$. So we have to check that
$w_{I}$ does not change when $a$ and $a+1$ are swapped. We claim that for any
$\tau\in S_{l}$, the sum $w_{I}^{\tau}+w_{I}^{(a,a+1)\tau}$ has this property.
Indeed, if $\tau(a)=b$, $\tau(a+1)=c$, and if $c>b$, then modulo $N$ we have

$\displaystyle(w_{I}^{\tau}+w_{I}^{(a,a+1)\tau})(t,z)$
$\displaystyle=\frac{h(z,t)}{(t_{b}-z_{a})(t_{c}-z_{a})}\left(\frac{t_{c}-z_{a}-\kappa
k}{t_{c}-z_{a+1}}+\frac{{t_{b}-z_{a}-\kappa
k}}{t_{b}-z_{a+1}}\,\frac{t_{b}-t_{c}+1-\kappa
k^{\prime}}{t_{b}-t_{c}+1}\right)$
$\displaystyle\equiv\frac{h(z,t)}{(t_{b}-z_{a+1})(t_{c}-z_{a+1})}\left(\frac{t_{c}-z_{a+1}-\kappa
k}{t_{c}-z_{a}}+\frac{{t_{b}-z_{a+1}-\kappa
k}}{t_{b}-z_{a}}\,\frac{t_{b}-t_{c}+1-\kappa
k^{\prime}}{t_{b}-t_{c}+1}\right)$
$\displaystyle=(w_{I}^{\tau}+w_{I}^{(a,a+1)\tau})(t,z_{1},\dots,z_{a+1},z_{a},\dots,z_{n}).$

Here the common factor $h(z,t)$ does not depend on $z_{a},z_{a+1}$. The case
$b>c$ is similar. Therefore, the term ${g}=w_{I}v_{I}$ satisfies (5.1).

Third, let $I$ be such that $a\in I$ and $a+1\not\in I$. We pair it up with
the set $J$ such that $a\not\in J$ and $a+1\in I$ in the most natural way.
Namely, let $J=(I\setminus a)\cup\\{a+1\\}$ be the set $I$ where $a$ is
replaced with $a+1$. Let $\tau\in S_{l}$ and $\tau(a)=b$. We claim that modulo
$N$,

$\displaystyle PR(z_{1}-z_{a+1})(w_{I}^{\tau}(t,z)\,v_{2}\otimes
v_{1}+w_{J}^{\tau}(t,z)\,v_{1}\otimes v_{2})$ $\displaystyle\hskip
30.0pt\equiv
w_{I}^{\tau}(t,z_{1},\dots,z_{a+1},z_{a},\dots,z_{n})\,v_{2}\otimes
v_{1}+w_{J}^{\tau}(t,z_{1},\dots,z_{a+1},z_{a},\dots,z_{n})\,v_{1}\otimes
v_{2}.$

Indeed, this is equivalent to

$\displaystyle
w_{I}^{\tau}(t,z_{1},\dots,z_{a+1},z_{a},\dots,z_{n})\equiv-\frac{1}{z_{a}-z_{a+1}-1}w_{I}^{\tau}(t,z)+\frac{z_{a}-z_{a+1}}{z_{a}-z_{a+1}-1}w_{J}^{\tau},$
$\displaystyle
w_{j}^{\tau}(t,z_{1},\dots,z_{a+1},z_{a},\dots,z_{n})\equiv\frac{z_{a}-z_{a+1}}{z_{a}-z_{a+1}-1}w_{I}^{\tau}(t,z)-\frac{1}{z_{a}-z_{a+1}-1}w_{J}^{\tau}.$

For the first equation we have

$\displaystyle-\frac{1}{z_{a}-z_{a+1}-1}w_{I}^{\tau}(t,z)+\frac{z_{a}-z_{a+1}}{z_{a}-z_{a+1}-1}w_{J}^{\tau}$
$\displaystyle=h(t,z)\left(\frac{1}{z_{a}-z_{a+1}-1}\frac{1}{t_{b}-z_{a}}-\frac{z_{a}-z_{a+1}}{z_{a}-z_{a+1}-1}\frac{t_{b}-z_{a}-\kappa
k^{\prime}}{t_{b}-z_{a}}\frac{1}{t_{b}-z_{a+1}}\right)$
$\displaystyle\equiv\frac{h(t,z)}{t_{b}-z_{a+1}}=w_{I}^{\tau}(t,z_{1},\dots,z_{a+1},z_{a},\dots,z_{n}).$

The second equation is similar. Therefore, the sum ${g}=w_{I}v_{I}+w_{J}v_{J}$
satisfies (5.1). Thus $g=w(t,z)$ satisfies (5.1). Equation (5.1) is proved.

We note that if one replaces $k\kappa^{\prime}$ in $w(t,z)$ by $2$, and
$\kappa k$ by $-1$ then ${w}(t,z)$ will satisfy (5.1) exactly (not only modulo
$N$). This property of the weight function $w(t,z)$ is known in large
generality and we could derive it, for example, from [TV, Theorem 4.9] instead
of checking it directly.

Also note that in the complex-valued case, there is an extra ingredient, as
one multiplies by a periodic function $W$ before integration, see Section A.3.
This function is usually not symmetric in $z$ and therefore, the integrand
loses symmetry (5.1).

###  5.3. The $n$-th equation

Clearly, for any $I\in\mathcal{I}_{l}$,

$\displaystyle(P^{(\mu)})^{-1}v_{I}=v_{\mu^{-1}I},\qquad\mu^{-1}I=\\{\mu^{-1}(a),\
a\in\ I\\}.$

To check the $n$-th equation

$\displaystyle(P^{(\mu)})^{-1}f_{r}(z_{1}-\kappa,z_{2},\dots,z_{n})$
$\displaystyle=$ $\displaystyle f_{r}(z_{2},\dots,z_{n},z_{1}),$

we show that if one takes $U_{I}^{\tau}(t,z_{1}-\kappa,z_{2},\dots,z_{n})$ and
changes $t_{i}\to t_{i}-\kappa$, where $i$ depends on $I$ and $\tau$, then the
result is $U_{\mu^{-1}I}^{\tau^{\prime}}(t,z_{2},\dots,z_{n},z_{1})$, where

$\displaystyle\tau^{\prime}=\begin{cases}\tau&1\not\in I,\\\
\tau\,(1,2,\dots,l)&1\in I.\end{cases}$

For that let us follow what happens to the strings when we shift $z_{1}$. The
change $z_{1}\to z_{1}-\kappa$ affects only strings of the form
$[t_{i}-z_{1}-\kappa]_{k}$, and $[t_{i}-z_{1}-\kappa]_{k-1}$ which are shifted
to $[t_{i}-z_{1}]_{k}$, and $[t_{i}-z_{1}]_{k-1}$ respectively. On Figure 2 it
corresponds to the shift of the corresponding strings to the left by one
position.

Let $I$ be such that $1\not\in I$. Then the strings depending on $z_{1}$ in
$U_{I}^{\tau}(t,z)$ are only $[t_{i}-z_{1}-\kappa]_{k}$, $i=1,\dots,l$. Note
that $n\not\in I^{\prime}=\mu^{-1}I$. Thus the strings depending on $z_{1}$
$U_{\mu^{-1}I}^{\tau}(t,z_{2},\dots,z_{n},z_{1})$ are $[t_{i}-z_{1}]_{k}$,
$i=1,\dots,l$. Therefore, the shift $z_{1}\to z_{1}\kappa$ takes the strings
in $U_{I}^{\tau}(t,z)$ with the strings in
$U_{\mu^{-1}I}^{\tau}(t,z_{2},\dots,z_{n},z_{1})$. The strings which do not
involve $z_{1}$ match as well. Thus we have
$U_{I}^{\tau}(t,z_{1}-\kappa,z_{2},\dots,z_{n})=U_{\mu^{-1}I}^{\tau}(t,z_{2},\dots,z_{n},z_{1})$.
Note that we did not shift $t_{i}$ in this case.

Let $I$ be such that $1\in I$. Let $\tau(1)=b$. Then the strings depending on
$z_{1}$ in $U_{I}^{\tau}(t,z)$ are $[t_{i}-z_{1}-\kappa]_{k}$, $i=1,\dots,l$,
$i\neq b$ and $[t_{b}-z_{1}-\kappa]_{k-1}$. Note that
$\tau^{\prime}(n)=\tau\mu(n)=\tau(1)=b$. Thus the strings depending on $z_{1}$
$U_{\mu^{-1}I}^{\tau}(t,z_{2},\dots,z_{n},z_{1})$ are $[t_{i}-z_{1}]_{k}$,
$i\neq b$ and $[t_{b}-z_{1}-\kappa]_{k-1}$.

We shift both $z_{1}\to z_{1}-\kappa$ and $t_{b}\to t_{b}-\kappa$ in
$U_{I}^{\tau}(t,z)$. Clearly, all strings are shifted appropriately to match
the strings in $U_{\mu^{-1}I}^{\tau^{\prime}}(t,z_{2},\dots,z_{n},z_{1})$.
Thus, in this case we have
$U_{I}^{\tau}(t_{1},\dots,t_{b}-\kappa,\dots,t_{l},z_{1}-\kappa,z_{2},\dots,z_{n})=U_{\mu^{-1}I}^{\tau}(t,z_{2},\dots,z_{n},z_{1})$.

Therefore, the $n$-th equation is valid for the integrand up to terms of the
form $D_{t_{b}}U_{I}^{\tau}(t,z_{1}-\kappa,z_{2},\dots,z_{n})$ which vanish
after taking the difference $r$-integral.

###  5.4. $N$-hypergeometric solutions are singular vectors modulo $N$

Finally, we claim that the vector $U(t,z)$ is a singular vector modulo $N$ up
to difference $t$-differentials, namely,

$\displaystyle
eU(t,z)\,\equiv\,\sum_{J\in\mathcal{I}_{l-1}}g_{J}(t,z)v_{J}\pmod{N},$

where $g_{J}(t,z)$ are suitable discrete $t$-differentials. This statement is
deduced directly from [TV, Lemma 2.21]. Hence taking a difference integral of
$eU(t,z)$ we obtain the zero vector modulo $N$.

######  Example 5.2.

Let $l=1$. Then
$eU(t,z)=\sum_{a=1}^{n}U_{\\{a\\}}(t_{1},z)\,v_{1}\otimes\dots\otimes
v_{1}\,,$ and modulo $N$ we have

$\displaystyle\Phi(t_{1}-\kappa,z)-\Phi(t_{1},z)=\Phi(t,z)\Big{[}\prod_{a=1}^{n}\frac{t_{1}-z_{a}-\kappa
k}{t_{1}-z_{l}}-1\Big{]}$ $\displaystyle=-\Phi(t,z)\kappa
k\sum_{a=1}^{n}\frac{1}{t_{1}-z_{a}}\prod_{l=1}^{a-1}\frac{t_{1}-z_{l}-\kappa
k}{t_{1}-z_{l}}=-\kappa
k\sum_{a=1}^{n}U_{\\{a\\}}(t_{1},z)\equiv\sum_{a=1}^{n}U_{\\{a\\}}(t_{1},z).$

Note, that one can modify solutions modulo $N$ by adding a vector valued
polynomial $f(z)$ of the form (2.7) such that the coordinate polynomials
$f_{I}(z)\in{\mathbb{Z}}[z]$ are all divisible by $N$. In many cases such a
modification allows us to obtain solutions which are singular vectors (not
only singular vectors modulo $N$), see Sections 5.5.

###  5.5. Example.

Let $n=3$, $l=1$, $\kappa=2$, $N=3$, $r=2$. Then $k=1$ and
$U(t_{1},z_{1},z_{2},z_{3})$ is a quadratic polynomial in $t_{1}$. We have

$\displaystyle\\{U(t_{1},z_{1},z_{2},z_{3})\\}^{t_{1}}_{2}=v_{2}\otimes
v_{1}\otimes v_{1}+v_{1}\otimes v_{2}\otimes v_{1}+v_{1}\otimes v_{1}\otimes
v_{2}.$

Note that the solution is a singular vector only modulo $3$. One can subtract,
for example, $3v_{2}\otimes v_{1}\otimes v_{1}$ to make it a singular vector.

Let $n=3$, $l=1$, $\kappa=2,$ $N=5$, $r=4$. Then $k=2$ and
$U(t_{1},z_{1},z_{2},z_{3})$ is a polynomial of degree $5$ in $t_{1}$. We have

$\displaystyle\\{U(t_{1},z_{1},z_{2},z_{3})\\}^{t_{1}}_{4}=(14-z_{1}-2z_{2}-2z_{3})v_{2}\otimes
v_{1}\otimes v_{1}$ $\displaystyle\hskip
70.0pt+(10-z_{2}-2z_{1}-2z_{3})v_{1}\otimes v_{2}\otimes
v_{1}+(6-z_{3}-2z_{1}-2z_{2})v_{1}\otimes v_{1}\otimes v_{2}.$

Again, the solution is a singular vector only modulo $5$ and one can subtract
a multiple of $5$, for example, $(30-5(z_{1}+z_{2}+z_{3}))v_{2}\otimes
v_{1}\otimes v_{1}$ to make the solution a singular vector.

More generally, for $n=3$, $l=1$, $\kappa=2$, $N=2k+1$, the vector
$U(t_{1},z_{1},z_{2},z_{3})$ is a polynomial of degree $3k-1$ in $t_{1}$ and
we have a difference $r$-integral with $r=2k$. The corresponding solution
modulo $N$ is a non-homogenous polynomial in $z_{1},\dots,z_{n}$ of degree
$k-1$. If $N=p$ is prime, then this is the only $N$-hypergeometric solution.

##  6\. From difference to differential equations

In this section we show that the solutions modulo $N$ constructed in Theorem
5.1 recover the solutions of the KZ differential equations modulo $N$
constructed in [SV]. For simplicity and following [SV], we restrict ourselves
only to maximal difference $r$-integrals, that is to the case of $r$, such
that $r_{i}\equiv-1\pmod{N}$.

###  6.1. $N$-hypergeometric solutions of KZ differential equations

In this section we remind the construction in [SV] of polynomial solutions
modulo an integer $N$ of the ${\mathfrak{sl}_{2}}$ differential KZ equations.

We consider the system of differential equations on a $V^{\otimes n}$-valued
function $f(z)$,

(6.1) $\displaystyle\kappa\,\frac{\partial f}{\partial z_{a}}=\sum_{s,\ s\neq
a}\frac{P^{(a,s)}-1}{z_{a}-z_{s}}\,f,\qquad a=1,\dots,n,$

where $z=(z_{1},\dots,z_{n})$ and $\kappa$ is a parameter of the system. This
system is called the KZ differential equations. This system commutes with the
${\mathfrak{sl}_{2}}$-action on $V^{\otimes n}$.

Let $0<\kappa<N$ be the same relatively prime integers as before. Let $l$ be a
positive integer, $2l\leqslant n$. Let $k,k^{\prime}$ be the same positive
integers as in (3.1). Let $t=(t_{1},\dots,t_{l})$.

Define the master polynomial

$\displaystyle\Phi^{0}(t,z)=\prod_{a=1}^{n}\prod_{i=1}^{l}(t_{i}-z_{a})^{k}\prod_{1\leqslant
i<j\leqslant l}(t_{i}-t_{j})^{k^{\prime}}\,.$

For a function $f(t)$ define

$\displaystyle\text{sym}_{t}\,f(t)$ $\displaystyle=$
$\displaystyle\sum_{\tau\in S_{l}}\,f(t_{\tau(1)},\dots,t_{\tau(l)})\,.$

For $I=\\{1\leqslant a_{1}<\dots<a_{l}\leqslant n\\}\in\mathcal{I}_{l}$,
define the weight functions

$\displaystyle
w_{I}^{0}(t,z)=\text{sym}_{t}\,\prod_{i=1}^{l}\frac{1}{t_{i}-z_{a_{i}}}\,,\qquad
U_{I}^{0}(t,z)=\Phi^{0}(t,z)\,w_{I}^{0}(t,z).$

Then $U_{I}^{0}(t,z)$ is a polynomial with integer coefficients. Expand
$U^{0}_{I}(t,z)$ with respect to the $t$-variables

$\displaystyle U_{I}^{0}(t,z)=\sum_{m_{1},\dots,m_{l}\geqslant
0}c^{0}_{I,m}(z)\,t_{1}^{m_{1}}\dots t_{l}^{m_{l}}\,,\qquad
c^{0}_{I,m}(z)\in{\mathbb{Z}}[z].$

For any $r=(r_{1},\dots,r_{l})\in{\mathbb{Z}}_{>0}^{l}$ such that
$r_{i}\equiv-1\pmod{N}$, for all $i$, denote

(6.2)
$\displaystyle{f}^{0}_{r}(z)=\sum_{I\in\mathcal{I}_{l}}c^{0}_{I,r}(z)\,v_{I}\,.$

######  Theorem 6.1.

[SV] For any $r=(r_{1},\dots,r_{l})\in{\mathbb{Z}}_{>0}^{l}$ such that
$r_{i}\equiv-1\pmod{N}$, for $i=1,\dots,l$, the polynomial ${f}^{0}_{r}(z)$
satisfies the KZ differential equations modulo $N$ and is a singular vector
modulo $N$,

(6.3) $\displaystyle e{f}_{r}^{0}(z)$ $\displaystyle\equiv$ $\displaystyle
0\phantom{aaaaaaaaaaaaa}\pmod{N},$ (6.4)
$\displaystyle\kappa\,\frac{\partial{f}_{r}^{0}}{\partial z_{a}}\,$
$\displaystyle\equiv$ $\displaystyle\,\sum_{s,\ s\neq
a}\frac{P^{(a,s)}-1}{z_{a}-z_{s}}\,{f}^{0}_{r}\pmod{N},\qquad
a=1,\dots,n.\ignorespaces$

The solutions ${f}^{0}_{r}$ of the KZ equations modulo $N$ given by Theorem
6.1 are called the $N$-hypergeometric solutions of the KZ equations.

Note that the polynomials $U_{I}^{0}(t,z)$ are homogeneous polynomials of
variables $t,z$ of degree $nlk+l(l-1)k^{\prime}/2-l$. Hence the
$N$-hypergeometric solution ${f}_{r}^{0}(z)$ is a homogeneous polynomial in
$z$ of degree

$\displaystyle d_{r}=nlk+l(l-1)k^{\prime}/2-l-\sum_{i-1}^{l}r_{i}.$

###  6.2. The top degree of the $N$-hypergeometric solutions

Unlike the solutions ${f}_{r}^{0}(z)$ of the KZ equations, the solutions
${f}_{r}(z)$ of the qKZ equations given by Theorem 5.1 are not homogeneous. It
turns out that the top degree part of these solutions coincides with
${f}_{r}^{0}(z)$.

Define degree of polynomials in ${\mathbb{Z}}[t,z]$ by setting

$\displaystyle\deg t_{i}=\deg z_{j}=1,\qquad i=1,\dots,l,\ j=1,\dots,n.$

Let $r=(r_{1},\dots,r_{l})\in{\mathbb{Z}}_{>0}^{l}$ be such that
$r_{i}\equiv-1\pmod{N}$, for all $i$. Let ${f}_{r}(z)$ be the
$N$-hypergeometric solution of the qKZ equations defined in Theorem 5.1. Let
${f}_{r}^{0}(z)$ be the $N$-hypergeometric solution of the KZ equations
defined in Theorem 6.1.

######  Proposition 6.2.

We have

$\displaystyle{f}_{r}(z)={f}^{0}_{r}(z)+\dots,$

where the dots denote the terms of degree less than $\deg{f}^{0}_{r}(z)$.

###### Proof.

For a polynomial $X$ in variables $t,z$ and integer coefficients, we write
$X=Y+\dots$ when $Y$ is a homogeneous polynomial in $t,z$ and all terms of
$X-Y$ have degree smaller than the degree of $Y$.

For any integer $a$, we have
$t_{i}-t_{j}-a\kappa]_{m}=(t_{i}-t_{j})^{m}+\dots$ and
$[t_{i}-z_{j}-a\kappa]_{m}=(t_{i}-z_{j})^{m}+\dots$. Hence
$U(t,z)=U^{0}(t,z)+\dots$. At the same time if

$\displaystyle
U(t,z)=\sum_{m}{c}_{m}(z)\prod_{i=1}^{l}[t_{i}]_{m_{i}},\qquad{c}_{m}(z)={c}_{m}^{0}(z)+\dots,$

then $U(t,z)=\sum_{m}{c}_{m}^{0}(z)\prod_{i=1}^{l}t_{i}^{m_{i}}+\dots$.
Therefore, $U^{0}(t,z)=\sum_{m}{c}_{m}^{0}(z)\prod_{i=1}^{l}t_{i}^{m_{i}}$ ,
and the proposition is proved. ∎

Proposition 6.2 says that there are at least as many $N$-hypergeometric
solutions of the qKZ equations as the $N$-hypergeometric solutions of the KZ
equations.

It is an interesting problem to describe the module of the $N$-hypergeometric
solutions in either case. The answer depends on arithmetic properties of
$N,n,\kappa,l$.

##  Appendix A Hypergeometric solutions

In this section we remind the limit which produces the KZ equations from the
qKZ equations. We also remind the formulas for the hypergeometric complex-
valued solutions of the ${\mathfrak{sl}_{2}}$ KZ differential and qKZ
difference equations with values in ${\operatorname{Sing}}\,V^{\otimes
n}[n-2]$, that is, in the case when $l=1$.

###  A.1. KZ equations as a limit of qKZ equations

It is well known, that the KZ equations can be obtained as a limit of the qKZ
equations which explains the origin of Proposition 6.2. For the reader’s
convenience, we recall the construction.

Let $f(z_{1},\dots,z_{n})$ satisfy the qKZ equations,

(A.1)
$f(z_{1},\dots,z_{a}-\kappa,\dots,z_{n})\,=\,K_{a}(z\>;\kappa)\,f(z),\qquad
a=1,\dots,n.\ignorespaces$

Define $g(w_{1},\dots,w_{n};\alpha)=f(w_{1}/\alpha,\dots,w_{n}/\alpha)$. Then

(A.2)
$g(w_{1},\dots,w_{a}-\alpha\kappa,\dots,w_{n})\,=\,K_{a}(w/\alpha\>;\kappa)\,g(w),\qquad
a=1,\dots,n,\ignorespaces$

where

$\displaystyle K_{a}(w/\alpha;\kappa)$ $\displaystyle=$ $\displaystyle
R^{(a,a-1)}(w_{a}-w_{a-1}-\alpha\kappa;\alpha)\,\dots\,R^{(a,1)}(w_{a}-w_{1}-\alpha\kappa;\alpha)$
$\displaystyle\times$ $\displaystyle
R^{(a,n)}(w_{a}-w_{n};\alpha)\,\dots\,R^{(a,a+1)}(w_{a}\mskip-0.99998mu-
w_{a+1};\alpha)\,,$

and

$\displaystyle R(u;\alpha)=\frac{u-\alpha
P}{u-\alpha}=1\,-\,\alpha\,\frac{P-1}{u-\alpha}\,.$

In the limit $\alpha\to 0$, the qKZ difference equations (A.1) turn into the
KZ differential equations (6.1).

###  A.2. Solutions of the KZ equations

Define the master function and weight functions

$\displaystyle\Phi(t_{1},z)=\prod_{a=1}^{n}(t_{1}-z_{a})^{-1/\kappa},\qquad
w_{\\{a\\}}(t_{1},z)=\frac{1}{t-z_{a}}\ ,\qquad a=1,\dots,n.$

Consider the vector of integrals

$\displaystyle{F}_{(\gamma)}^{0}(z)\ =\ \sum_{a=1}^{n}\
\int_{\gamma(z)}\Phi(t_{1},z)\,w_{\\{a\\}}(t_{1},z)\,dt_{1}\,v_{\\{a\\}}\,\in\
V^{\otimes n}[n-2]\,,$

where $\gamma(z)$ is a flat family of 1-cycles associated with the multi-
valued master function $\Phi(t_{1},z)$. Then ${F}_{(\gamma)}^{0}(z)$ is a
solution of the KZ differential equations (6.1) and
$e{F}_{(\gamma)}^{0}(z)=0$.

In this case the solutions are labeled by the families of cycles $\gamma(z)$.

###  A.3. Solutions of the qKZ equations

Define the master function and weight functions

$\displaystyle\Phi(t_{1},z)=\prod_{a=1}^{n}\frac{\Gamma\big{(}\frac{t_{1}-z_{a}+1}{-\kappa}\big{)}}{\Gamma\big{(}\frac{t_{1}-z_{a}}{-\kappa}\big{)}},\qquad
w_{\\{a\\}}(t_{1},z)=\frac{1}{t_{1}-z_{a}}\prod_{l=1}^{a-1}\frac{t_{1}-z_{l}+1}{t-z_{l}}\
,\qquad a=1,\dots,n.$

Define the trigonometric weight functions

$\displaystyle W_{\\{b\\}}(t_{1},z)\,=\,\frac{e^{\pi
i(t-z_{b})/\kappa}}{\sin(\pi(t-z_{b})/\kappa)}\,\prod_{l=1}^{b-1}\,\frac{\sin(\pi(t-z_{l}+1)/\kappa)}{\sin(\pi(t-z_{l})/\kappa)}\,,$
$\displaystyle
W_{\\{b\\}}^{\text{sing}}(t_{1},z)=W_{\\{b\\}}(t_{1},z)-W_{\\{b+1\\}}(t_{1},z)\,,\qquad
b=1,\dots,n-1.$

Consider the vector of integrals

$\displaystyle{F}_{\\{b\\}}(z)\ =\ \sum_{a=1}^{n}\
\int_{i{\mathbb{R}}}\Phi(t_{1},z)\,w_{\\{a\\}}(t_{1},z)\,W_{\\{b\\}}^{\text{sing}}(t_{1},z)\,dt_{1}\,v_{\\{a\\}}\,\in\
V^{\otimes n}[n-2]\,,$

$b=1,\dots,n-1$, where $i{\mathbb{R}}$ is the imaginary axis properly deformed
depending on values of $z$. Then ${F}_{\\{b\\}}(z)$ is a solution of the qKZ
equations (2.4) and $e{F}_{\\{b\\}}(z)=0$, see [TV, Section 8].

In this case the solutions are labeled by the trigonometric weight functions
$W_{\\{b\\}}^{\text{sing}}(t_{1},z)$.

One may study the functions ${F}_{\\{b\\}}(z)$ under the limit $\alpha\to 0$
as in Section A.1 and show that the limit gives solutions
${F}_{(\gamma)}^{0}(z)$ of Section A.2, see [TV, Section 8]. The proof of this
fact is rather delicate, if compared with the trivial observation in
Proposition 6.2.

## References

  * [1]
  * [FR] I. Frenkel and N. Reshetikhin, Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146 (1992), no. 1, 1–60
  * [2]
  * [MV] E. Mukhin and A. Varchenko, The qKZ equation in tensor products of irreducible ${\mathfrak{sl}_{2}}$-modules, in ”Calogero-Moser-Sutherland Models”, CRM Series in Mathematical Physics, Springer, 2000, 347–384
  * [3]
  * [SV] V. Schechtman and A. Varchenko, Solutions of KZ differential equations modulo $p$, The Ramanujan Journal, 48 (3), 2019, 655–683
  * [4]
  * [TV] V. Tarasov and A. Varchenko, Geometry of q-hypergeometric functions as a bridge between Yangians and quantum affine algebras, Invent. Math. 128 (1997), 501–588
  * [5]
  * [V1] A. Varchenko, Hyperelliptic integrals modulo $p$ and Cartier-Manin matrices, Pure and Applied Math Quarterly, 16 (2020), n. 3, 315–336
  * [6]
  * [V2] A. Varchenko, On the number of p-hypergeometric solutions of KZ equations, arXiv:2203.03039, 1–19
  * [7]
  * [V3] A. Varchenko, Notes on solutions of KZ equations modulo $p^{s}$ and $p$-adic limit $s\to\infty$, with Appendix written jointly with S. Sperber, arXiv:2103.01725, 1–42
  * [8]