# Dwork–type congruences and $p$-adic KZ connection

Alexander Varchenko

###### Abstract.

We show that the $p$-adic KZ connection associated with the family of curves
$y^{q}=(t-z_{1})\dots(t-z_{qg+1})$ has an invariant subbundle of rank $g$,
while the corresponding complex KZ connection has no nontrivial proper
subbundles due to the irreducibility of its monodromy representation. The
construction of the invariant subbundle is based on new Dwork–type congruences
for associated Hasse–Witt matrices.

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

Key words: KZ equations; Dwork–type congruences; Hasse–Witt matrices.

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

††footnotetext: ${}^{\star}\mskip-0.99998mu$E-mail<EMAIL_ADDRESS>supported in part by NSF grant DMS-1954266

###### Contents

  1. 1 Introduction
  2. 2 Dwork–type congruences
    1. 2.1 Definition of ghosts
    2. 2.2 Convex polytopes
    3. 2.3 Hasse–Witt matrices
    4. 2.4 Congruences
    5. 2.5 Derivations
  3. 3 Convergence
    1. 3.1 Unramified extensions of ${\mathbb{Q}}_{p}$
    2. 3.2 Domain $\mathfrak{D}_{B}$
    3. 3.3 Uniqueness theorem
    4. 3.4 Infinite tuples
    5. 3.5 Domain of convergence
  4. 4 KZ equations and complex solutions
    1. 4.1 KZ equations
    2. 4.2 Solutions over ${\mathbb{C}}$
    3. 4.3 Solutions as vectors of first derivatives
  5. 5 Solutions modulo powers of $p$
    1. 5.1 Assumptions
    2. 5.2 Polynomial solutions
    3. 5.3 Coefficients of solutions
  6. 6 Congruences for solutions of KZ equations
    1. 6.1 Congruences for Hasse–Witt matrices of KZ equations
    2. 6.2 Congruences for frames of solutions of KZ equations
    3. 6.3 Domain of convergence
    4. 6.4 Vector bundle $\mathcal{L}\,\to\,\mathfrak{D}^{(m),o}_{\operatorname{KZ}}$
    5. 6.5 Remarks
      1. 6.5.1
      2. 6.5.2
      3. 6.5.3
      4. 6.5.4

##  1\. Introduction

The Knizhnik–Zamolodchikov (KZ) differential equations are objects of
conformal field theory, representation theory, enumerative geometry, see for
example [KZ, Dr, EFK, MO, V2]. The solutions of the KZ equations have the form
of multidimensional hypergeometric functions, see [SV1]. In this paper we
discuss the analog of hypergeometric solutions of the KZ equations considered
over a $p$-adic field instead of the field of complex numbers.

More precisely, we consider the KZ equations in the special case, in which the
complex hypergeometric solutions are given by the integrals of the form

(1.1) $\displaystyle
I(z_{1},\dots,z_{qg+1})=\int_{C}\frac{R(t,z_{1},\dots,z_{qg+1})\,dt}{\sqrt[1/q]{(t-z_{1})\dots(t-z_{qg+1})}}$

where $q,g$ are positive integer parameters, and $R(t,z)$ are suitable
rational functions.

In this case the space of solutions of the KZ equations is a $qg$-dimensional
complex vector space. We also consider the $p$-adic version of the same
differential equations. We assume that $q$ is a prime number (that is a
technical assumption) and show that the $qg$-dimensional space of local
solutions of these $p$-adic KZ equations has a remarkable $g$-dimensional
subspace of solutions which can be $p$-adic analytically continued as a
subspace to a large domain $\mathfrak{D}_{\operatorname{KZ}}^{(m),o}$ in the
space where the KZ equations are defined, see Theorems 6.10 and 6.12 for
precise statements. This $g$-dimensional global subspace of solutions is
defined as the uniform $p$-adic limit of a $g$-dimensional space of polynomial
solutions of these KZ equations modulo $p^{s}$ as $s\to\infty$. For $q=2$ and
$g=1$ this construction was deduced in [V5] from the classical B. Dwork’s
paper [Dw], see also [VZ1]. For $q=2$ and any $g$ the corresponding
construction was developed in [VZ2].

In [SV2] general KZ equations were considered over the field
${\mathbb{F}}_{p}$ and their polynomial solutions were constructed as
$p$-approximations of hypergeometric integrals. In the current paper that
construction is modified to obtain polynomial solutions modulo $p^{s}$ of the
KZ equations related to the integrals in formula (1.1). The polynomial
solutions are vectors of polynomials with integer coefficients. We call them
the $p^{s}$-hypergeometric solutions. While the complex analytic integrals in
(1.1) give the whole $qg$-dimensional space of all solutions of the complex KZ
equations, the $p^{s}$-hypergeometric solutions span only a $g$-dimensional
subspace. Then the $p$-adic limit of that subspace as $s\to\infty$ gives the
desired globally defined subspace of solutions.

On other $p$-approximations of hypergeometric periods see [SV2, RV1, RV2, VZ1,
VZ2].

In order to prove Theorems 6.10 and 6.12 we develop new matrix Dwork–type
congruences in Section 2. In Section 3 we show how our Dwork–type congruences
imply the uniform $p$-adic convergence of certain sequences of matrices on
suitable domains of the space of their parameters. In Section 4 we define our
KZ equations and construct their complex holomorphic solutions. In Section 5
we describe the $p^{s}$-hypergeometric solutions of the same equations. In
Section 6 we formulate and prove the main Theorems 6.10 and 6.12.

This paper may be viewed as a continuation of the paper [VZ2] where the case
$q=2$ is developed.

The author thanks Louis Funar, Toshitake Kohno, Nick Salter, and Wadim Zudilin
for useful discussions. The author thanks Max Planck Institute for Mathematics
in Bonn for hospitality in May-June of 2022.

##  2\. Dwork–type congruences

The Dwork–type congruences were originated by B. Dwork in the classical paper
[Dw]. On Dwork–type congruences see for example [Dw, Me, MV, Vl, VZ1, VZ2].

In this paper $p$ is an odd prime. We denote by ${\mathbb{Z}}_{p}[w^{\pm 1}]$
the ring of Laurent polynomials in variables $w$ with coefficients in
${\mathbb{Z}}_{p}$. A congruence $F(w)\equiv G(w)\pmod{p^{s}}$ for two Laurent
polynomials from the ring is understood as the divisibility by $p^{s}$ of all
coefficients of $F(w)-\nobreak G(w)$.

For a Laurent polynomial $G(w)$ we define $\sigma(G(w))=G(w^{p})$.

We denote $x=(t,z)$, where $t=(t_{1},\dots,t_{r})$ and $z=(z_{1},\dots,z_{n})$
are two groups of variables.

###  2.1. Definition of ghosts

Let ${\bf e}=(e_{1},\dots,e_{l})$ be a tuple of positive integers and
$\Lambda=(\Lambda_{0}(x),\Lambda_{1}(x),\dots,\Lambda_{l}(x))$ a tuple of
Laurent polynomials in ${\mathbb{Z}}_{p}[x^{\pm 1}]$.

Define $V_{0}(x)=\Lambda_{0}(x)$. For $s=1,\dots,l$, define $V_{s}(x)$ by the
recursive formula

(2.1)
$\displaystyle\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\dots\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}}=V_{s}(x)+V_{s-1}(x)\Lambda_{s}(x^{p^{e_{1}+\dots+e_{s}}})+$
$\displaystyle\phantom{aaa}+V_{s-2}(x)\Lambda_{s-1}(x^{p^{e_{1}+\dots+e_{s-1}}})\Lambda_{s}(x^{p^{e_{1}+\dots+e_{s-1}}})^{p^{e_{s}}}+\dots+$
$\displaystyle\phantom{aaaaaa}+V_{0}(x)\Lambda_{1}(x^{p^{e_{1}}})\Lambda_{2}(x^{p^{e_{1}}})^{p^{e_{2}}}\cdots\Lambda_{s}(x^{p^{e_{1}}})^{p^{e_{2}+\dots+e_{s}}},$

The Laurent polynomials $V_{0}(x),\dots,V_{l}(x)\in{\mathbb{Z}}_{p}[x^{\pm
1}]$ are called the ghosts associated with the tuples ${\bf e}$ and $\Lambda$.

For every $0\leqslant j\leqslant s\leqslant l$, denote

$\displaystyle W_{s}(x)$ $\displaystyle:=$
$\displaystyle\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\cdots\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}},$
$\displaystyle W_{s}^{(j)}(x)$ $\displaystyle:=$
$\displaystyle\Lambda_{j}(x)\Lambda_{j+1}(x)^{p^{e_{j+1}}}\cdots\Lambda_{s}(x)^{p^{e_{j+1}+\dots+e_{s}}}.$

Then (2.1) can be formulates as

(2.2)
$W_{s}(x)=V_{s}(x)+\sum_{j=1}^{s}V_{j-1}(x)W_{s}^{(j)}(x^{p^{e_{1}+\dots+e_{j}}}),$

or as

(2.3)
$W_{s}(x)=V_{s}(x)+\sum_{j=1}^{s}V_{j-1}(x)\sigma^{e_{1}+\dots+e_{j}}(W_{s}^{(j)}(x)).$

######  Lemma 2.1.

For $s=0,1,\dots,l$, we have $V_{s}(x)\equiv 0\pmod{p^{s}}$.

###### Proof.

In the proof we use the congruence $F(x^{p})^{p^{i-1}}\equiv
F(x)^{p^{i}}\pmod{p^{i}}$ valid for $i>0$.

For $s=0$ we have $V_{0}(x)=\Lambda_{0}(x)$ and no requirements on
divisibility. For $s=1$, we have

$\displaystyle
V_{1}(x)=\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}-V_{0}(x)\Lambda_{1}(x^{p^{e_{1}}})=\Lambda_{0}(x)(\Lambda_{1}(x)^{p^{e_{1}}}-\Lambda_{1}(x^{p^{e_{1}}})),$

and

(2.4)
$\displaystyle\Lambda_{1}(x^{p^{e_{1}}})\stackrel{{\scriptstyle\pmod{p}}}{{\equiv}}\Lambda_{1}(x^{p^{e_{1}-1}})^{p}\stackrel{{\scriptstyle\pmod{p^{2}}}}{{\equiv}}\Lambda_{1}(x^{p^{e_{1}-2}})^{p^{2}}\stackrel{{\scriptstyle\pmod{p^{3}}}}{{\equiv}}\dots\stackrel{{\scriptstyle\pmod{p^{e_{1}}}}}{{\equiv}}\Lambda_{1}(x)^{p^{e_{1}}}.$

This proves the lemma for $s=1$.

For $s>1$ the proof is by induction on $s$. Assume that the lemma is proved
for all $j<s$. Then similarly to (2.4) we obtain
$\Lambda_{s}(x^{p^{e_{1}+\dots+e_{j}}})^{p^{e_{j+1}+\dots+e_{s}}}{\equiv}\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}}\pmod{p^{1+e_{j+1}+\dots+e_{s}}}$
and hence

$\displaystyle
V_{j-1}(x)\Lambda_{s}(x^{p^{e_{1}+\dots+e_{j}}})^{p^{e_{j+1}+\dots+e_{s}}}$
$\displaystyle\equiv$ $\displaystyle
V_{j-1}(x)\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}}\pmod{p^{j+e_{j+1}+\dots+e_{s}}}$
$\displaystyle\equiv$ $\displaystyle
V_{j-1}(x)\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}}\pmod{p^{s}}$

since $e_{i}\geqslant 1$ for all $i$. Then we deduce modulo $p^{s}$:

$\displaystyle V_{s}(x)$
$\displaystyle=W_{s-1}(x)\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}}-\sum_{j=1}^{s-1}V_{j-1}(x)W_{s-1}^{(j)}(x^{p^{e_{1}+\dots+e_{j}}})\Lambda_{s}(x^{p^{e_{1}+\dots+e_{j}}})^{p^{e_{j+1}+\dots+e_{s}}}$
$\displaystyle-V_{s-1}(x)\Lambda_{s}(x^{p^{e_{1}+\dots+e_{s}}})\equiv$
$\displaystyle\equiv\bigg{(}W_{s-1}(x)-\sum_{j=1}^{s-1}V_{j-1}(x)W_{s-1}^{(j)}(x^{p^{e_{1}+\dots+e_{j}}})-V_{s-1}(x)\bigg{)}\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}}=0,$

obtaining the required statement. ∎

For a Laurent polynomial $F(t,z)$ in $t,z$, let $N(F)\subset{\mathbb{R}}^{r}$
be the Newton polytope of $F(t,z)$ with respect to the $t$ _variables only_.

######  Lemma 2.2.

For $s=0,1,\dots,l$, we have

$\displaystyle N(V_{s})\subset
N(\Lambda_{0})+p^{e_{1}}N(\Lambda_{1})+\dots+p^{e_{1}+\dots+e_{s}}N(\Lambda_{s})\,.$

###### Proof.

This follows from (2.2) by induction on $s$. ∎

###  2.2. Convex polytopes

Let $\Delta=(\Delta_{0},\dots,\Delta_{l})$ be a tuple of nonempty finite
subsets of ${\mathbb{Z}}^{r}$ of the same size $\\#\Delta_{j}=g$ for some
positive integer $g$.

######  Definition 2.3.

A tuple $(N_{0},N_{1},\dots,N_{l})$ of convex polytopes in ${\mathbb{R}}^{r}$
is called $(\Delta,{\bf e})$-_admissible_ if for any $0\leqslant i\leqslant
j<l$ we have

(2.5)
$\displaystyle\phantom{aaaaaa}\big{(}\Delta_{i}+N_{i}+p^{e_{i+1}}N_{i+1}+\dots+p^{e_{i+1}+\dots+e_{j}}N_{j}\big{)}\cap
p^{e_{i+1}+\dots+e_{j+1}}{\mathbb{Z}}^{r}\subset
p^{e_{i+1}+\dots+e_{j+1}}\Delta_{j+1}\,.$

Notice that any sub-tuple $(N_{i},N_{i+1},\dots,N_{j})$ of a $(\Delta,{\bf
e})$-admissible tuple $(N_{0},N_{1},\dots,N_{l})$ is $(\Delta^{\prime},{\bf
e}^{\prime})$-admissible where $\Delta^{\prime}=(\Delta_{i},\dots,\Delta_{j})$
and ${\bf e}^{\prime}=(e_{i+1},\dots,e_{j})$.

######  Definition 2.4.

A tuple $(\Lambda_{0}(t,z),\Lambda_{1}(t,z),\dots,\Lambda_{l}(t,z))$ of
Laurent polynomials is called $(\Delta,{\bf e})$-_admissible_ if the tuple
$\big{(}N(\Lambda_{0}),N(\Lambda_{1}),\dots,N(\Lambda_{l})\big{)}$ is
$(\Delta,{\bf e})$-admissible.

######  Example.

Let $r=1$, $n=13$, ${\bf e}=(2,2,\dots,2)$,
$\Gamma=\\{1,2,3,4\\}\subset{\mathbb{Z}}$,
$\Delta=(\Gamma,\Gamma,\dots,\Gamma)$,
$N=[0,13(p^{2}-1)/3]\subset{\mathbb{R}}$,
$F(t_{1},z)=\prod_{i=1}^{13}(t_{1}-z_{i})^{(p^{2}-1)/3}$. Then the tuple
$(N,N,\dots,N)$ of intervals in ${\mathbb{R}}$ and the tuple of polynomials
$(F(t_{1},z),F(t_{1},z),\dots,F(t_{1},z))$ are $(\Delta,{\bf e})$-admissible.

###  2.3. Hasse–Witt matrices

For $v\in{\mathbb{Z}}^{r}$ denote by $\operatorname{Coeff}_{v}F(t,z)$ the
coefficient of $t^{v}$ in the Laurent polynomial $F(t,z)$. This is a Laurent
polynomial in $z$.

Given $m\geqslant 1$ and finite subsets
$\Delta^{\prime},\Delta^{\prime\prime}\subset{\mathbb{Z}}^{r}$, define the
_Hasse–Witt matrix_ of the Laurent polynomial $F(t,z)$ by the formula

(2.6) $\displaystyle
A(m,\Delta^{\prime},\Delta^{\prime\prime},F(t,z)):=\big{(}\operatorname{Coeff}_{p^{m}v-u}F(t,z)\big{)}_{u\in\Delta^{\prime},v\in\Delta^{\prime\prime}}\,.$

######  Lemma 2.5.

Let $\Lambda$ be a $(\Delta,{\bf e})$-admissible tuple of Laurent polynomials
in ${\mathbb{Z}}_{p}[x^{\pm 1}]={\mathbb{Z}}_{p}[t^{\pm 1},z^{\pm 1}]$. Then
for $0\leqslant s\leqslant l$ we have

(i)
$\displaystyle\;\;A(e_{1}+\dots+e_{s+1},\Delta_{0},\Delta_{s+1},V_{s})\equiv
0\pmod{p^{s}};$ (ii)
$\displaystyle\;\;A\big{(}e_{1}+\dots+e_{s+1},\Delta_{0},\Delta_{s+1},W_{s}\big{)}=$
$\displaystyle=A\big{(}e_{1},\Delta_{0},\Delta_{1},V_{0})\cdot\sigma^{e_{1}}\big{(}A\big{(}e_{2}+\dots+e_{s+1},\Delta_{1},\Delta_{s+1},W_{s}^{(1)}\big{)}\big{)}+$
$\displaystyle+A\big{(}e_{1}+e_{2},\Delta_{0},\Delta_{2},V_{1})\cdot\sigma^{e_{1}+e_{2}}\big{(}A\big{(}e_{3}+\dots+e_{s+1},\Delta_{2},\Delta_{s+1},W_{s}^{(2)}\big{)}\big{)}+\dots+$
$\displaystyle+A\big{(}e_{1}+\dots+e_{s},\Delta_{0},\Delta_{s},V_{s-1})\cdot\sigma^{e_{1}+\dots+e_{s}}\big{(}A\big{(}e_{s+1},\Delta_{s},\Delta_{s+1},W_{s}^{(s)}\big{)}\big{)}+$
$\displaystyle+A\big{(}e_{1}+\dots+e_{s+1},\Delta_{0},\Delta_{s+1},V_{s}).$

Notice that all these matrices are $g\times g$-matrices.

###### Proof.

Part (i) follows from Lemma 2.1. To prove (ii) consider the identity

(2.7)
$\displaystyle\Lambda_{0}(t,z)\Lambda_{1}(t,z)^{p^{e_{1}}}\dots\Lambda_{s}(t,z)^{p^{e_{1}+\dots+e_{s}}}=\sum_{j=1}^{s}V_{j-1}(t,z)\Lambda_{j}(t^{p^{e_{1}+\dots+e_{j}}},z^{p^{e_{1}+\dots+e_{j}}})\times$
$\displaystyle\qquad\times\Lambda_{j+1}(t^{p^{e_{1}+\dots+e_{j}}},z^{p^{e_{1}+\dots+e_{j}}})^{p^{e_{j+1}}}\dots\Lambda_{s}(t^{p^{e_{1}+\dots+e_{j}}},z^{p^{e_{1}+\dots+e_{j}}})^{p^{e_{j+1}+\dots+e_{s}}}+V_{s}(t,z),$

which is nothing else but (2.1). Let $u\in\Delta_{0},v\in\Delta_{s+1}$. In
order to calculate the coefficient of $t^{p^{e_{1}+\dots+e_{s+1}}v-u}$ in the
$j$-th summand on the right-hand side of (2.7), we look for all pairs of
vectors $w\in N(V_{j-1})$ and $y\in
N(\Lambda_{j}(t,z)\dots\Lambda_{s}(t,z)^{p^{e_{j+1}+\dots+e_{s+1}}})$ such
that

$\displaystyle w+p^{e_{1}+\dots+e_{j}}y=p^{e_{1}+\dots+e_{s+1}}v-u.$

Hence $u+w\in p^{e_{1}+\dots+e_{j}}{\mathbb{Z}}^{r}$. On the other hand, it
follows from Lemma 2.2 that $w\in
N(\Lambda_{0})+p^{e_{1}}N(\Lambda_{1})+\dots+p^{e_{1}+\dots+e_{j-1}}N(\Lambda_{j-1})$,
so that

$\displaystyle
u+w\in\Delta_{0}+N(\Lambda_{0})+pN(\Lambda_{1})+\dots+p^{e_{1}+\dots+e_{j-1}}N(\Lambda_{j-1}).$

From the $(\Delta,{\bf e})$-admissibility we deduce that
$u+w=p^{e_{1}+\dots+e_{j}}\delta$ for some $\delta\in\Delta_{j}$, thus
$w=p^{e_{1}+\dots+e_{j}}\delta-\nobreak u$,
$y=p^{e_{j+1}+\dots+e_{s+1}}v-\delta$ and

$\displaystyle\operatorname{Coeff}_{p^{e_{1}+\dots+e_{s+1}}v-u}\big{(}V_{j-1}(t,z)\Lambda_{j}(t^{p^{e_{1}+\dots+e_{j}}},z^{p^{e_{1}+\dots+e_{j}}})\dots\Lambda_{s}(t^{p^{e_{1}+\dots+e_{j}}},z^{p^{e_{1}+\dots+e_{j}}})^{p^{e_{j+1}+\dots+e_{s}}}\big{)}=$
$\displaystyle\phantom{aaa}=\sum_{\delta\in\Delta_{j}}\operatorname{Coeff}_{p^{e_{1}+\dots+e_{j}}\delta-u}(V_{j-1}(t,z))\,\cdot$
$\displaystyle\phantom{aaaaaa}\cdot\,\sigma^{e_{1}+\dots+e_{j}}\big{(}\operatorname{Coeff}_{p^{e_{j+1}+\dots+e_{s+1}}v-\delta}\big{(}\Lambda_{j}(t,z)\Lambda_{j+1}(t,z)^{p^{e_{j+1}}}\dots\Lambda_{s}(t,z)^{p^{e_{j+1}+\dots+e_{s}}}\big{)}\big{)}.$

This proves (ii). ∎

###  2.4. Congruences

The next results discuss congruences of the type
$F_{1}(z)F_{2}(z)^{-1}\equiv G_{1}(z)G_{2}(z)^{-1}\pmod{p^{s}}$, where
$F_{1},F_{2},G_{1},G_{2}$ are $g\times g$ matrices whose entries are Laurent
polynomials in $z$. We consider such congruences when the determinants $\det
F_{2}(z)$ and $\det G_{2}(z)$ are Laurent polynomials both nonzero modulo $p$.
Using Cramer’s rule we write the entries of the inverse matrix $F_{2}(z)^{-1}$
in the form $f_{ij}(z)/\det F_{2}(z)$ for $f_{ij}(z)\in{\mathbb{Z}}_{p}[z^{\pm
1}]$ and do a similar computation for $G_{2}(z)$. This presents the congruence
$F_{1}(z)F_{2}(z)^{-1}\equiv G_{1}(z)G_{2}(z)^{-1}\pmod{p^{s}}$ in the form

(2.8) $\displaystyle\frac{1}{\det F_{2}(z)}\cdot F(z)\ \equiv\ \frac{1}{\det
G_{2}(z)}\cdot G(z)\pmod{p^{s}}$

for some $g\times g$ matrices $F(z),G(z)$ with entries in
${\mathbb{Z}}_{p}[z^{\pm 1}]$, while (2.8) is nothing else but the congruence
$F(z)\cdot\det G_{2}(z)\equiv G(z)\cdot\det F_{2}(z)\pmod{p^{s}}$.

######  Theorem 2.6.

Let $(\Lambda_{0}(t,z),\Lambda_{1}(t,z),\dots,\Lambda_{l}(t,z))$ be a
$(\Delta,{\bf e})$-admissible tuple of Laurent polynomials in
${\mathbb{Z}}_{p}[x^{\pm 1}]={\mathbb{Z}}_{p}[t^{\pm 1},z^{\pm 1}]$.

  1. (i)

For $0\leqslant s\leqslant l$ we have

$\displaystyle
A\big{(}e_{1}+\dots+e_{s+1},\Delta_{0},\Delta_{s+1},\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\cdots\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}}\big{)}\equiv$
$\displaystyle\equiv
A\big{(}e_{1},\Delta_{0},\Delta_{1},\Lambda_{0}(x)\big{)}\cdot\sigma^{e_{1}}\big{(}A\big{(}e_{2},\Delta_{1},\Delta_{2},\Lambda_{1}(x)\big{)}\big{)}\cdots\sigma^{e_{1}+\dots+e_{s}}\big{(}A\big{(}e_{s+1},\Delta_{s},\Delta_{s+1},\Lambda_{s}(x)\big{)}\big{)}$

modulo $p$.

  2. (ii)

Assume that the determinants of the matrices
$A\big{(}e_{i+1},\Delta_{i},\Delta_{i+1},\Lambda_{i}(t,z)\big{)}$,
$i=0,1,\dots,l$, are Laurent polynomials all nonzero modulo $p$. Then for
$1\leqslant s\leqslant l$ the determinant of the matrix
$A\big{(}e_{2}+\dots+e_{s+1},\Delta_{1},\Delta_{s+1},\Lambda_{1}(x)\Lambda_{2}(x)^{p^{e_{2}}}\cdots\Lambda_{s}(x)^{p^{e_{2}+\dots+e_{s}}}\big{)}$
is a Laurent polynomial nonzero modulo $p$ and we have modulo $p^{s}$ :

(2.9) $\displaystyle
A\big{(}e_{1}+\dots+e_{s+1},\Delta_{0},\Delta_{s+1},\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\cdots\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}}\big{)}\cdot$
$\displaystyle\cdot\sigma^{e_{1}}\big{(}A\big{(}e_{2}+\dots+e_{s+1},\Delta_{1},\Delta_{s+1},\Lambda_{1}(x)\Lambda_{2}(x)^{p^{e_{2}}}\cdots\Lambda_{s}(x)^{p^{e_{2}+\dots+e_{s}}}\big{)}\big{)}^{-1}\equiv$
$\displaystyle\equiv
A\big{(}e_{1}+\dots+e_{s},\Delta_{0},\Delta_{s},\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\cdots\Lambda_{s-1}(x)^{p^{e_{1}+\dots+e_{s-1}}}\big{)}\cdot$
$\displaystyle\cdot\sigma^{e_{1}}\big{(}A\big{(}e_{2}+\dots+e_{s},\Delta_{1},\Delta_{s},\Lambda_{1}(x)\Lambda_{2}(x)^{p^{e_{2}}}\cdots\Lambda_{s-1}(x)^{p^{e_{2}+\dots+e_{s-1}}}\big{)}\big{)}^{-1},$

where in this congruence for $s=1$ we understand the second factor on the
right-hand side as the $g\times g$ identity matrix, see formula (2.10) below.

###### Proof.

By Lemma 2.5 we have

$\displaystyle
A\big{(}e_{1}+\dots+e_{s+1},\Delta_{0},\Delta_{s+1},\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\dots\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}}\big{)}\equiv$
$\displaystyle\equiv
A\big{(}e_{1},\Delta_{0},\Delta_{1},\Lambda_{0}(x))\cdot\sigma^{e_{1}}\big{(}A\big{(}e_{2}+\dots+e_{s+1},\Delta_{1},\Delta_{s+1},\Lambda_{1}(x)\Lambda_{2}(x)^{p^{e_{2}}}\dots\Lambda_{s}(x)^{p^{e_{2}+\dots+e_{s}}}\big{)}\big{)}$

modulo $p$. Iteration gives part (i) of the theorem.

If the determinants of the matrices
$A\big{(}e_{i+1},\Delta_{i},\Delta_{i+1},\Lambda_{i}(t,z)\big{)}$,
$i=0,1,\dots,l$, are Laurent polynomials all nonzero modulo $p$, then part (i)
implies that the determinant

$\displaystyle\det
A\big{(}e_{2}+\dots+e_{s+1},\Delta_{1},\Delta_{s+1},\Lambda_{1}(x)\Lambda_{2}(x)^{p^{e_{2}}}\cdots\Lambda_{s}(x)^{p^{e_{2}+\dots+e_{s}}}\big{)}\equiv$
$\displaystyle\equiv\prod_{j=1}^{s}\det\sigma^{e_{2}+\dots+e_{j}}\big{(}A\big{(}e_{j+1},\Delta_{j},\Delta_{j+1},\Lambda_{j}(t,z)\big{)}\big{)}\pmod{p},$

is a Laurent polynomial nonzero modulo $p$. This proves the first statement of
part (ii) of the theorem and allows us to consider the inverse matrices in the
congruence of part (ii).

We prove part (ii) by induction on $s$. For $s=1$, congruence (2.9) takes the
form

(2.10) $\displaystyle
A\big{(}e_{1}+e_{2},\Delta_{0},\Delta_{2},\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\big{)}\cdot\sigma^{e_{1}}\big{(}A\big{(}e_{2},\Delta_{1},\Delta_{2},\Lambda_{1}(x)\big{)}\big{)}^{-1}\equiv
A\big{(}e_{1},\Delta_{2},\Delta_{1},\Lambda_{0}(x)\big{)}$

modulo $p$. This congruence follows from part (i).

For $1<s<l$ we substitute the expressions for
$A\big{(}e_{1}+\dots+e_{s+1},\Delta_{0},\Delta_{s+1},\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\cdots$
$\cdots\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}}\big{)}$ and
$A\big{(}e_{1}+\dots+e_{s},\Delta_{0},\Delta_{s},\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\cdots\Lambda_{s-1}(x)^{p^{e_{1}+\dots+e_{s-1}}}\big{)}$
from part (ii) of Lemma 2.5 into the two sides of the desired congruence:

(2.11) $\displaystyle
A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\cdots\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}}\big{)}\cdot$
$\displaystyle\cdot\sigma^{e_{1}}\big{(}A\big{(}\sum_{a=2}^{s+1}e_{a},\Delta_{1},\Delta_{s+1},\Lambda_{1}(x)\Lambda_{2}(x)^{p^{e_{2}}}\cdots\Lambda_{s}(x)^{p^{e_{2}+\dots+e_{s}}}\big{)}\big{)}^{-1}=A\big{(}e_{1},\Delta_{0},\Delta_{1},V_{0})+$
$\displaystyle+\sum_{j=2}^{s}A\big{(}\sum_{a=1}^{j}e_{a},\Delta_{0},\Delta_{j+1},V_{j-1})\cdot\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}\big{)}\big{)}\cdot$
$\displaystyle\cdot\,\sigma^{e_{1}}\big{(}A\big{(}\sum_{a=2}^{s+1}e_{a},\Delta_{1},\Delta_{s+1},W_{s}^{(1)}\big{)}\big{)}^{-1}+$
$\displaystyle+\,A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},V_{s}\big{)}\cdot\sigma^{e_{1}}\big{(}A\big{(}\sum_{a=2}^{s+1}e_{a},\Delta_{1},\Delta_{s+1},W_{s}^{(1)}\big{)}\big{)}^{-1}$

and

(2.12) $\displaystyle
A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\cdots\Lambda_{s-1}(x)^{p^{e_{1}+\dots+e_{s-1}}}\big{)}\cdot$
$\displaystyle\cdot\sigma^{e_{1}}\big{(}A\big{(}\sum_{a=2}^{s}e_{a},\Delta_{1},\Delta_{s},\Lambda_{1}(x)\Lambda_{2}(x)^{p^{e_{2}}}\cdots\Lambda_{s-1}(x)^{p^{e_{2}+\dots+e_{s-1}}}\big{)}\big{)}^{-1}=A\big{(}e_{1},\Delta_{0},\Delta_{1},V_{0})+$
$\displaystyle+\sum_{j=2}^{s}A\big{(}\sum_{a=1}^{j}e_{a},\Delta_{0},\Delta_{j+1},V_{j-1})\cdot\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s}e_{a},\Delta_{j},\Delta_{s},W_{s-1}^{(j)}\big{)}\big{)}\cdot$
$\displaystyle\cdot\,\sigma^{e_{1}}\big{(}A\big{(}\sum_{a=2}^{s}e_{a},\Delta_{1},\Delta_{s},W_{s-1}^{(1)}\big{)}\big{)}^{-1}.$

Since we want to compare these two expressions modulo $p^{s}$, the last term
in (2.11) containing $V_{s}\equiv 0\pmod{p^{s}}$ can be ignored.

Given $j=2,\dots,s$, we use the inductive hypothesis as follows:

$\displaystyle
A\big{(}\sum_{a=i+1}^{s+1}e_{a},\Delta_{i},\Delta_{s+1},W^{(i)}_{s}\big{)}\cdot\sigma^{e_{i+1}}\big{(}A\big{(}\sum_{a=i+2}^{s+1}e_{a},\Delta_{i+1},\Delta_{s+1},W^{(i+1)}_{s}\big{)}\big{)}^{-1}\equiv$
$\displaystyle\equiv
A\big{(}\sum_{a=i+1}^{s}e_{a},\Delta_{i},\Delta_{s},W^{(i)}_{s-1}\big{)}\cdot\sigma^{e_{i+1}}\big{(}A\big{(}\sum_{a=i+2}^{s}e_{a},\Delta_{i+1},\Delta_{s},W^{(i+1)}_{s-1}\big{)}\big{)}^{-1}\pmod{p^{s-i}}$

for $i=1,\dots,j-1$. Applying $\sigma^{\sum_{a=1}^{i}e_{a}}$ to the $i$-th
congruence and multiplying them out lead to telescoping products on both
sides:

$\displaystyle\sigma^{e_{1}}\big{(}A\big{(}\sum_{a=2}^{s+1}e_{a},\Delta_{1},\Delta_{s+1},W_{s}^{(1)}\big{)}\big{)}\cdot\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}\big{)}\big{)}^{-1}\equiv$
$\displaystyle\equiv\sigma^{e_{1}}\big{(}A\big{(}\sum_{a=2}^{s}e_{a},\Delta_{1},\Delta_{s},W_{s-1}^{(1)}\big{)}\big{)}\cdot\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s}e_{a},\Delta_{j},\Delta_{s},W_{s-1}^{(j)}\big{)}\big{)}^{-1}$

modulo $p^{s-j+1}$. By our assumptions these four matrices are invertible.
Therefore, we can invert them to obtain the congruence

(2.13)
$\displaystyle\phantom{aaa}\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}\big{)}\big{)}\cdot\sigma^{e_{1}}\big{(}A\big{(}\sum_{a=2}^{s+1}e_{a},\Delta_{1},\Delta_{s+1},W_{s}^{(1)}\big{)}\big{)}^{-1}\equiv$
$\displaystyle\phantom{aaa}\equiv\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s}e_{a},\Delta_{j},\Delta_{s},W_{s-1}^{(j)}\big{)}\big{)}\cdot\sigma^{e_{1}}\big{(}A\big{(}\sum_{a=2}^{s}e_{a},\Delta_{1},\Delta_{s},W_{s-1}^{(1)}\big{)}\big{)}^{-1}$

modulo $p^{s-j+1}$. Since $V_{j-1}\equiv 0\pmod{p^{j-1}}$, we obtain the
congruence

$\displaystyle
A\big{(}\sum_{a=1}^{j}e_{a},\Delta_{0},\Delta_{j+1},V_{j-1})\cdot$
$\displaystyle\cdot\,\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}\big{)}\big{)}\cdot\sigma^{e_{1}}\big{(}A\big{(}\sum_{a=2}^{s+1}e_{a},\Delta_{1},\Delta_{s+1},W_{s}^{(1)}\big{)}\big{)}^{-1}\equiv$
$\displaystyle\equiv
A\big{(}\sum_{a=1}^{j}e_{a},\Delta_{0},\Delta_{j+1},V_{j-1})\cdot$
$\displaystyle\cdot\,\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s}e_{a},\Delta_{j},\Delta_{s},W_{s-1}^{(j)}\big{)}\big{)}\cdot\sigma^{e_{1}}\big{(}A\big{(}\sum_{a=2}^{s}e_{a},\Delta_{1},\Delta_{s},W_{s-1}^{(1)}\big{)}\big{)}^{-1}$

modulo $p^{s}$. This shows that the $j$-th summands in (2.11) and (2.12) are
congruent modulo $p^{s}$. The theorem is proved. ∎

######  Corollary 2.7.

Under the assumptions of part (ii) of Theorem 2.6 for $1\leqslant s\leqslant
l$ we have :

$\displaystyle\det
A\big{(}e_{1}+\dots+e_{s+1},\Delta_{0},\Delta_{s+1},\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\cdots\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}}\big{)}\cdot$
$\displaystyle\cdot\det\sigma^{e_{1}}\big{(}A\big{(}e_{2}+\dots+e_{s},\Delta_{1},\Delta_{s},\Lambda_{1}(x)\Lambda_{2}(x)^{p^{e_{2}}}\cdots\Lambda_{s-1}(x)^{p^{e_{2}+\dots+e_{s-1}}}\big{)}\big{)}\equiv$
$\displaystyle\equiv\det
A\big{(}e_{1}+\dots+e_{s},\Delta_{0},\Delta_{s},\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\cdots\Lambda_{s-1}(x)^{p^{e_{1}+\dots+e_{s-1}}}\big{)}\cdot$
$\displaystyle\cdot\det\sigma^{e_{1}}\big{(}A\big{(}e_{2}+\dots+e_{s+1},\Delta_{1},\Delta_{s+1},\Lambda_{1}(x)\Lambda_{2}(x)^{p^{e_{2}}}\cdots\Lambda_{s}(x)^{p^{e_{2}+\dots+e_{s}}}\big{)}\big{)}$

modulo $p^{s}$.

###  2.5. Derivations

Recall that $z=(z_{1},\dots,z_{n})$. Denote

$\displaystyle D_{v}=\frac{\partial}{\partial z_{v}},\quad v=1,\dots,n.$

Let $F_{1}(z),F_{2}(z),G_{1}(z),G_{2}(z)\in{\mathbb{Z}}_{p}[z^{\pm 1}]$ and
$\ell\geqslant 1$. If

$\displaystyle D_{v}(F_{1}(z))\cdot F_{2}(z)\equiv D_{v}(G_{1}(z))\cdot
G_{2}(z)\pmod{p^{s}}\,,$

then

(2.14) $\displaystyle
D_{v}(\sigma^{\ell}(F_{1}(z)))\cdot\sigma^{\ell}(F_{2}(z))-D_{v}(\sigma^{\ell}(G_{1}(z)))\cdot\sigma^{\ell}(G_{2}(z))=$
$\displaystyle\qquad=D_{v}(F_{1}(z^{p^{\ell}}))\cdot
F_{2}(z^{p^{\ell}})-D_{v}(G_{1}(z^{p^{\ell}}))\cdot G_{2}(z^{p^{\ell}})=$
$\displaystyle\qquad=p^{\ell}z_{v}^{p^{\ell}-1}\big{(}D_{v}(F_{1}(z))\cdot
F_{2}(z)-D_{v}(G_{1}(z))\cdot G_{2}(z)\big{)}\big{|}_{z\to
z^{p^{\ell}}}\equiv$ $\displaystyle\qquad\equiv 0\pmod{p^{s+\ell}}.$

######  Theorem 2.8.

Let $(\Lambda_{0}(t,z),\Lambda_{1}(t,z),\dots,\Lambda_{l}(t,z))$ be a
$(\Delta,{\bf e})$-admissible tuple of Laurent polynomials in
${\mathbb{Z}}_{p}[x^{\pm 1}]={\mathbb{Z}}_{p}[t^{\pm 1},z^{\pm 1}]$. Let
$D=D_{v}$ for some $v=1,\dots,n$. Then under the assumptions of part (ii) of
Theorem 2.6 we have

(2.15) $\displaystyle
D\big{(}\sigma^{\ell}\big{(}A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}\big{)}\big{)}\cdot\sigma^{\ell}\big{(}A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}\big{)}^{-1}\equiv$
$\displaystyle\equiv
D\big{(}\sigma^{\ell}\big{(}A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}\big{)}\big{)}\cdot\sigma^{\ell}\big{(}A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}\big{)}^{-1}\pmod{p^{s+\ell}}$

for $1\leqslant s\leqslant l$ and $0\leqslant\ell$.

###### Proof.

Notice that it is sufficient to establish the congruences (2.15) for $\ell=0$,
as the general $\ell$ case follows from (2.14). So, we assume that $\ell=0$
and proceed by induction on $s\geqslant 0$. For $s=0$ the statement is
trivially true.

Using part (ii) of Lemma 2.5 we can write

(2.16) $\displaystyle
D\big{(}A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}\big{)}\cdot
A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}^{-1}=$
$\displaystyle=\sum_{j=1}^{s+1}D\big{(}A\big{(}\sum_{a=1}^{j}e_{a},\Delta_{0},\Delta_{j+1},V_{j-1}\big{)}\big{)}\cdot\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}\big{)}\big{)}\cdot$
$\displaystyle\cdot\,A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}^{-1}+$
$\displaystyle+\sum_{j=1}^{s+1}A\big{(}\sum_{a=1}^{j}e_{a},\Delta_{0},\Delta_{j+1},V_{j-1})\cdot
D\big{(}\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}\big{)}\big{)}\big{)}\cdot$
$\displaystyle\cdot\,A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}^{-1}$

and

(2.17) $\displaystyle
D\big{(}A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}\big{)}\cdot
A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}^{-1}=$
$\displaystyle=\sum_{j=1}^{s}D\big{(}A\big{(}\sum_{a=1}^{j}e_{a},\Delta_{0},\Delta_{j+1},V_{j-1}\big{)}\big{)}\cdot\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s}e_{a},\Delta_{j},\Delta_{s},W_{s-1}^{(j)}\big{)}\big{)}\cdot$
$\displaystyle\cdot\,A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}^{-1}+$
$\displaystyle+\sum_{j=1}^{s}A\big{(}\sum_{a=1}^{j}e_{a},\Delta_{0},\Delta_{j+1},V_{j-1})\cdot
D\big{(}\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s}e_{a},\Delta_{j},\Delta_{s},W_{s-1}^{(j)}\big{)}\big{)}\big{)}\cdot$
$\displaystyle\cdot\,A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}^{-1}.$

The summands corresponding to $j=s+1$ in (2.16) vanish modulo $p^{s}$ and can
be ignored since $V_{s}\equiv 0\pmod{p^{s}}$.

For the same reason

(2.18) $\displaystyle
D\big{(}A\big{(}\sum_{a=1}^{j}e_{a},\Delta_{0},\Delta_{j+1},V_{j-1}\big{)}\big{)}\equiv
0\pmod{p^{j-1}}.$

We also have

(2.19)
$\displaystyle\phantom{aaa}\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}\big{)}\big{)}\cdot
A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}^{-1}\equiv$
$\displaystyle\phantom{aaa}\equiv\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s}e_{a},\Delta_{j},\Delta_{s},W_{s-1}^{(j)}\big{)}\big{)}\cdot
A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}^{-1}\pmod{p^{s-j+1}}.$

This follows from (2.13), in which we take $j+1$ and $s+1$ for $j$ and $s$ and
use $W_{s}$ instead of $W_{s+1}^{(1)}$.

Multiplying congruences (2.18) and (2.19) we get

(2.20) $\displaystyle
D\big{(}A\big{(}\sum_{a=1}^{j}e_{a},\Delta_{0},\Delta_{j+1},V_{j-1}\big{)}\big{)}\cdot\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}\big{)}\big{)}\cdot$
$\displaystyle\cdot\,A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}^{-1}\equiv$
$\displaystyle\equiv
D\big{(}A\big{(}\sum_{a=1}^{j}e_{a},\Delta_{0},\Delta_{j+1},V_{j-1}\big{)}\big{)}\cdot\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s}e_{a},\Delta_{j},\Delta_{s},W_{s-1}^{(j)}\big{)}\big{)}\cdot$
$\displaystyle\cdot\,A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}^{-1}\pmod{p^{s}}.$

Congruence (2.20) implies that the first sum in (2.16) is congruent to the
first sum in (2.17) modulo $p^{s}$.

To match the second sums we recall the inductive hypothesis in the form

(2.21) $\displaystyle
D\big{(}\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}\big{)}\big{)}\big{)}\cdot\phantom{aaaaaaaaaaaa}$
$\displaystyle\cdot\,\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}\big{)}\big{)}^{-1}\equiv$
$\displaystyle\equiv\,D\big{(}\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s}e_{a},\Delta_{j},\Delta_{s},W_{s-1}^{(j)}\big{)}\big{)}\big{)}\cdot\phantom{aaaaaaaaa}$
$\displaystyle\phantom{aaaaaaaaa}\cdot\,\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s}e_{a},\Delta_{j},\Delta_{s},W_{s-1}^{(j)}\big{)}\big{)}^{-1}\pmod{p^{s}},$

and notice that both sides in (2.21) are congruent to zero modulo
$\sigma^{\sum_{a=1}^{j}e_{a}}$ by formula (2.14). Therefore, multiplying
congruences (2.21) and (2.19) we obtain

$\displaystyle
D\big{(}\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}\big{)}\big{)}\big{)}\cdot
A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}^{-1}\equiv$
$\displaystyle\equiv
D\big{(}\sigma^{\sum_{a=1}^{j}e_{a}}\big{(}A\big{(}\sum_{a=j+1}^{s}e_{a},\Delta_{j},\Delta_{s},W_{s-1}^{(j)}\big{)}\big{)}\big{)}\cdot
A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}^{-1}\pmod{p^{s}}.$

Multiplying both sides of this congruence by
$A\big{(}\sum_{a=1}^{j}e_{a},\Delta_{0},\Delta_{j+1},V_{j-1})$ we conclude
that the second sum in (2.16) is congruent to the second sum in (2.17) modulo
$p^{s}$. The theorem is proved. ∎

There are similar congruences for higher order derivatives of the matrices
$A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}$. We
restrict ourselves with the second order derivatives.

######  Theorem 2.9.

Let $(\Lambda_{0}(t,z),\Lambda_{1}(t,z),\dots,\Lambda_{l}(t,z))$ be a
$(\Delta,{\bf e})$-admissible tuple of Laurent polynomials in
${\mathbb{Z}}_{p}[x^{\pm 1}]={\mathbb{Z}}_{p}[t^{\pm 1},z^{\pm 1}]$. Then
under the assumptions of part (ii) of Theorem 2.6 we have

(2.22) $\displaystyle
D_{u}\big{(}D_{v}\big{(}A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}\big{)}\big{)}\,\cdot\,A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}^{-1}\equiv$
$\displaystyle\equiv
D_{u}\big{(}D_{v}\big{(}A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}\big{)}\big{)}\cdot
A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}^{-1}\pmod{p^{s}}$

for all $1\leqslant u,v\leqslant n$ and $0\leqslant s\leqslant l$.

###### Proof.

Notice that, for an invertible matrix $F(z)$ and a derivation $D$, we have
$D(F^{-1})=-F^{-1}\,D(F)\,F^{-1}$.

We apply the derivation $D_{u}$ to congruence (2.15) with $D=D_{v}$ :

$\displaystyle
D_{u}\big{(}D_{v}\big{(}A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}\big{)}\big{)}\cdot
A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}^{-1}+$
$\displaystyle+\,D_{v}\big{(}A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}\big{)}\cdot
A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}^{-1}\cdot$
$\displaystyle\cdot\,D_{u}\big{(}A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}\big{)}\cdot
A\big{(}\sum_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\big{)}^{-1}\equiv$
$\displaystyle\equiv
D_{u}\big{(}D_{v}\big{(}A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}\big{)}\big{)}\cdot
A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}^{-1}+$
$\displaystyle+\,D_{v}\big{(}A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}\big{)}\cdot
A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}^{-1}\cdot$
$\displaystyle\cdot\,D_{u}\big{(}A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}\big{)}\cdot
A\big{(}\sum_{a=1}^{s}e_{a},\Delta_{0},\Delta_{s},W_{s-1}\big{)}^{-1}$

modulo $p^{s}$. It remains to apply (2.15) with $D=D_{u}$ and $D=D_{v}$ and
$\ell=0$ to see that the second terms on both sides agree modulo $p^{s}$.
After their cancellation we are left with the required congruences in (2.22).
∎

######  Remark.

The results of Section 2 in the case ${\bf e}=(e_{1},\dots,e_{l})=(1,\dots,1)$
and $\Delta=(\Delta_{0},\dots,\Delta_{l})$ such that
$\Delta_{0}=\dots=\Delta_{l}$ were obtained in [VZ2].

##  3\. Convergence

###  3.1. Unramified extensions of ${\mathbb{Q}}_{p}$

We fix an algebraic closure $\overline{{\mathbb{Q}}_{p}}$ of
${\mathbb{Q}}_{p}$. For every $m$, there is a unique unramified extension of
${\mathbb{Q}}_{p}$ in $\overline{{\mathbb{Q}}_{p}}$ of degree $m$, denoted by
${\mathbb{Q}}_{p}^{(m)}$. This can be obtained by attaching to
${\mathbb{Q}}_{p}$ a primitive root of $1$ of order $p^{m}-1$. The norm
$|\cdot|_{p}$ on ${\mathbb{Q}}_{p}$ extends to a norm $|\cdot|_{p}$ on
${\mathbb{Q}}_{p}^{(m)}$. Let

$\displaystyle{\mathbb{Z}}_{p}^{(m)}=\\{a\in{\mathbb{Q}}_{p}^{(m)}\mid|a|_{p}\leqslant
1\\}$

denote the ring of integers in ${\mathbb{Q}}_{p}^{(m)}$. The ring
${\mathbb{Z}}_{p}^{(m)}$ has the unique maximal ideal

$\displaystyle\mathbb{M}_{p}^{(m)}=\\{a\in{\mathbb{Q}}_{p}^{(m)}\mid|a|_{p}<1\\},$

such that $\mathbb{Z}_{p}^{(m)}\big{/}\mathbb{M}_{p}^{(m)}$ is isomorphic to
the finite field ${\mathbb{F}}_{p^{m}}$.

For every $u\in{\mathbb{F}}_{p^{m}}$ there is a unique
$\tilde{u}\in\mathbb{Z}_{p}^{(m)}$ that is a lift of $u$ and such that
$\tilde{u}^{p^{m}}=\tilde{u}$. The element $\tilde{u}$ is called the
Teichmuller lift of $u$.

###  3.2. Domain $\mathfrak{D}_{B}$

For $u\in{\mathbb{F}}_{p^{m}}$ and $r>0$ denote

$\displaystyle
D_{u,r}=\\{a\in{\mathbb{Z}}_{p}^{(m)}\mid|a-\tilde{u}|_{p}<r\\}\,.$

We have the partition

$\displaystyle{\mathbb{Z}}_{p}^{(m)}=\bigcup_{u\in{\mathbb{F}}_{p^{m}}}D_{u,1}\,.$

Recall $z=(z_{1},\dots,z_{n})$. For $B(z)\in{\mathbb{Z}}[z]$, define

$\displaystyle\mathfrak{D}_{B}\ =\ \\{a\in({\mathbb{Z}}_{p}^{(m)})^{n}\,\mid\
|B(a)|_{p}=1\\}.$

Let $\bar{B}(z)$ be the projection of $B(z)$ to
${\mathbb{F}}_{p}[z]\subset{\mathbb{F}}_{p^{m}}[z]$. Then $\mathfrak{D}_{B}$
is the union of unit polydiscs,

$\displaystyle\mathfrak{D}_{B}=\bigcup_{\begin{subarray}{c}u_{1},\dots,u_{n}\in{\mathbb{F}}_{p^{m}}\\\
\bar{B}(u_{1},\dots,u_{n})\neq 0\end{subarray}}\ D_{u_{1},1}\times\dots\times
D_{u_{n},1}\,.$

For any $k$ we have

$\displaystyle\\{a\in({\mathbb{Z}}_{p}^{(m)})^{n}\mid\
|B(a^{p^{k}})|_{p}=1\\}$
$\displaystyle=\bigcup_{\begin{subarray}{c}u_{1},\dots,u_{n}\in{\mathbb{F}}_{p^{m}}\\\
\sigma^{k}(\bar{B}(u_{1},\dots,u_{n}))\neq 0\end{subarray}}\
D_{u_{1},1}\times\dots\times D_{u_{n},1}=$
$\displaystyle=\bigcup_{\begin{subarray}{c}u_{1},\dots,u_{n}\in F_{p^{m}}\\\
\bar{B}(u_{1},\dots,u_{n})\neq 0\end{subarray}}\ D_{u_{1},1}\times\dots\times
D_{u_{n},1}=\mathfrak{D}_{B}\,.$

######  Lemma 3.1 ([VZ2, Lemma 6.1]).

Let $\bar{B}_{1}(z),\dots,\bar{B}_{k}(z)\in{\mathbb{F}}_{p}[z]$ be nonzero
polynomials such that $\deg\bar{B}_{j}(z)\leqslant d$, $j=1,\dots,k$, for some
$d$. If $kd+1<p^{m}$, then the set

$\displaystyle\\{a\in({\mathbb{F}}_{p^{m}})^{n}\mid\bar{B}_{j}(a)\neq
0,\,j=1,\dots,f\\}$

is nonempty.

###  3.3. Uniqueness theorem

Let $\mathfrak{D}\subset({\mathbb{Z}}_{p}^{(m)})^{n}$ be the union of some of
the unit polydiscs $D_{u_{1},1}\times\dots\times D_{u_{n},1}$ , where
$u_{1},\dots,u_{n}\in{\mathbb{F}}_{p^{m}}$.

Let $(F_{i}(z))_{i=1}^{\infty}$ and $(G_{i}(z))_{i=1}^{\infty}$ be two
sequences of rational functions on $({\mathbb{F}}_{p^{m}})^{n}$. Assume that
each of the rational functions has the form $P(z)/Q(z)$, where
$P(z),Q(z)\in{\mathbb{Z}}[z]$, and for any polydisc
$D_{u_{1},1}\times\dots\times D_{u_{n},1}\,\subset\mathfrak{D}$, we have
$|Q(\tilde{u}_{1},\dots,\tilde{u}_{n})|_{p}=1$, which implies that

$\displaystyle|Q(a_{1},\dots,a_{n})|_{p}=1,\qquad\forall\
(a_{1},\dots,a_{n})\in\mathfrak{D}.$

Assume that the sequences $(F_{i}(z))_{i=1}^{\infty}$ and
$(G_{i}(z))_{i=1}^{\infty}$ uniformly converge on $\mathfrak{D}$ to analytic
functions, which we denote by $F(z)$ and $G(z)$, respectively.

######  Theorem 3.2 ([VZ2]).

Under these assumptions, if $F(z)=G(z)$ on an open nonempty subset of
$\mathfrak{D}$. Then $F(z)=G(z)$ on $\mathfrak{D}$.

###  3.4. Infinite tuples

Let ${\bf e}=(e_{1},e_{2},\dots)$ be an infinite tuple of positive integers.
Let $\Delta=(\Delta_{0},\dots,\Delta_{l})$ be an infinite tuple of nonempty
finite subsets of ${\mathbb{Z}}^{r}$ of the same size $\\#\Delta_{j}=g$ for
some positive integer $g$. Let $\Lambda=(\Lambda_{0}(x),\Lambda_{1}(x),\dots)$
be an infinite tuple of Laurent polynomials in ${\mathbb{Z}}_{p}[x^{\pm
1}]={\mathbb{Z}}_{p}[t^{\pm 1},z^{\pm 1}]$.

Assume that the tuple $\Lambda$ is $(\Delta,{\bf e})$-admissible.

Assume that each of the tuples ${\bf e},\Delta,\Lambda$ have only finitely
many distinct elements. This means that there is a finite set of 4-tuples

(3.1)
$\displaystyle\mathcal{T}=\\{(e^{j},\bar{\Delta}^{j},\tilde{D}^{j},\Lambda^{j})\mid
j=1,\dots,k\\}$

such that for any $l\geqslant 0$ the 4-tuple
$(e_{l+1},\Delta_{l},\Delta_{l+1},\Lambda_{l})$ equals one of the 4-tuples in
$\mathcal{T}$.

######  Definition 3.3.

The $(\Delta,{\bf e})$-admissible tuple $\Lambda$ is called _nondegenerate_ ,
if for any $i=1,\dots,k$, the Laurent polynomial

$\displaystyle\det
A\big{(}e^{j},\bar{\Delta}^{j},\tilde{\Delta}^{j},\Lambda^{j}\big{)}\ \in\
{\mathbb{Z}}_{p}[z^{\pm 1}]$

is nonzero modulo $p$.

Recall the notation:

$\displaystyle W_{s}(x)$ $\displaystyle:=$
$\displaystyle\Lambda_{0}(x)\Lambda_{1}(x)^{p^{e_{1}}}\cdots\Lambda_{s}(x)^{p^{e_{1}+\dots+e_{s}}},$
$\displaystyle W_{s}^{(j)}(x)$ $\displaystyle:=$
$\displaystyle\Lambda_{j}(x)\Lambda_{j+1}(x)^{p^{e_{j+1}}}\cdots\Lambda_{s}(x)^{p^{e_{j+1}+\dots+e_{s}}}.$

If a $(\Delta,{\bf e})$-admissible tuple $\Lambda$ is nondegenerate, then for
any $0\leqslant j\leqslant s$, the Laurent polynomials $\det
A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}\big{)}\in{\mathbb{Z}}_{p}[z^{\pm
1}]$ are not congruent to zero modulo $p$ and we may consider congruences
involving the inverse matrices
$A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}\big{)}^{-1}$.

###  3.5. Domain of convergence

Assume that $\Lambda$ is an infinite nondegenerate $(\Delta,{\bf
e})$-admissible tuple and $m$ is a positive integer. Denote

$\displaystyle\mathfrak{D}^{(m)}=\\{a\in({\mathbb{Z}}_{p}^{(m)})^{n}\ \mid\
|\det
A\big{(}e^{j},\bar{\Delta}^{j},\tilde{\Delta}^{j},\Lambda^{j}(t,a)\big{)}|_{p}=1,\,\,j=1,\dots,k\\}.$

######  Lemma 3.4.

For any $0\leqslant j\leqslant s$ and $a\in\mathfrak{D}^{(m)}$ we have

$\displaystyle\Big{|}\det
A\Big{(}{\sum}_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}(t,a)\Big{)}\Big{|}_{p}=1.$

∎

######  Corollary 3.5.

All entries of
$A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}(t,z)\big{)}^{-1}$
are rational functions in $z$ regular on $\mathfrak{D}^{(m)}$. For every
$a\in\mathfrak{D}^{(m)}$ all entries of
$A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}(t,a)\big{)}$
and
$A\big{(}\sum_{a=j+1}^{s+1}e_{a},\Delta_{j},\Delta_{s+1},W_{s}^{(j)}(t,a)\big{)}^{-1}$
are elements of ${\mathbb{Z}}_{p}^{(m)}$. ∎

######  Theorem 3.6.

Let $\Lambda$ be an infinite nondegenerate $(\Delta,{\bf e})$-admissible
tuple. Consider the sequence of $g\times g$ matrices

(3.2)
$\displaystyle\phantom{aaa}\Big{(}A\Big{(}{\sum}_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}(t,z)\Big{)}\cdot\sigma^{e_{1}}\Big{(}A\Big{(}{\sum}_{a=2}^{s+1}e_{a},\Delta_{1},\Delta_{s+1},W_{s}^{(1)}(t,z)\Big{)}\Big{)}^{-1}\Big{)}_{s\geqslant
0}$

whose entries are rational functions in $z$ regular on the domain
$\mathfrak{D}^{(m)}$. This sequence uniformly converges on
$\mathfrak{D}^{(m)}$ as $s\to\infty$ to an analytic $g\times g$ matrix with
values in ${\mathbb{Z}}_{p}^{(m)}$. Denote this matrix by
$\mathcal{A}_{\Lambda}(z)$. For $a\in\mathfrak{D}^{(m)}$ we have

(3.3) $\displaystyle\Big{|}\det\mathcal{A}_{\Lambda}(a)\Big{|}_{p}=1\,$

and the matrix $\mathcal{A}_{\Lambda}(a)$ is invertible.

###### Proof.

By part (i) of Theorem 2.6 we have
$|\det\sigma^{e_{1}}\big{(}A\big{(}{\sum}_{a=2}^{s+1}e_{a},\Delta_{1},\Delta_{s+1},W_{s}^{(1)}(t,a)\big{)}\big{)}|_{p}=1$
for $a\in\mathfrak{D}^{(m)}$. Hence the matrix in (3.2) is a matrix of
rational functions in $z$ regular on $\mathfrak{D}^{(m)}$. Moreover, if
$a\in\mathfrak{D}^{(m)}$, then every entry of this matrix is an element of
${\mathbb{Z}}_{p}^{(m)}$. The uniform convergence on $\mathfrak{D}^{(m)}$ of
the sequence (3.2) is a corollary of part (ii) of Theorem 2.6. Equation (3.3)
follows from part (i) of Theorem 2.6. The theorem is proved. ∎

######  Theorem 3.7.

Let $\Lambda$ be an infinite nondegenerate $(\Delta,{\bf e})$-admissible
tuple, and $D=D_{v}$, $v=1,\dots,n$. Given $\ell\geqslant 0$ consider the
sequence of $g\times g$ matrices

$\displaystyle\Big{(}\,D\Big{(}\sigma^{\ell}\Big{(}A\Big{(}{\sum}_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\Big{)}\Big{)}\Big{)}\cdot\sigma^{\ell}\Big{(}A\Big{(}{\sum}_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\Big{)}\Big{)}^{-1}\,\Big{)}_{s\geqslant
0}$

whose entries are rational functions in $z$ regular on the domain
$\mathfrak{D}$. This sequence uniformly converges on $\mathfrak{D}$ as
$s\to\infty$ to an analytic $g\times g$ matrix with values in
${\mathbb{Z}}_{p}^{(m)}$. Denote this matrix by
$\mathcal{A}_{\Lambda,D\sigma^{\ell}}(z)$.

###### Proof.

The theorem is a corollary of Theorem 2.8. ∎

######  Theorem 3.8.

Let $\Lambda=(\Lambda_{0}(x),\Lambda_{1}(x),\Lambda_{2}(x),\dots)$ be an
infinite nondegenerate $(\Delta,{\bf e})$-admissible tuple. Given
$\ell\geqslant 0$ and $1\leqslant u,v\leqslant n$, consider the sequence of
$g\times g$ matrices

$\displaystyle\Big{(}\,D_{u}\Big{(}D_{v}\Big{(}A\Big{(}{\sum}_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\Big{)}\Big{)}\Big{)}\cdot
A\Big{(}{\sum}_{a=1}^{s+1}e_{a},\Delta_{0},\Delta_{s+1},W_{s}\Big{)}^{-1}\,\Big{)}_{s\geqslant
0}$

whose entries are rational functions in $z$ regular on the domain
$\mathfrak{D}$. This sequence uniformly converges on $\mathfrak{D}$ as
$s\to\infty$ to an analytic $g\times g$ matrix with values in
${\mathbb{Z}}_{p}^{(m)}$. Denote this matrix by
$\mathcal{A}_{\Lambda,D_{u}D_{v}}(z)$.

###### Proof.

The theorem is a corollary of Theorem 2.9. ∎

Let $\Lambda=(\Lambda_{0}(x),\Lambda_{1}(x),\Lambda_{2}(x),\dots)$ be an
infinite nondegenerate $(\Delta,{\bf e})$-admissible tuple. Consider the
$g\times g$ matrix valued functions
$\mathcal{A}_{\Lambda,\frac{\partial}{\partial z_{u}}\sigma^{0}}(z)$,
$\mathcal{A}_{\Lambda,\frac{\partial}{\partial z_{v}}\sigma^{0}}(z)$ in
Theorem 3.7 and denote them by $\mathcal{A}_{u}(z)$, $\mathcal{A}_{v}(z)$,
respectively. Consider the $g\times g$ matrix valued function
$\mathcal{A}_{\Lambda,\frac{\partial}{\partial z_{u}}\frac{\partial}{\partial
z_{v}}}(z)$ in Theorem 3.8 and denote it by $\mathcal{A}_{u,v}(z)$. All the
three functions are analytic on $\mathfrak{D}^{(m)}$.

######  Lemma 3.9 ([VZ2, Lemma 3.7]).

We have

$\displaystyle\frac{\partial}{\partial
z_{u}}\mathcal{A}_{v}=\mathcal{A}_{u,v}-\mathcal{A}_{v}\mathcal{A}_{u}\,.$

∎

##  4\. KZ equations and complex solutions

###  4.1. KZ equations

Let ${\mathfrak{g}}$ be a simple Lie algebra with an invariant scalar product.
The Casimir element is $\Omega={\sum}_{i}\,h_{i}\otimes
h_{i}\in{\mathfrak{g}}\otimes{\mathfrak{g}}$, where
$(h_{i})\subset{\mathfrak{g}}$ is an orthonormal basis. Let
$V=\otimes_{i=1}^{n}V_{i}$ be a tensor product of ${\mathfrak{g}}$-modules,
$\kappa\in{\mathbb{C}}^{\times}$ a nonzero number. The KZ equations is the
system of differential equations on a $V$-valued function
$I(z_{1},\dots,z_{n})$,

$\displaystyle\frac{\partial I}{\partial z_{i}}\ =\
\frac{1}{\kappa}\,{\sum}_{j\neq i}\,\frac{\Omega_{i,j}}{z_{i}-z_{j}}I,\qquad
i=1,\dots,n,$

where $\Omega_{i,j}:V\to V$ is the Casimir operator acting in the $i$th and
$j$th tensor factors, see [KZ, EFK].

This system is a system of Fuchsian first order linear differential equations.
The equations are defined on the complement in ${\mathbb{C}}^{n}$ to the union
of all diagonal hyperplanes.

The object of our discussion is the following particular case. Let $n,q$ be
positive integers. We consider the following system of differential and
algebraic equations for a column $n$-vector $I=(I_{1},\dots,I_{n})$ depending
on variables $z=(z_{1},\dots,z_{n})$ :

(4.1) $\displaystyle\phantom{aaa}\frac{\partial I}{\partial z_{i}}\ =\
{\frac{1}{q}}\sum_{j\neq i}\frac{\Omega_{ij}}{z_{i}-z_{j}}I,\quad
i=1,\dots,n,\qquad I_{1}+\dots+I_{n}=0,$

where $z=(z_{1},\dots,z_{n})$; the $n\times n$-matrices $\Omega_{ij}$ have the
form

$\displaystyle\Omega_{ij}\ =\
\begin{pmatrix}&\vdots^{\kern-3.41432pti}&&\vdots^{\kern-3.41432ptj}&\\\
{\scriptstyle i}\cdots&{-1}&\cdots&1&\cdots\\\ &\vdots&&\vdots&\\\
{\scriptstyle j}\cdots&1&\cdots&-1&\cdots\\\ &\vdots&&\vdots&\end{pmatrix},$

and all other entries are zero. This joint system of differential and
algebraic equations will be called the system of KZ equations in this paper.

For $i=1,\dots,n$ denote

(4.2) $\displaystyle H_{i}(z)={\frac{1}{q}}\sum_{j\neq
i}\frac{\Omega_{ij}}{z_{i}-z_{j}}\,,\qquad\nabla_{i}^{\operatorname{KZ}}=\frac{\partial}{\partial
z_{i}}-H_{i}(z),\qquad i=1,\dots,n.$

The linear operators $H_{i}(z)$ are called the Gaudin Hamiltonians. The KZ
equations can be written as the system of equations,

$\displaystyle\nabla_{i}^{\operatorname{KZ}}I=0,\quad i=1,\dots,n,\qquad
I_{1}+\dots+I_{n}=0.$

System (4.1) is the system of the differential KZ equations with parameter
$\kappa=q$ associated with the Lie algebra ${\mathfrak{sl}}_{2}$ and the
subspace of singular vectors of weight $n-2$ of the tensor power
$({\mathbb{C}}^{2})^{\otimes{n}}$ of two-dimensional irreducible
${\mathfrak{sl}}_{2}$-modules, up to a gauge transformation, see this example
in [V2, Section 1.1], see also [V3].

###  4.2. Solutions over ${\mathbb{C}}$

Define the master function

$\displaystyle\Phi(t,z)=(t-z_{1})^{-1/q}\dots(t-z_{n})^{-1/q}$

and the column $n$-vector

(4.3) $\displaystyle
I^{(C)}(z)=(I_{1},\dots,I_{n}):=\int_{C}\Big{(}\frac{\Phi(t,z)}{t-z_{1}},\dots,\frac{\Phi(t,z)}{t-z_{n}}\Big{)}dt\,,$

where $C\subset{\mathbb{C}}-\\{z_{1},\dots,z_{n}\\}$ is a contour on which the
integrand takes its initial value when $t$ encircles $C$.

######  Theorem 4.1.

The function $I^{(C)}(z)$ is a solution of system (4.1).

This theorem is a very particular case of the results in [SV1].

###### Proof.

The theorem follows from Stokes’ theorem and the two identities:

(4.4)
$\displaystyle-\frac{1}{q}\,\Big{(}\frac{\Phi(t,z)}{t-z_{1}}+\dots+\frac{\Phi(t,z)}{t-z_{n}}\Big{)}\,=\,\frac{\partial\Phi}{\partial
t}(t,z)\,,$ (4.5) $\displaystyle\Big{(}\frac{\partial}{\partial
z_{i}}-\frac{1}{q}\sum_{j\neq
i}\frac{\Omega_{i,j}}{z_{i}-z_{j}}\Big{)}\Big{(}\frac{\Phi(t,z)}{t-z_{1}},\dots,\frac{\Phi(t,z)}{t-z_{n}}\Big{)}\,=\frac{\partial\Psi^{i}}{\partial
t}(t,z),$

where $\Psi^{i}(t,z)$ is the column $n$-vector
$(0,\dots,0,-\frac{\Phi(t,z)}{t-z_{i}},0,\dots,0)$ with the nonzero element at
the $i$-th place. ∎

######  Theorem 4.2 (cf. [V1, Formula (1.3)]).

All solutions of system (4.1) have this form. Namely, the complex vector space
of solutions of the form (4.3) is $(n-1)$-dimensional.

###  4.3. Solutions as vectors of first derivatives

Consider the integral

$\displaystyle T(z)=T^{(C)}(z)=\int_{C}\Phi(t,z)\,dt.$

Then

$\displaystyle I^{(C)}(z)=\,q\,\Big{(}\frac{\partial T^{(C)}}{\partial
z_{1}},\dots,\frac{\partial T^{(C)}}{\partial z_{n}}\Big{)}.$

Denote $\nabla T=\Big{(}\frac{\partial T}{\partial z_{1}},\dots,\frac{\partial
T}{\partial z_{n}}\Big{)}$. Then the column gradient vector $\nabla T$ of the
function $T(z)$ satisfies the following system of KZ equations

$\displaystyle\nabla_{i}^{\operatorname{KZ}}\nabla T=0,\quad
i=1,\dots,n,\qquad\frac{\partial T}{\partial z_{1}}+\dots+\frac{\partial
T}{\partial z_{n}}=0.$

This is a system of second order linear differential equations on the function
$T(z)$.

##  5\. Solutions modulo powers of $p$

###  5.1. Assumptions

Let $p,q$, $p>q$, be prime numbers. Let $e$ be the order of $p$ modulo $q$,
that is, the least positive integer such that $p^{e}\equiv 1\pmod{q}$. Hence
$(p^{e}-1)/q$ is a positive integer. Let $n=gq+1$ for some positive integer
$g$. Assume that $p^{e}>n$ and $p\geqslant n+q-2$.

In this paper we consider the system of KZ equations (4.1) with $n=gq+1$ and
$\kappa=q$ and study polynomial solutions of the KZ equations modulo powers of
$p$.

###  5.2. Polynomial solutions

For an integer $s\geqslant 1$ define the master polynomial

$\displaystyle\Phi_{s}(t,z)=\big{(}(t-z_{1})\dots(t-z_{n})\big{)}^{(p^{es}-1)/q}.$

For $\ell=1,\dots,g$ define the column $n$-vector

$\displaystyle I_{s,\ell}(z)=(I_{s,\ell,1},\dots,I_{s,\ell.n})$

as the coefficient of $t^{\ell p^{es}-1}$ in the column $n$-vector of
polynomials
$\big{(}\frac{\Phi_{s}(t,z)}{t-z_{1}},\dots,\frac{\Phi_{s}(t,z)}{t-z_{n}}\big{)}$.
Notice that

$\displaystyle\deg_{t}\frac{\Phi_{s}(t,z)}{t-z_{i}}=(gq+1)\frac{p^{es}-1}{q}-1=gp^{es}-1+\frac{p^{es}-1}{q}-g.$

If $\ell>g$, then the polynomial $\frac{\Phi_{s}(t,z)}{t-z_{i}}$ does not have
the monomial $t^{\ell p^{es}-1}$.

######  Theorem 5.1 (cf. [V5, VZ2]).

The column $n$-vector $I_{s,\ell}(z)$ of polynomials in $z$ is a solution of
the system of KZ equations (4.1) modulo $p^{es}$.

We call the column $n$-vectors $I_{s,\ell}(z)$, $\ell=1,\dots,g$, the
$p^{es}$-hypergeometric solutions of the KZ equations (4.1).

###### Proof.

We have the following modifications of identities (4.4), (4.5) :

$\displaystyle\frac{p^{es}-1}{q}\,\Big{(}\frac{\Phi_{s}(t,z)}{t-z_{1}}+\dots+\frac{\Phi_{s}(t,z)}{t-z_{n}}\Big{)}\,=\,\frac{\partial\Phi_{s}}{\partial
t}(t,z)\,,$ $\displaystyle\Big{(}\frac{\partial}{\partial
z_{i}}+\frac{p^{es}-1}{q}\sum_{j\neq
i}\frac{\Omega_{i,j}}{z_{i}-z_{j}}\Big{)}\Big{(}\frac{\Phi_{s}(t,z)}{t-z_{1}},\dots,\frac{\Phi_{s}(t,z)}{t-z_{n}}\Big{)}\,=\frac{\partial\Psi_{s}^{i}}{\partial
t}(t,z),$

where $\Psi_{s}^{i}(t,z)$ is the column $n$-vector
$(0,\dots,0,-\frac{\Phi_{s}(t,z)}{t-z_{i}},0,\dots,0)$ with the nonzero
element at the $i$-th place. Theorem 5.1 follows from these identities. ∎

Consider the $n\times g$ matrix

$\displaystyle
I_{s}(z)=(I_{s,1},\dots,I_{s,g})=\big{(}I_{s,\ell,i}\big{)}_{\ell=1,\dots,g}^{i=1,\dots,n}\
,$

where $I_{s,\ell,i}$ stays at the $\ell$-th column and $i$-th row. The matrix
$I_{s}(z)$ satisfies the KZ equations,

$\displaystyle\nabla_{i}^{\operatorname{KZ}}I_{s}(z)=0,\quad
i=1,\dots,n,\qquad I_{s,\ell,1}+\dots+I_{s,\ell,n}(z)=0,\quad\ell=1,\dots,g,$

modulo $p^{es}$.

###  5.3. Coefficients of solutions

Consider the lexicographical ordering of monomials $z_{1}^{d_{1}}\dots
z_{n}^{d_{n}}$. We have $z_{1}>\dots>z_{n}$ and so on. For a nonzero Laurent
polynomial $f(z)=\sum_{d_{1},\dots,d_{n}}a_{d_{1},\dots,d_{n}}z_{1}^{d_{1}}$
…$z_{n}^{d_{n}}$ with coefficients in ${\mathbb{Z}}$ , the nonzero summand
$a_{d_{1},\dots,d_{n}}z_{1}^{d_{1}}\dots z_{n}^{d_{n}}$ with the largest
monomial $z_{1}^{d_{1}}$ …$z_{n}^{d_{n}}$ is called the leading term of
$f(z)$.

If $f(z)$ and $g(z)$ are two nonzero Laurent polynomials, then the leading
term of $f(z)g(z)$ equals the product of the leading terms of $f(z)$ and
$g(z)$.

######  Lemma 5.2.

For $l=1,\dots,g$, the leading term of the vector-polynomial $I_{1,\ell}$
equals $C_{\ell}\cdot(z_{1}\dots
z_{q(g-\ell)+1})^{(p^{e}-1)/q}/z_{q(g-\ell)+1}^{\ell}$,

(5.1) $\displaystyle
C_{\ell}\,=\,\pm\binom{(p^{e}-1)/q-1}{\ell-1}\Big{(}0,\dots,0,1,\frac{p^{e}-1}{q\ell},\dots,\frac{p^{e}-1}{q\ell}\Big{)},$

where $\frac{p^{e}-1}{q\ell}$ is repeated $q\ell$ times, and

(5.2) $\displaystyle\binom{(p^{e}-1)/q-1}{\ell-1}\not\equiv 0\pmod{p}.$

###### Proof.

Formula (5.1) is obtained by inspection. To prove (5.2) consider the $p$-ary
presentation $(p^{e}-1)/q-1=a_{0}+a_{1}p+\dots$ with $0\leqslant
a_{i}\leqslant p-1$. The inequality (5.2) follows from the inequality
$a_{0}\geqslant g-1$ and Lucas theorem.

We prove that $a_{0}\geqslant g-1$ under our assumption $p\geqslant n+q-2$.
Indeed, $p^{e}=1+q(1+a_{0})+qa_{1}p+\dots$. Hence $1+q(1+a_{0})\geqslant p$.
Let $p=qk+r$ for some integers $k,r$, $1\leqslant r\leqslant q-1$. Then
$1+q(1+a_{0})\geqslant qk+r$ or $q(1+a_{0})\geqslant qk+r-1\geqslant kq$ or
$a_{0}\geqslant k-1$. Hence $a_{0}\geqslant g-1$ if $k\geqslant g$.

The inequality $p\geqslant n+q-2$ can be written as $kq+r\geqslant gq+1+q-2$
or $kq\geqslant gq+q-r-1$. Hence $k\geqslant g$. The lemma is proved. ∎

######  Lemma 5.3.

Consider the $n\times g$ matrix $I_{1}(z)=(I_{1,1},\dots,I_{1,g})$ and its
$g\times g$ minor $M(z)$ in rows with indices $q(g-\ell)+1$ where
$\ell=1,\dots,g$. Then $M(z)$ is a homogeneous polynomial of degree

(5.3) $\displaystyle
d_{M}=\frac{p^{e}-1}{q}\cdot\frac{qg^{2}+2g-qg}{2}-\frac{g(g+1)}{2}\,,$

and the polynomial $M(z)$ is nonzero modulo $p$.

###### Proof.

Every column of $I_{1,\ell}$ is a homogeneous polynomial. Hence $M(z)$ is a
homogeneous polynomial. By Lemma 5.2 the leading term of $M(z)$ equals

(5.4)
$\displaystyle\pm\prod_{\ell=1}^{g}\binom{(p^{e}-1)/q-1}{\ell-1}(z_{1}\dots
z_{q(g-\ell)+1})^{(p^{e}-1)/q}/z_{q(g-\ell)+1}^{\ell}\,.$

This expression is nonzero modulo $p$ by Lemma 5.2. Formula (5.4) implies
(5.3). ∎

##  6\. Congruences for solutions of KZ equations

###  6.1. Congruences for Hasse–Witt matrices of KZ equations

Let $r=1$, $n=gq+1$, ${\bf e}=(e,e,\dots)$, where $e$ is defined in Section
5.1. Let

(6.1)
$\displaystyle\Gamma=\\{1,\dots,g\\}\subset{\mathbb{Z}},\qquad\Delta=(\Gamma,\Gamma,\dots),$
$\displaystyle N=[0,gp^{e}+(p^{e}-1)/q-g]\subset{\mathbb{R}}.$

The infinite tuple $(N,N,\dots)$ of intervals is $(\Delta,{\bf
e})$-admissible, see Definition 2.3.

Recall the polynomial

$\displaystyle\Phi_{1}(t,z)=\big{(}(t-z_{1})\dots(t-z_{n})\big{)}^{(p^{e}-1)/q}.$

The Newton polytope of $\Phi_{1}(t,z)$ with respect to variable $t$ is the
interval $N=[0,gp+(p-1)/q-g]$. We also have

$\displaystyle\Phi_{s}(t,z)=\Phi_{1}(t,z)\cdot\Phi_{1}(t,z)^{p^{e}}\dots\Phi_{1}(t,z)^{p^{e(s-1)}}\,.$

The infinite tuple $(\Phi_{1}(t,z),\Phi_{1}(t,z),\dots)$ is $(\Delta,{\bf
e})$-admissible, see Definition 2.4.

For $s\geqslant 1$ consider the Hasse–Witt $g\times g$ matrix

$\displaystyle
A(\Phi_{s}(t,z)):=A(es,\Gamma,\Gamma,\Phi_{s}(t,z))=\big{(}\operatorname{Coeff}_{p^{es}v-u}(\Phi_{s}(t,z))\big{)}_{u,v=1,\dots,g}\,,$

see (2.6). The entries of this matrix are polynomials in $z$.

######  Theorem 6.1.

The determinant $\det A(\Phi_{1}(t,z))$ is a homogeneous polynomial in $z$ of
degree

(6.2) $\displaystyle d_{\Phi}=\frac{p^{e}-1}{q}\cdot\frac{qg^{2}+2g-qg}{2}\,,$

and the determinant is nonzero modulo $p$.

###### Proof.

Denote $A(\Phi_{1}(t,z))=:(A_{u,v}(z))_{u,v=1,\dots,g}$ .

######  Lemma 6.2.

The leading term of $A_{u,v}(z)$ equals

$\displaystyle\pm\binom{(p^{e}-1)/q}{v-u}(z_{1}z_{2}\dots
z_{qg+1-qv})^{(p^{e}-1)/q}/z_{qg+1-qv}^{v-u}\,,\qquad\text{if}\;\;v\geqslant
u,$ $\displaystyle\pm\binom{(p^{e}-1)/q}{u-v}(z_{1}z_{2}\dots
z_{qg+1-qv})^{(p^{e}-1)/q}z_{qg+2-qv}^{u-v}\,,\qquad\quad\
\text{if}\;\;v\leqslant u.$

For example, for $g=2$ the matrix of leading terms is

(6.3) $\displaystyle\begin{pmatrix}\pm(z_{1}\dots
z_{g+1})^{(p^{e}-1)/q}&\pm\binom{(p^{e}-1)/q}{1}z_{1}^{(p^{e}-1)/q}/z_{1}\\\
\pm\binom{(p^{e}-1)/q}{1}(z_{1}\dots z_{q+1})^{(p^{e}-1)/q}z_{q+2}&\pm
z_{1}^{(p^{e}-1)/q}\end{pmatrix}.$

###### Proof.

The proof is by inspection. ∎

The fact that $\det A(\Phi_{1}(t,z))$ is a homogeneous polynomial easily
follows from the definition of $A(\Phi_{1}(t,z))$. It is also easy to see that
the leading term of the determinant of the matrix of leading terms of
$A_{u,v}(z)$ equals the product of diagonal elements,

(6.4) $\displaystyle\pm\,\prod_{v=1}^{g}(z_{1}\dots
z_{qg+1-qv})^{(p^{e}-1)/q}.$

This expression is not congruent to zero modulo $p$. Counting the degree of
the monomial in (6.4) we obtain (6.2). This proves Theorem 6.1. ∎

######  Corollary 6.3.

The infinite nondegenerate $(\Delta,{\bf e})$-admissible tuple
$(\Phi_{1}(t,z),\Phi_{1}(t,z),\dots)$ satisfies the assumptions of Theorem
2.6. Therefore,

  1. (i)

for $s\geqslant 1$ we have

(6.5) $\displaystyle A(\Phi_{s}(t,z))\equiv
A(\Phi_{1}(t,z))\cdot\sigma^{e}(A(\Phi_{1}(t,z)))\cdots\sigma^{e(s-1)}(A\big{(}\Phi_{1}(t,z)))\pmod{p}\,;$

  2. (ii)

for $s\geqslant 1$ the determinant of the matrix $A(\Phi_{s}(t,z))$ is a
polynomial, which is nonzero modulo $p$, and we have modulo $p^{s}$ :

$\displaystyle A(\Phi_{s+1}(t,z))\cdot\sigma^{e}(A(\Phi_{s}(t,z)))^{-1}\equiv
A(\Phi_{s}(t,z))\cdot\sigma^{e}(A(\Phi_{s-1}(t,z)))^{-1},$

where for $s=1$ we understand the second factor on the right-hand side as the
$g\times g$ identity matrix.

###### Proof.

The corollary follows from Theorems 6.1 and 2.6. ∎

###  6.2. Congruences for frames of solutions of KZ equations

######  Theorem 6.4.

We have the following congruences of $n\times g$ matrices.

  1. (i)

For $s\geqslant 1$,

$\displaystyle I_{s+1}(z)\cdot A(\Phi_{s+1}(t,z))^{-1}\equiv I_{s}(z)\cdot
A(\Phi_{s}(t,z))^{-1}\pmod{p^{s}}\,.$

  2. (ii)

For $s\geqslant 1$ and $j=1,\dots,n$,

$\displaystyle\frac{\partial I_{s+1}}{\partial z_{j}}(z)\cdot
A(\Phi_{s+1}(t,z))^{-1}\equiv\frac{\partial I_{s}}{\partial z_{j}}(z)\cdot
A(\Phi_{s}(t,z))^{-1}\pmod{p^{s}}\,.$

###### Proof.

Consider the first row of the Hasse–Witt matrix $A(\Phi_{s}(t,z))$,

$\displaystyle\big{(}A_{1,1}(\Phi_{s}(t,z)),\dots,A_{1,g}(\Phi_{s}(t,z))\big{)},\quad
A_{1,\ell}(\Phi_{s}(t,z))=\operatorname{Coeff}_{\ell p^{s}-1}(\Phi_{s}(t,z)).$

For each $A_{1,\ell}(\Phi_{s}(t,z))$ we view the gradient

$\displaystyle\nabla A_{1,\ell}(\Phi_{s}(t,z))=\Big{(}\frac{\partial
A_{1,\ell}(s)}{\partial z_{1}},\dots,\frac{\partial A_{1,\ell}(s)}{\partial
z_{n}}\Big{)}$

as a column $n$-vector. The resulting $n\times g$ matrix of gradients

$\displaystyle\nabla A(s,z):=(\nabla A_{1,1}(\Phi_{s}(t,z)),\dots,\nabla
A_{1,g}(\Phi_{s}(t,z)))$

is proportion to the matrix $I_{s}(z)$, $\nabla
A(s,z)=\frac{1-p^{es}}{q}I_{s}(z)$. By Theorems 2.8 and 2.9 we have modulo
$p^{s}$,

$\displaystyle\nabla A(s+1,z)\cdot A(\Phi_{s+1}(t,z))^{-1}\equiv\nabla
A(s,z)\cdot A(\Phi_{s}(t,z))^{-1},$ $\displaystyle\frac{\partial}{\partial
z_{j}}\big{(}\nabla A(s+1,z)\big{)}\cdot
A(\Phi_{s+1}(t,z))^{-1}\equiv\frac{\partial}{\partial z_{j}}\big{(}\nabla
A(s,z)\big{)}\cdot A(\Phi_{s}(t,z))^{-1}.$

These congruences imply the theorem. ∎

######  Corollary 6.5.

For $s\geqslant 1$ we have

$\displaystyle I_{s}(z)\cdot A(\Phi_{s}(t,z))^{-1}\equiv I_{1}(z)\cdot
A(\Phi_{1}(t,z))^{-1}\pmod{p}.$

###  6.3. Domain of convergence

By Theorem 6.1 the polynomial $\det A(\Phi_{1}(t,z))\in{\mathbb{Z}}[z]$ is of
degree $d_{\Phi}$ and this polynomial is nonzero modulo $p$. For a positive
integer $m$ define

$\displaystyle\mathfrak{D}^{(m)}_{\operatorname{KZ}}=\\{a\in({\mathbb{Z}}_{p}^{(m)})^{n}\mid\
|\det A(\Phi_{1}(t,a))|_{p}=1\\}\,.$

By Lemma 3.1 the domain $\mathfrak{D}^{(m)}_{\operatorname{KZ}}$ is nonempty
if $p^{m}>d_{\Phi}$. In what follows we assume that $p^{m}>d_{\Phi}$.

We have $|\det A(\Phi_{s}(t,a))\big{|}_{p}=1$ for
$a\in\mathfrak{D}^{(m)}_{\operatorname{KZ}}$. All entries of
$A(\Phi_{s}(t,z))^{-1}$ are rational functions in $z$ regular on
$\mathfrak{D}^{(m)}_{\operatorname{KZ}}$. For every
$a\in\mathfrak{D}^{(m)}_{\operatorname{KZ}}$ all entries of $A(\Phi_{s}(t,a))$
and $A(\Phi_{s}(t,a))^{-1}$ are elements of ${\mathbb{Z}}_{p}^{(m)}$.

######  Theorem 6.6.

The sequence of $g\times g$ matrices

$\displaystyle\big{(}A\big{(}\Phi_{s}(t,z)\big{)}\cdot\sigma^{e}\big{(}A\big{(}\Phi_{s-1}(t,z)\big{)}\big{)}^{-1}\big{)}_{s\geqslant
1}\,,$

whose entries are rational functions in $z$ regular on
$\mathfrak{D}^{(m)}_{\operatorname{KZ}}$, uniformly converges on
$\mathfrak{D}^{(m)}_{\operatorname{KZ}}$ as $s\to\infty$ to an analytic
$g\times g$ matrix which will be denoted by $\mathcal{A}(z)$. For
$a\in\mathfrak{D}^{(m)}_{\operatorname{KZ}}$ we have

$\displaystyle\big{|}\det\mathcal{A}(a)\big{|}_{p}=1\,$

and the matrix $\mathcal{A}(a)$ is invertible.

###### Proof.

The theorem follows from Theorem 3.6. ∎

######  Theorem 6.7.

For $i=1,\dots,n$ the sequence of $g\times g$ matrices

$\displaystyle\Big{(}\Big{(}\frac{\partial}{\partial
z_{i}}A\big{(}\Phi_{s}(t,z)\big{)}\Big{)}\cdot
A\big{(}\Phi_{s}(t,z)\big{)}^{-1}\Big{)}_{s\geqslant 1}\,,$

whose entries are rational functions in $z$ regular on
$\mathfrak{D}^{(m)}_{\operatorname{KZ}}$, uniformly converges on
$\mathfrak{D}^{(m)}_{\operatorname{KZ}}$ as $s\to\infty$ to an analytic
$g\times g$ matrix, which will be denoted by $\mathcal{A}^{(i)}(z)$.

The sequence of $n\times g$ matrices

$\displaystyle\big{(}I_{s}(z)\cdot
A\big{(}\Phi_{s}(t,z)\big{)}^{-1}\big{)}_{s\geqslant 1}\,,$

whose entries are rational functions in $z$ regular on
$\mathfrak{D}^{(m)}_{\operatorname{KZ}}$, uniformly converges on
$\mathfrak{D}^{(m)}_{\operatorname{KZ}}$ as $s\to\infty$ to an analytic
$n\times g$ matrix which will be denoted by $\mathcal{I}(z)$.

For $i=1,\dots,n$ the sequence of $n\times g$ matrices

$\displaystyle\Big{(}\frac{\partial I_{s}}{\partial z_{i}}(z)\cdot
A\big{(}s,\Phi_{s}(t,z)\big{)}^{-1}\Big{)}_{s\geqslant 1}\,,$

whose entries are rational functions in $z$ regular on
$\mathfrak{D}^{(m)}_{\operatorname{KZ}}$, uniformly converges on
$\mathfrak{D}^{(m)}_{\operatorname{KZ}}$ as $s\to\infty$ to an analytic
$n\times g$ matrix which will be denoted by $\mathcal{I}^{(i)}(z)$.

We have

$\displaystyle\frac{\partial\mathcal{I}}{\partial
z_{i}}=\mathcal{I}^{(i)}-\mathcal{I}\cdot\mathcal{A}^{(i)}\,.$

###### Proof.

The theorem follows from Theorems 3.7, 3.8, and Lemma 3.9. ∎

######  Theorem 6.8.

We have the following system of equations on
$\mathfrak{D}^{(m)}_{\operatorname{KZ}}$ :

$\displaystyle\mathcal{I}^{(i)}=H_{i}\cdot\mathcal{I},\qquad i=1,\dots,n,$

where $H_{i}$ are the Gaudin Hamiltonians defined in (4.2).

###### Proof.

The theorem is a corollary of Theorem 5.1. ∎

######  Corollary 6.9.

For $a\in\mathfrak{D}^{(m)}_{\operatorname{KZ}}$ we have

$\displaystyle\mathcal{I}(a)\equiv I_{1}(a)\cdot
A\big{(}\Phi_{1}(t,a)\big{)}^{-1}\pmod{p}.$

###### Proof.

The corollary follows from Corollary 6.5 and Theorem 6.7. ∎

###  6.4. Vector bundle
$\mathcal{L}\,\to\,\mathfrak{D}^{(m),o}_{\operatorname{KZ}}$

Denote

$\displaystyle W=\\{(I_{1},\dots,I_{n})\in({\mathbb{Q}}_{p}^{(m)})^{n}\ |\
I_{1}+\dots+I_{n}=0\\}.$

We consider vectors $(I_{1},\dots,I_{n})$ as column vectors. The differential
operators $\nabla^{\operatorname{KZ}}_{i}$, $i=1,\dots,n$, define a connection
on the trivial bundle
$W\times\mathfrak{D}^{(m)}_{\operatorname{KZ}}\to\mathfrak{D}^{(m)}_{\operatorname{KZ}}$,
called the KZ connection. The connection has singularities at the diagonal
hyperplanes in $({\mathbb{Z}}_{p}^{(m)})^{n}$ and is well-defined over

$\displaystyle\mathfrak{D}^{(m),o}_{\operatorname{KZ}}=\\{a=(a_{1},\dots,a_{n})\in({\mathbb{Z}}_{p}^{(m)})^{n}\mid|\det
A(\Phi_{1}(t,a))|_{p}=1,\,a_{i}\neq a_{j}\,\ \forall i,j\\}.$

The KZ connection is flat,

$\displaystyle\big{[}\nabla^{\operatorname{KZ}}_{i},\nabla^{\operatorname{KZ}}_{j}\big{]}=0\qquad\forall\,i,j,$

see [EFK]. The flat sections of the KZ connection are solutions of system
(4.1) of KZ equations.

For any $a\in\mathfrak{D}^{(m)}_{\operatorname{KZ}}$ let
$\mathcal{L}_{a}\subset W$ be the vector subspace generated by columns of the
$n\times g$ matrix $\mathcal{I}(a)$. Then

$\displaystyle\mathcal{L}:=\bigcup\nolimits_{a\in\mathfrak{D}^{(m)}_{\operatorname{KZ}}}\,\mathcal{L}_{a}\,\to\,\mathfrak{D}^{(m)}_{\operatorname{KZ}}$

is an analytic distribution of vector subspaces in the fibers of the trivial
bundle
$W\times\mathfrak{D}^{(m)}_{\operatorname{KZ}}\to\mathfrak{D}^{(m)}_{\operatorname{KZ}}$.

######  Theorem 6.10 ([VZ2, Theorem 6.7]).

The distribution $\mathcal{L}\,\to\,\mathfrak{D}^{(m)}_{\operatorname{KZ}}$ is
invariant with respect to the KZ connection. In other words, if $s(z)$ is a
local section of $\mathcal{L}\,\to\,\mathfrak{D}^{(m)}_{\operatorname{KZ}}$,
then the sections $\nabla_{i}^{\operatorname{KZ}}s(z)$, $i=1,\dots,n$, also
are sections of $\mathcal{L}\,\to\,\mathfrak{D}^{(m)}_{\operatorname{KZ}}$.

###### Proof.

Let $\mathcal{I}(z)=(\mathcal{I}_{1}(z),\dots,\mathcal{I}_{g}(z))$ be columns
of the $n\times g$ matrix $\mathcal{I}(z)$. Let
$a\in\mathfrak{D}^{(m)}_{\operatorname{KZ}}$. Let
$c(z)=(c_{1}(z),\dots,c_{g}(z))$ be a column vector of analytic functions at
$a$. Consider a local section of the distribution
$\mathcal{L}\,\to\,\mathfrak{D}^{(m)}_{\operatorname{KZ}}$,
$s(z)\,=\,\sum_{j=1}^{g}\,c_{j}(z)\mathcal{I}_{j}(z)\,=:\,\mathcal{I}\cdot c$.
Then

$\displaystyle\nabla^{\operatorname{KZ}}_{i}s(z)$ $\displaystyle=$
$\displaystyle-H_{i}\cdot\mathcal{I}\cdot
c+\frac{\partial\mathcal{I}}{\partial z_{i}}\cdot
c+\mathcal{I}\cdot\frac{\partial c}{\partial z_{i}}$ $\displaystyle=$
$\displaystyle-H_{i}\cdot\mathcal{I}\cdot
c+(\mathcal{I}^{(i)}-\mathcal{I}\cdot\mathcal{A}^{(i)})\cdot
c+\mathcal{I}\cdot\frac{\partial c}{\partial z_{i}}$ $\displaystyle=$
$\displaystyle-H_{i}\cdot\mathcal{I}\cdot
c+(H_{i}\cdot\mathcal{I}-\mathcal{I}\cdot\mathcal{A}^{(i)})\cdot
c+\mathcal{I}\cdot\frac{\partial c}{\partial z_{i}}$ $\displaystyle=$
$\displaystyle-\mathcal{I}\cdot\mathcal{A}^{(i)}\cdot
c+\mathcal{I}\cdot\frac{\partial c}{\partial z_{i}}\,.$

Clearly, the last expression is a local section of
$\mathcal{L}\,\to\,\mathfrak{D}^{(m)}_{\operatorname{KZ}}$. ∎

######  Theorem 6.11.

The function $a\mapsto\dim_{{\mathbb{Q}}_{p}^{(m)}}\mathcal{L}_{a}$ is
constant on $\mathfrak{D}^{(m),o}_{\operatorname{KZ}}$, in other words,
$\mathcal{L}\,\to\,\mathfrak{D}^{(m)}_{\operatorname{KZ}}$ is a vector bundle
over
$\mathfrak{D}^{(m),o}_{\operatorname{KZ}}\subset\mathfrak{D}^{(m)}_{\operatorname{KZ}}$.

The proof coincides with the proof of Theorem 6.8 in [VZ2].

Recall that $d_{\Phi}$ is the degree of the polynomial $\det A(\Phi_{1}(t,z))$
and $d_{M}$ is the degree of the minor defined in Lemma 5.3.

######  Theorem 6.12.

If $p^{m}>d_{\Phi}+d_{M}$, then the analytic vector bundle
$\mathcal{L}\to\mathfrak{D}^{(m),o}_{\operatorname{KZ}}$ is of rank $g$.

###### Proof.

If $p^{m}>d_{\Phi}+d_{M}$, then the minor $M(z)$ defines a function on
$\mathfrak{D}^{(m),o}_{\operatorname{KZ}}$ nonzero modulo $p$ by Lemma 3.1.
Then by Corollary 6.9, the $n\times g$ matrix valued function $\mathcal{I}(z)$
has a $g\times g$ minor nonzero on $\mathfrak{D}^{(m)}_{\operatorname{KZ}}$.
This proves the theorem. ∎

###  6.5. Remarks

#### 6.5.1.

One may expect that the subbundle
$\mathcal{L}\to\mathfrak{D}^{(m),o}_{\operatorname{KZ}}$ can be extended to a
rank $g$ subbundle over
$\mathfrak{D}^{(m)}_{\operatorname{KZ}}-\mathfrak{D}^{(m),o}_{\operatorname{KZ}}$,
the union of the diagonal hyperplanes in
$\mathfrak{D}^{(m)}_{\operatorname{KZ}}$.

#### 6.5.2.

Following Dwork we may expect that locally at any point
$a\in\mathfrak{\mathfrak{missing}}D^{(m),o}_{\operatorname{KZ}}$, the
solutions of the KZ equations with values in
$\mathcal{L}\to\mathfrak{D}^{(m),o}_{\operatorname{KZ}}$ are given at $a$ by
power series in $z_{i}-a_{i}$, $i=1,\dots,n$, bounded in their polydiscs of
convergence, while any other local solution at $a$ is given by a power series
unbounded in its polydisc of convergence, cf. [Dw] and [V5, Theorem A.4].

#### 6.5.3.

The KZ connection $\nabla^{\operatorname{KZ}}_{i}$, $i=1,\dots,n$, over
${\mathbb{C}}$ has no nontrivial proper invariant subbundles due to the
irreducibility of its monodromy representation, see [Fo, Lemma 6]. Thus the
existence of the invariant subbundle
$\mathcal{L}\to\mathfrak{D}^{(m),o}_{\operatorname{KZ}}$ is a $p$-adic
feature.

#### 6.5.4.

The invariant subbundles of the KZ connection over ${\mathbb{C}}$ usually are
related to some additional conformal block constructions, for example see
[FSV, SV2, V3, V4]. Apparently our subbundle
$\mathcal{L}\to\mathfrak{D}^{(m),o}_{\operatorname{KZ}}$ is of a different
$p$-adic nature.

## References

  * [1]
  * [Dr] V. G. Drinfeld, Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations, Problems of modern quantum field theory (Alushta, 1989), 1–13, Res. Rep. Phys., Springer 1989.
  * [2]
  * [Dw] B. Dwork, $p$-adic cycles, Publ. Math. de lHÉS 37 (1969), 27–115
  * [3]
  * [EFK] P. Etingof, I. Frenkel, A. Kirillov, Lectures on representation theory and Knizhnik–Zamolodchikov equations, Mathematical Surveys and Monographs, vol. 58 (AMS, Providence, RI, 1998), xiv+198 pp. ISBN: 0-8218-0496-0
  * [4]
  * [FSV] B. Feigin, V. Schechtman, A. Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models, I, Comm. Math. Phys. 163 (1994), 173–184; II, Comm. In Math. Phys. 70 (1995), 219–247
  * [5]
  * [Fo] E. Formanek, Braid group representations of low degree, Proc. London Math. Soc. (3) 73 (1996), no. 2, 279–322
  * [6]
  * [KZ] V. Knizhnik, A. Zamolodchikov, Current algebra and the Wess–Zumino model in two dimensions, Nucl. Phys. B247 (1984), 83–103
  * [7]
  * [MO] D. Maulik, A. Okounkov, Quantum groups and quantum cohomology, Astérisque, t. 408 (Société Mathématique de France, 2019), 1–277;
  * [8] https://doi.org/10.24033/ast.1074
  * [9]
  * [Me] A. Mellit, A proof of Dwork’s congruences, unpublished (October 20, 2009), 1–3
  * [10]
  * [MV] A. Mellit, M. Vlasenko, Dwork’s congruences for the constant terms of powers of a Laurent polynomial, Int. J. Number Theory 12 (2016), no. 2, 313–321
  * [11]
  * [RV1] R. Rimányi, A. Varchenko, The ${\mathbb{F}}_{p}$-Selberg integral, arXiv:2011.14248, 1–19
  * [12]
  * [RV2] R. Rimányi, A. Varchenko, The $\mathbb{F}_{p}$-Selberg integral of type $A_{n}$, arXiv:2012.01391, 1–21
  * [13]
  * [SV1] V. Schechtman, A. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139–194
  * [14]
  * [SV2] V. Schechtman, A. Varchenko, Solutions of KZ differential equations modulo $p$, Ramanujan J. 48 (2019), no. 3, 655–683;
  * [15] https://doi.org/10.1007/s11139-018-0068-x, arXiv:1707.02615
  * [16]
  * [V1] A. Varchenko, Beta-function of Euler, Vandermonde determinant, Legendre equation and critical values of linear functions of configuration of hyperplanes, I. Izv. Akademii Nauk USSR, Seriya Mat. 53 (1989), no. 6, 1206–1235; II, Izv. Akademii Nauk USSR, Seriya Mat. 54 (1990), no. 1, 146–158
  * [17]
  * [V2] A. Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Advanced Series in Mathematical Physics, Vol. 21, World Scientific, 1995
  * [18]
  * [V3] A. Varchenko, Special functions, KZ type equations, and representation theory, CBMS Regional Conference Series in Math., vol. 98 (AMS, Providence, RI, 2003), viii+118 pp. ISBN: 0-8218-2867-3
  * [19]
  * [V4] A. Varchenko, An invariant subbundle of the KZ connection mod $p$ and reducibility of $\widehat{{\mathfrak{sl}_{2}}}$ Verma modules mod $p$, Math. Notes 109 (2021), no. 3, 386–397
  * [20]
  * [V5] 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
  * [21]
  * [VZ1] A. Varchenko and W. Zudilin, Ghosts and congruences for $p^{s}$-appoximations of hypergeometric periods, arXiv:2107.08548, 1–29
  * [22]
  * [VZ2] A. Varchenko and W. Zudilin, Congruences for Hasse–Witt matrices and solutions of $p$-adic KZ equations, arXiv:2108.12679, 1–26
  * [23]
  * [Vl] M. Vlasenko, Higher Hasse–Witt matrices, Indag. Math. 29 (2018), 1411–1424
  * [24]