# Ghosts and congruences for $p^{s}$-approximations
of hypergeometric periods

Alexander Varchenko⋆ and Wadim Zudilin⋄

###### Abstract.

We prove general Dwork-type congruences for constant terms attached to tuples
of Laurent polynomials. We apply this result to establishing arithmetic and
$p$-adic analytic properties of functions originating from polynomial
solutions modulo $p^{s}$ of hypergeometric and KZ equations, solutions which
come as coefficients of master polynomials and whose coefficients are
integers.

As an application we show that the simplest example of a $p$-adic KZ
connection has an invariant line subbundle while its complex analog has no
nontrivial subbundles due to the irreducibility of its monodromy
representation.

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

⋆ Faculty of Mathematics and Mechanics, Lomonosov Moscow State University
Leninskiye Gory 1, 119991 Moscow GSP-1, Russia

⋆ Moscow Center of Fundamental and Applied Mathematics
Leninskiye Gory 1, 119991 Moscow GSP-1, Russia

⋄ Institute for Mathematics, Astrophysics and Particle Physics, Radboud
University,
PO Box 9010, 6500 GL, Nijmegen, The Netherlands

Key words: Hypergeometric equation; KZ equations; Dwork congruences; master
polynomials; $p^{s}$-approximation polynomials.

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
${}^{\diamond}\mskip-0.99998mu$E-mail<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 On ghosts
    1. 2.1 Mellit’s theorem
    2. 2.2 Convex polytopes
    3. 2.3 Definition of ghosts
    4. 2.4 Composed ghosts
    5. 2.5 Indecomposable tuples
    6. 2.6 Polynomials $I_{\lambda}$
    7. 2.7 Admissible tuples of Laurent polynomials
    8. 2.8 Dwork congruence for tuples of Laurent polynomials
  3. 3 $p^{s}$-Approximation of a hypergeometric integral
  4. 4 Function ${}_{2}F_{1}\Big{(}\frac{1}{2},\frac{1}{2};1,x\Big{)}$
    1. 4.1 Polynomials $P_{s}(x)$
    2. 4.2 Baby congruences
    3. 4.3 Congruences for $P_{s}(x)$
    4. 4.4 Coefficients of master polynomials
    5. 4.5 Limits of $P_{s}(x)$
  5. 5 Function ${}_{2}F_{1}\Big{(}\frac{2}{3},\frac{1}{3};1;x\Big{)}$
    1. 5.1 Two hypergeometric integrals
    2. 5.2 The case $p=3\ell+1$
    3. 5.3 The case $p=3\ell+2>2$
    4. 5.4 Limits of $\bar{Q}_{s}(x)$
    5. 5.5 Remark
    6. 5.6 Congruences related to $-\frac{1}{5}$, $-\frac{2}{5}$, $-\frac{3}{5}$, $-\frac{4}{5}$
  6. 6 KZ equations
    1. 6.1 KZ equations
    2. 6.2 Solutions over ${\mathbb{C}}$
    3. 6.3 Solutions as vectors of first derivatives
    4. 6.4 Solutions modulo $p^{s}$
    5. 6.5 $p^{s}$-Approximation polynomials of $T(z)$
    6. 6.6 Limits of $T_{s}(z)$
    7. 6.7 Subbundle $\mathcal{L}\,\to\,\mathfrak{D}$
    8. 6.8 Other definitions of subbundle $\mathcal{L}\to\mathfrak{D}$
      1. 6.8.1 Line subbundle $\mathcal{M}\to\mathfrak{D}$
      2. 6.8.2 Line subbundle $\mathcal{N}\to\hat{\mathfrak{D}}$
  7. 7 Concluding remarks
    1. 7.1 Conjectural stronger congruences for $\bar{P}_{s}(x)$
    2. 7.2 Dwork crystals

##  1\. Introduction

In the seminal work [Dw] Dwork laid the foundation of the theory of $p$-adic
hypergeometric differential equations. His principal working example was the
differential equation

(1.1) $\displaystyle x(1-x)I^{\prime\prime}+(1-2x)I^{\prime}-\frac{1}{4}I=0,$

whose analytic at the origin solution

(1.2)
${}_{2}F_{1}\Big{(}\frac{1}{2},\frac{1}{2};1;x\Big{)}=\frac{1}{\pi}\,\int_{1}^{\infty}t^{-1/2}(t-1)^{-1/2}(t-x)^{-1/2}dt=\sum_{k=0}^{\infty}\binom{-1/2}{k}^{2}x^{k}$

encodes periods of the Legendre family $y^{2}=t(t-1)(t-x)$. Dwork used the
approximations

$\displaystyle
F_{s}(x)=\sum_{k=0}^{p^{s}-1}\binom{-1/2}{k}^{2}x^{k}\qquad\text{for}\quad
s=1,2,\dots,$

which are nothing but truncations of the infinite sum in (1.2) and clearly
converge to it in the disk $D_{0,1}=\\{x\mid|x|_{p}<1\\}$, to show that the
uniform limit $F_{s+1}(x)/F_{s}(x^{p})$ as $s\to\infty$ exists in a larger
domain $\mathfrak{D}^{\text{Dw}}$ and this limit, dubbed as the “unit root”,
corresponds to a root of the local zeta function of the $x$-fiber in the
family. Dwork’s work boosted the whole body of research in the area; we limit
ourselves to mentioning some recent contributions on the theme [AS, BV, LTYZ].

Interestingly enough, Dwork himself indicates in [Dw] that he adopts a similar
point of view in the $p$-adic case to the one Igusa had in [Ig] for modulo $p$
solution of (1.1). Namely, the cycles of the elliptic curve
$y^{2}=t(t-1)(t-x)$ for a given $x$ can be thought of as the local at $x$
analytic solutions of the differential equation (1.1). At the same time,
Igusa’s modulo $p$ solution

$\displaystyle g(x)=\sum_{k=0}^{(p-1)/2}\binom{(p-1)/2}{k}^{2}x^{k}$

of (1.1), though indeed coinciding with Dwork’s $F_{1}(x)$ modulo $p$, hints
at a somewhat different way for approximating (1.2) $p$-adically through
_natural_ truncations of the sum. This recipe seems to escape its own
development until very recently Schechtman and Varchenko gave solutions to
general Knizhnik–Zamolodchikov (KZ) equations modulo $p$ in [SV2], recovering
Igusa’s polynomial as a particular case. It has been realized in [V4] that
this approach goes in parallel with that of Dwork in [Dw]; explicitly, the
polynomial solutions of hypergeometric equation and of a certain class of KZ
equations modulo $p^{s}$ are shown in [V4] to give rise to $p$-adic
approximations of the corresponding unit roots. The principal goal of this
paper is to prove such expectations, at least for the cases in which the
technicality of proofs do not overshadow the beauty of outcomes.

In this paper we study certain $p^{s}$-approximation polynomials of
hypergeometric periods. We consider an integral of hypergeometric type like in
(1.1) without specifying the cycle of integration. For any positive integer
$s$ we replace the integrand by a polynomial $\Phi_{s}(t,x)$ with integer
coefficients called the master polynomial and define the $p^{s}$-approximation
polynomial as the coefficient of $t^{p^{s}-1}$ in the master polynomial. This
is our $p^{s}$-analog of the initial integral. In the example of (1.1) the
master polynomial is
$\Phi_{s}(t,x)=t^{(p^{s}-1)/2}(t-1)^{(p^{s}-1)/2}(t-x)^{(p^{s}-1)/2}$ and the
$p^{s}$-approximation polynomial is

(1.3) $\displaystyle
P_{s}(x)=(-1)^{(p^{s}-1)/2}\sum_{k=0}^{(p^{s}-1)/2}\binom{(p^{s}-1)/2}{k}^{2}x^{k}.$

We prove the Dwork-type congruence,

(1.4) $\displaystyle P_{s+1}(x)P_{s-1}(x^{p})\equiv
P_{s}(x)P_{s}(x^{p})\pmod{p^{s}},$

in Theorem 4.2. More general $p^{s}$-approximation constructions are discussed
in [SV2, V4, RV1, RV2].

In Section 6 we consider the simplest example of the KZ connection. The KZ
connections are objects of conformal field theory, representation theory,
enumerative geometry, see [KZ, EFK, MO]. In our example, the KZ connection is
identified with the Gauss–Manin connection of the family of elliptic curves
$y^{2}=(t-z_{1})(t-z_{2})(t-z_{3})$. We study the $p^{s}$-approximation
polynomials to the elliptic period $\int\\!dt/y$ and show that the $p$-adic KZ
connection of our example has an invariant line subbundle. This is a $p$-adic
feature since the corresponding complex KZ connection has no proper invariant
subbundle due to the irreducibility of its monodromy representation.

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]. Apparently our subbundle is of a different $p$-adic nature.

The results above require proving $p$-adic convergence, which in turn rests
upon establishing certain special congruences. Considering more general
hypergeometric series $F(x)=\sum_{n=0}^{\infty}A(n)x^{n}$ and its
$p^{s}$-truncations $F_{s}(x)=\sum_{n=0}^{p^{s}-1}A(n)x^{n}$, Dwork showed
that

(1.5) $\displaystyle F_{s+1}(x)F_{s-1}(x^{p})\equiv
F_{s}(z)F_{s}(x^{p})\pmod{p^{s}}\quad\text{for}\;s=1,2,\dots,$

in [Dw, Theorem 2]; this allowed him to conclude the existence of the $p$-adic
limit $F_{s+1}(x)/F_{s}(x^{p})$ as $s\to\infty$ in [Dw, Theorem 3]. As an
auxiliary component of Dwork’s argument the other set of congruences, a là
Lucas, was used for the sequence of coefficients $A(n)$:

(1.6)
$\displaystyle\frac{A(n+mp^{s})}{A([n/p]+mp^{s-1})}\equiv\frac{A(n)}{A([n/p])}\pmod{p^{s}}\quad\text{for}\;m,n\in{\mathbb{Z}}_{\geqslant
0}\;\text{and}\;s=1,2,\dots$

(see Corollary 1(ii) on p. 36 in [Dw]). These two different-looking families
of congruences (1.5) and (1.6) are both known as Dwork congruences, and to
distinguish between the two we dub them type I and type II, respectively. Our
Igusa-inspired $p^{s}$-aproximations of solutions of hypergeometric equation,
like (1.3) above, are of dual nature. Although congruences (1.4) look like
congruences (1.5) of type I we may view the sequence $(P_{s}(x))_{s\geqslant
1}$ as a subsequence of a suitable polynomial sequence $A(n;x)$ depending on
the extra parameter $x$ and satisfying

(1.7)
$\displaystyle\frac{A(n+mp^{s};x)}{A([n/p]+mp^{s-1};x^{p})}\equiv\frac{A(n;x)}{A([n/p];x^{p})}\pmod{p^{s}}\qquad\text{for}\quad
m,n\in{\mathbb{Z}}_{\geqslant 0}\quad\text{and}\quad s=1,2,\dots\,.$

Then the restriction of these type II Dwork congruences to the subsequence
$(P_{s}(x))_{s\geqslant 1}$ reads as type I congruences for the
$p^{s}$-aproximation polynomials in parameter $x$. (We make this explicit in
the remark after Theorem 4.4.) To summarize, our principal tool for
establishing the existence of $p$-adic convergence are Dwork-type congruences
(1.7), of which required type I congruences are particular instances. General
theorems towards Dwork’s congruences of type II were given by Mellit in [Me]
and, independently, by Samol and van Straten in [SvS]. As these results are
insufficiently general and only for (1.6) rather than (1.7), we extend them
further using the method from Mellit’s unpublished preprint [Me] (see also
[MV] for another application). Our result based on Mellit’s elegant approach
is displayed in Section 2, and its power is illustrated by the congruences
(1.4) from Theorem 4.2 and by several other quite different applications in
later sections.

In Section 7 we conjecture some stronger congruences for the polynomials
$P_{s}(x)$.

### Acknowledgements

The authors thank Frits Beukers, Andrew Granville, Anton Mellit, Richárd
Rimányi, Steven Sperber, and Masha Vlasenko for useful discussions.

##  2\. On ghosts

In this paper $p$ is an odd prime.

###  2.1. Mellit’s theorem

Let $\Lambda(t)$ be a Laurent polynomial in variables $t=(t_{1},\dots,t_{r})$
with coefficients in ${\mathbb{Z}}_{p}$ and constant term
$\operatorname{CT}_{t}(\Lambda)$. Assume that the Newton polytope of
$\Lambda(t)$ contains only one interior point $\\{0\\}$.

For a tuple $a=(a_{0},a_{1},\dots,a_{l-1})$, denote by $l(a)=l$ its length.
For two tuples $a$ and $b$, the concatenation product $a*b$ is the tuple of
length $l(a)+l(b)$ obtained by gluing $a$ and $b$ together. For $a=(a_{0}$, …,
$a_{l-1})$ of length $l$, denote by $a^{\prime}=(a_{1},\dots,a_{l-1})$ the
“derivative” tuple of length $l-1$. If $a$ is a tuple of numbers, denote
$|a|=\sum_{i=0}^{l-1}a_{i}$.

For a tuple $m=(m_{0},\dots,m_{l})$ of integers from $\\{1,\dots,p-1\\}$
denote by $\operatorname{CT}_{t}(\Lambda^{m})$ the constant term of the
Laurent polynomial
$\Lambda(t)^{m_{0}+m_{1}p+m_{2}p^{2}+\dots+m_{l-1}p^{l-1}}$.

######  Theorem 2.1 (Anton Mellit, 2009, [Me], unpublished).

Let $a,b,c$ be tuples of integers from $\\{1,\dots,p-1\\}$, where
$b,c,a^{\prime}$ can be empty, that is, of length 0. Then

(2.1)
$\displaystyle\operatorname{CT}_{t}(\Lambda^{a*b})\operatorname{CT}_{t}(\Lambda^{a^{\prime}*c})\equiv\operatorname{CT}_{t}(\Lambda^{a^{\prime}*b})\operatorname{CT}_{t}(\Lambda^{a*c})\pmod{p^{l(a)}}.$

We modify the statement and three-page Mellit’s proof of Theorem 2.1 and prove
Theorem 2.9 below.

###  2.2. Convex polytopes

Given a positive integer $r$ we consider convex polytopes, which are convex
hulls of finite subsets of ${\mathbb{Z}}^{r}\subset{\mathbb{R}}^{r}$.

######  Definition 2.2.

A tuple $(N_{0},N_{1},\dots,N_{l-1})$ of convex polytopes is called
$\operatorname{admissible}$ if for any $0\leqslant i\leqslant j\leqslant l-1$
we have

$\displaystyle\big{(}N_{i}+pN_{i+1}+\dots+p^{j-i}N_{j}\big{)}\cap
p^{j-i+1}{\mathbb{Z}}^{r}=\\{0\\}.$

###  2.3. Definition of ghosts

Let $\Lambda(t,z)$ be a Laurent polynomial in some variables
$t=(t_{1},\dots,t_{r})$, $z=(z_{1},\dots,z_{r^{\prime}})$ with coefficients in
${\mathbb{Z}}_{p}$. We define the ghost terms $R_{m}(\Lambda)$, $m\geqslant
0,$ as the unique sequence of Laurent polynomials in $t,z$ satisfying the
following two properties:

  1. (i)

For each $m$ we have

$\displaystyle\Lambda(t,z)^{p^{m}}=R_{0}(\Lambda)(t^{p^{m}},z^{p^{m}})+R_{1}(\Lambda)(t^{p^{m-1}},z^{p^{m-1}})+\dots+R_{m}(\Lambda)(t,z).$

  2. (ii)

For each $m$ the coefficients of $R_{m}(\Lambda)(t,z)$ are divisible by
$p^{m}$ in ${\mathbb{Z}}_{p}$.

Properties (i)–(ii) recursively determine the polynomials
$R_{m}(\Lambda)(t,z)$. Namely,

(2.2) $\displaystyle
R_{m}(\Lambda)(t,z)=\Lambda(t,z)^{p^{m}}-\Lambda(t^{p},z^{p})^{p^{m-1}}\,,\quad
R_{0}(\Lambda)(t,z)=\Lambda(t,z).$

Let $F(t,z)$ be a Laurent polynomial in $t,z$. Let $N(F)$ be the Newton
polytope of $F(t,z)$ with respect to the $t$ variables only. Clearly, we have

(2.3) $\displaystyle N(R_{m}(\Lambda))\subset p^{m}N(\Lambda).$

###  2.4. Composed ghosts

Let $\lambda=(\Lambda_{0}(t,x),\dots,\Lambda_{l-1}(t,z))$ be a tuple of
Laurent polynomials with coefficients in ${\mathbb{Z}}_{p}$. We decompose the
product

$\displaystyle\tilde{\lambda}(t,z):=\Lambda_{0}(t,x)(\Lambda_{1}(t,z))^{p}\cdots(\Lambda_{l-1}(t,z))^{p^{l-1}}$

into the sum of ghost terms of $\Lambda_{0},\dots,\Lambda_{l-1}$. As the
result we obtain that $\tilde{\lambda}$ is the sum of the products

$\displaystyle R_{m,\lambda}(t,z)$ $\displaystyle:=$ $\displaystyle
R_{m_{0}}(\Lambda_{0})(t,z)\cdot
R_{m_{1}}(\Lambda_{1})(t^{p^{1-m_{1}}},z^{p^{1-m_{1}}})\cdot
R_{m_{2}}(\Lambda_{2})(t^{p^{2-m_{2}}},z^{p^{2-m_{2}}})\cdots$
$\displaystyle\cdots
R_{m_{l-1}}(\Lambda_{l-1})(t^{p^{l-1-m_{l-1}}},z^{p^{l-1-m_{l-1}}}),$

where $m=(m_{0},\dots,m_{l-1})$ runs over the set of all $l$-tuples of
integers satisfying $0\leqslant m_{i}\leqslant i$. Clearly, we have

$\displaystyle R_{m,\lambda}(t,z)\equiv 0\pmod{p^{|m|}}$

and

$\displaystyle N(R_{m,\lambda}(t,z))\subset
N(\Lambda_{0}(t,z))+pN(\Lambda_{1}(t,z))+\dots+p^{l-1}N(\Lambda_{l-1}(t,z)).$

###  2.5. Indecomposable tuples

Denote by $S_{k}$ the set of all $k$-tuples $m=(m_{0},\dots,m_{k-1})$ of
integers such that $0\leqslant m_{i}\leqslant i$. Put
$S=\bigcup_{k=1}^{\infty}S_{k}$. A tuple $m\in S$ is called indecomposable if
it cannot be presented as $m^{\prime}*m^{\prime\prime}$ for
$m^{\prime},m^{\prime\prime}\in S$. Denote by $S_{k}^{\operatorname{ind}}$ the
set of all indecomposable $k$-tuples and put
$S^{\operatorname{ind}}=\bigcup_{k=1}^{\infty}S_{k}^{\operatorname{ind}}$.

######  Lemma 2.3.

If $m\in S_{k}^{\operatorname{ind}}$, then $|m|\geqslant k-1$.

###### Proof.

If $m$ is indecomposable, then for each $i\in\\{1,\dots,k-1\\}$ there exists
$j\geqslant i$ such that $m_{j}>j-i$, i.e. $j\geqslant i>j-m_{j}$. The number
of such $i$ for a given $j$ is $m_{j}$. The total number of $i$ is $k-1$,
therefore the sum of $m_{j}$ is at least $k-1$. ∎

######  Lemma 2.4.

For each $m\in S$, there exist unique indecomposable $m^{1},\dots,m^{r}$ such
that $m=m^{1}*\dots*m^{r}$.

###### Proof.

The proof is by induction on $l(m)$. If $l(m)=1$, then $m=(m_{0})=(0)$ and $m$
is indecomposable. Let us prove the induction step. Let

$\displaystyle m=m^{1}*\dots*m^{r}=n^{1}*\dots*n^{s}$

be two decompositions into indecomposable factors. We may assume that
$l(n^{s})\geqslant l(m^{r})$. If $l(n^{s})=l(m^{r})$, then $n^{s}=m^{r}$. In
this case we can conclude that $m^{1}*\dots*m^{r-1}=n^{1}*\dots*n^{s-1}$, and
the statement follows from the induction hypothesis. If $l(n^{s})>l(m^{r})$,
then the sequence $n^{s}$ contains the sequence $m^{r}$ as its last
$l(m^{r})$-part. This contradicts to the indecomposability of $n^{s}$. The
lemma is proved. ∎

###  2.6. Polynomials $I_{\lambda}$

For an $l$-tuple
$\lambda=(\Lambda_{0}(t,z),\Lambda_{1}(t,z),\dots,\Lambda_{l-1}(t,z))$ of
Laurent polynomials with coefficients in ${\mathbb{Z}}_{p}$ define

$\displaystyle I_{\lambda}(t,z)=\sum_{m\in
S_{l}^{\operatorname{ind}}}R_{m,\lambda}(t,z).$

We have

(2.4) $\displaystyle I_{\lambda}(x,z)\equiv 0\qquad\pmod{p^{l-1}}$

by Lemma 2.3 and

$\displaystyle N(I_{\lambda}(t,z))\subset
N(\Lambda_{0}(t,z))+pN(\Lambda_{1}(t,z))+\dots+p^{l-1}N(\Lambda_{l-1}(t,z))\,.$

######  Lemma 2.5.

We have

(2.5)
$\displaystyle\tilde{\lambda}(t,z)=\sum_{\lambda=\lambda^{1}*\dots*\lambda^{s}}I_{\lambda^{1}}(t,z)I_{\lambda^{2}}(t^{p^{l(\lambda^{1})}},z^{p^{l(\lambda^{1})}})\cdots
I_{\lambda^{s}}(t^{p^{l(\lambda^{1})+\dots+l(\lambda^{s-1})}},z^{p^{l(\lambda^{1})+\dots+l(\lambda^{s-1})}}),$

where the sum is over the set of all possible decompositions of the tuple
$\lambda$ into a product of tuples.

###### Proof.

We have

$\displaystyle\tilde{\lambda}(t,z)=\sum_{m\in S_{l}}R_{m,\lambda}(t,z).$

For any $m\in S_{l}$, let $m=m^{1}*\dots*m^{s}$ be its unique idecomposition
into indecomposable factors. Let $\lambda=\lambda^{1}*\dots*\lambda^{s}$ be
the corresponding factorization of the sequence $\lambda$. Then

(2.6) $\displaystyle
R_{m,\lambda}(t,z)=R_{m^{1},\lambda^{1}}(t,z)R_{m^{2},\lambda^{2}}(t^{p^{l(\lambda^{1})}},z^{p^{l(\lambda^{1})}})\cdots
R_{m^{s},\lambda^{s}}(t^{p^{l(\lambda^{1})+\dots+l(\lambda^{s-1})}},z^{p^{l(\lambda^{1})+\dots+l(\lambda^{s-1})}}).$

This product contributes to the the expansion of the product

(2.7) $\displaystyle
I_{\lambda^{1}}(t,z)I_{\lambda^{2}}(t^{p^{l(\lambda^{1})}},z^{p^{l(\lambda^{1})}})\cdots
I_{\lambda^{s}}(t^{p^{l(\lambda^{1})+\dots+l(\lambda^{s-1})}},z^{p^{l(\lambda^{1})+\dots+l(\lambda^{s-1})}})$

into the sum, and conversely each summand in the expansion of (2.7) comes from
(2.6) for a unique indecomposable factorization $m=m^{1}*\dots*m^{s}$. ∎

###  2.7. Admissible tuples of Laurent polynomials

######  Definition 2.6.

A tuple $\lambda=(\Lambda_{0}(t,z),\Lambda_{1}(t,z),\dots,\Lambda_{l-1}(t,z))$
of Laurent polynomials is called $\operatorname{admissible}$ if the tuple
$(N(\Lambda_{0}(t,z)),N(\Lambda_{1}(t,z)),\dots,N(\Lambda_{l-1}(t,z)))$ of its
Newton polytopes with respect to variables $t$ is admissible.

Denote by $\operatorname{CT}_{t}(\Lambda)(z)$ the constant term of the Laurent
polynomial $\Lambda(t,z)$ with respect to the variables $t$. The constant term
$\operatorname{CT}_{t}(\Lambda)(z)$ is a Laurent polynomial in $z$.

######  Lemma 2.7.

Let $\lambda=(\Lambda_{0}(t,z),\Lambda_{1}(t,z),\dots,\Lambda_{l-1}(t,z))$ be
an admissible tuple of Laurent polynomials with coefficients in
${\mathbb{Z}}_{p}$ and $\lambda=\lambda^{1}*\dots*\lambda^{s}$. Then

(2.8)
$\displaystyle\operatorname{CT}_{t}\\!\bigg{(}\prod_{i=1}^{s}I_{\lambda^{i}}\big{(}t^{p^{l(\lambda^{1})+\dots+l(\lambda^{i-1})}},z^{p^{l(\lambda^{1})+\dots+l(\lambda^{i-1})}}\big{)}\bigg{)}\\!(z)=$
$\displaystyle\phantom{aaaaaaaaaaaaaaaa}=\,\prod_{i=1}^{s}\operatorname{CT}_{t}\big{(}I_{\lambda^{i}}(t,z)\big{)}(z^{p^{l(\lambda^{1})+\dots+l(\lambda^{i-1})}}).$

###### Proof.

We have

$\displaystyle N(I_{\lambda^{1}}(t,z))\subset
N(\Lambda_{0}(t,z))+pN(\Lambda_{1}(t,z))+\dots+p^{l(\lambda^{1})-1}N(\Lambda_{l(\lambda^{1})-1}(t,z)).$

Hence

$\displaystyle N(I_{\lambda^{1}}(t,z))\cap
p^{l(\lambda^{1})}{\mathbb{Z}}^{r}=\\{0\\}$

and

$\displaystyle\operatorname{CT}_{t}\\!\Big{(}\prod_{i=1}^{s}I_{\lambda^{i}}(t^{p^{l(\lambda^{1})+\dots+l(\lambda^{i-1})}},z^{p^{l(\lambda^{1})+\dots+l(\lambda^{i-1})}})\Big{)}\\!(z)=$
$\displaystyle\phantom{aaaaaa}=\operatorname{CT}_{t}\big{(}I_{\lambda^{1}}(t,z)\big{)}(z)\operatorname{CT}_{t}\\!\Big{(}\prod_{i=2}^{s}I_{\lambda^{i}}\big{(}t^{p^{l(\lambda^{2})+\dots+l(\lambda^{i-1})}},z^{p^{l(\lambda^{2})+\dots+l(\lambda^{i-1})}}\big{)}\Big{)}\big{(}z^{p^{l(\lambda^{1})}}\big{)}.$

Thus by induction on $s$ we prove the statement. ∎

######  Corollary 2.8.

We have

(2.9)
$\displaystyle\operatorname{CT}_{t}\big{(}\tilde{\lambda}\big{)}(z)=\sum_{\lambda=\lambda^{1}*\dots*\lambda^{s}}\operatorname{CT}_{t}(I_{\lambda^{1}})(z)\cdot\operatorname{CT}_{t}(I_{\lambda^{2}})(z^{p^{l(\lambda^{1})}})\cdots\operatorname{CT}_{t}(I_{\lambda^{s}})(z^{p^{l(\lambda^{1})+\dots+l(\lambda^{s-1})}}),$

where the sum is over the set of all decompositions of $\lambda$ into a
product of tuples.

###  2.8. Dwork congruence for tuples of Laurent polynomials

######  Theorem 2.9.

Let $a,b,c$ be tuples of Laurent polynomials in $t,z$ with coefficients in
${\mathbb{Z}}_{p}$, where $b,c,a^{\prime}$ can be empty. Assume that the
tuples $a*b,a*c,a^{\prime}*b,a^{\prime}*c$ of Laurent polynomials are
admissible. Then

(2.10)
$\displaystyle\phantom{aaaaaa}\operatorname{CT}_{t}\big{(}\widetilde{a*b}\big{)}\\!(z)\,\operatorname{CT}_{t}\big{(}\widetilde{a^{\prime}*c}\big{)}\\!(z^{p})\,\equiv\,\operatorname{CT}_{t}\big{(}\widetilde{a^{\prime}*b}\big{)}\\!(z^{p})\,\operatorname{CT}_{t}\big{(}\widetilde{a*c}\big{)}\\!(z)\pmod{p^{l(a)}}.$

###### Proof.

The left-hand side and right-hand side of (2.10) are

(2.11) $\displaystyle\sum_{{a*b=x^{1}*\dots*x^{q}\atop
a^{\prime}*c=y^{1}*\dots*y^{s}}}\prod_{i=1}^{q}\operatorname{CT}_{t}(I_{x^{i}})(z^{p^{l(x^{1})+\dots+l(x^{i-1})}})\prod_{i=1}^{s}\operatorname{CT}_{t}(I_{y^{j}})(z^{p^{1+l(y^{1})+\dots+l(y^{j-1})}})$

and

(2.12) $\displaystyle\sum_{{a^{\prime}*b=x^{1}*\dots*x^{q}\atop
a*c=y^{1}*\dots*y^{s}}}\prod_{i=1}^{q}\operatorname{CT}_{t}(I_{x^{i}})(z^{p^{1+l(x^{1})+\dots+l(x^{i-1})}})\prod_{i=1}^{s}\operatorname{CT}_{t}(I_{y^{j}})(z^{p^{l(y^{1})+\dots+l(y^{j-1})}}),$

respectively. Since we work modulo $p^{l(a)}$, all the terms with

$\displaystyle\sum_{i=1}^{q}l(x^{i})+\sum_{j=1}^{s}l(y^{j})-q-s\geqslant l(a)$

may be dropped off from consideration. That inequality can be reformulated as
$l(a)+l(b)+l(a)+l(c)-1-q-s\geqslant l(a)$, equivalently, as

(2.13) $\displaystyle l(a)+l(b)+l(c)\geqslant q+s+1.$

Let us prove that the remaining terms in both expressions are in a bijective
correspondence such that the corresponding terms are _equal_.

Namely, take one of the remaining summands on the left-hand side:

(2.14)
$\displaystyle\prod_{i=1}^{q}\operatorname{CT}_{t}(I_{x^{i}})(z^{p^{l(x^{1})+\dots+l(x^{i-1})}})\prod_{i=1}^{s}\operatorname{CT}_{t}(I_{y^{j}})(z^{p^{1+l(y^{1})+\dots+l(y^{j-1})}}),$

the summand corresponding to the presentation $a*b=x^{1}*\dots*x^{q}$,
$a^{\prime}*c=y^{1}*\dots*y^{s}$.

######  Lemma 2.10.

There exist indices $i\geqslant 1$ and $j\geqslant 0$ such that

(2.15) $\displaystyle
l(x^{1})+\dots+l(x^{i})=l(y^{1})+\dots+l(y^{j})+1\leqslant l(a).$

###### Proof.

If $l(x^{1})=1$, then $i=1$ and $j=0$ are the required indices.

Assume that $l(x^{1})>1$ and the required $i$, $j$ do not exist. Then each
number in $\\{2,\dots,l(a)\\}$ cannot be represented simultaneously as
$l(x^{1})+\dots+l(x^{i})$ and $l(y^{1})+\dots+l(y^{j})+1$. Therefore the sum
of the total number of $i\geqslant 1$, such that
$l(x^{1})+\dots+l(x^{i})\leqslant l(a)$, and the total number of $j\geqslant
1$, such that $l(y^{1})+\dots+l(y^{j})+1\leqslant l(a),$ is at most $l(a)-1$.
The number of remaining $i$ is at most $l(b)$ and the number of remaining $j$
is at most $l(c)$. Therefore, $q+s\leqslant l(a)-1+l(b)+l(c)$, which is the
same as (2.13). Hence the corresponding summand must have been dropped off.
This establishes the existence of indices $i$ and $j$ required. ∎

Now we return to the remaining summand (2.14). Choose the minimal indices
$i\geqslant 1$ and $j\geqslant 0$ such that (2.15) holds. Then it is easy to
see that

(2.16)
$\displaystyle\phantom{aaa}a^{\prime}*b=y^{1}*\dots*y^{j}*x^{i+1}*\dots*x^{q},\qquad
a*c=x^{1}*\dots*x^{i}*y^{j+1}*\dots*x^{s},$

and the summand in (2.12) corresponding to the presentations in (2.16) equals
the product in (2.14). This clearly gives the desired bijection. ∎

##  3\. $p^{s}$-Approximation of a hypergeometric integral

Let $\alpha,\beta,\gamma$ be rational numbers with
$|\alpha|_{p}=|\beta|_{p}=|\gamma|_{p}=1$. Consider a hypergeometric integral

(3.1) $\displaystyle
I^{(C)}(x)=\int_{C}t^{\alpha}(t-1)^{\beta}(t-x)^{\gamma}dt$

where $C\subset{\mathbb{C}}-\\{0,1,x\\}$ is a contour on which the integrand
takes its initial value when $t$ encircles $C$. The function $I^{(C)}(x)$
satisfies the hypergeometric differential equation

(3.2) $\displaystyle
x(1-x)I^{\prime\prime}+((\alpha+\beta+2\gamma)x-(\alpha+\gamma))I^{\prime}-\gamma(\alpha+\beta+\gamma+1)I=0.$

This follows from Stokes’ theorem and the following identity of differential
forms. Denote $\Phi(t,x)=t^{\alpha}(t-1)^{\beta}(t-x)^{\gamma}$,

$\displaystyle\mathcal{D}=x(1-x)\frac{d^{2}}{dx^{2}}+((\alpha+\beta+2\gamma)x-(\alpha+\gamma))\frac{d}{dx}-\gamma(\alpha+\beta+\gamma+1).$

Then

(3.3) $\displaystyle
d_{t}\Big{(}\gamma\frac{t(t-1)}{t-x}\Phi(t,x)\Big{)}=\mathcal{D}\,\Phi(t,x)dt.$

The differential equation (3.2) turns into the standard hypergeometric
differential equation

(3.4) $\displaystyle x(1-x)I^{\prime\prime}+(c-(a+b+1)x)I^{\prime}-abI=0$

if $\alpha=a-c$, $\beta=c-b-1$, $\gamma=-a$. For a suitable choice of $C$ and
multiplication of the integral by a constant, the integral in (3.1) can be
expanded as a power series

${}_{2}F_{1}(a,b;c;x)={}_{2}F_{1}\biggl{(}\begin{matrix}a,\,b\\\
c\end{matrix}\biggm{|}x\biggr{)}=\sum_{k=0}^{\infty}\frac{(a)_{k}(b)_{k}}{k!(c)_{k}}x^{k}\,.$

Here $(a)_{n}=\Gamma(a+n)/\Gamma(a)=\prod_{k=0}^{n-1}(a+k)$ stands for
Pochhammer’s symbol.

We consider the following $p^{s}$-approximation of the integral in (3.1).
Given a positive integer $s$, let
$1\leqslant\alpha_{s},\beta_{s},\gamma_{s}\leqslant p^{s}$ be the unique
positive integers such that

(3.5)
$\displaystyle\alpha_{s}\equiv\alpha,\quad\beta_{s}\equiv\beta,\quad\gamma_{s}\equiv\gamma\qquad\pmod{p^{s}}.$

Define the master polynomial

(3.6)
$\displaystyle\Phi_{s}(t,x)=t^{\alpha_{s}}(t-1)^{\beta_{s}}(t-x)^{\gamma_{s}}$

and the $p^{s}$-approximation polynomial $I_{s}(x)$ as the coefficient of
$t^{p^{s}-1}$ in the master polynomial $\Phi_{s}(t,x)$. Then

(3.7) $\displaystyle
I_{s}(x)=(-1)^{\alpha_{s}+\beta_{s}+\gamma_{s}-p^{s}+1}\sum_{k_{1}+k_{2}=\alpha_{s}+\beta_{s}+\gamma_{s}-p^{s}+1}\binom{\beta_{s}}{k_{1}}\binom{\gamma_{s}}{k_{2}}x^{k_{2}}.$

The polynomial $I_{s}(x)$ has integer coefficients.

######  Theorem 3.1.

The polynomial $I_{s}(x)$ is a solution of the hypergeometric differential
equation (3.2) modulo $p^{s}$,

(3.8) $\displaystyle\mathcal{D}I_{s}(x)\,\in\,p^{s}{\mathbb{Z}}_{p}[x]\,.$

###### Proof.

The theorem follows from formula (3.3). ∎

In this paper we prove Dwork-type congruences for the $p^{s}$-approximation
polynomials $I_{s}(x)$ in several basic examples and leave general
considerations for another occasion.

For more general versions of the $p^{s}$-approximation construction see in
[SV2, V4].

##  4\. Function ${}_{2}F_{1}\Big{(}\frac{1}{2},\frac{1}{2};1,x\Big{)}$

###  4.1. Polynomials $P_{s}(x)$

The function

(4.1)
${}_{2}F_{1}\Big{(}\frac{1}{2},\frac{1}{2};1;x\Big{)}=\frac{1}{\pi}\,\int_{1}^{\infty}t^{-1/2}(t-1)^{-1/2}(t-x)^{-1/2}dt=\sum_{k=0}^{\infty}\binom{-1/2}{k}^{2}x^{k}$

satisfies the hypergeometric differential equation

(4.2) $\displaystyle x(1-x)I^{\prime\prime}+(1-2x)I^{\prime}-\frac{1}{4}I=0.$

Define the master polynomial

(4.3)
$\displaystyle\Phi_{p^{s}}(t,x)=t^{(p^{s}-1)/2}(t-1)^{(p^{s}-1)/2}(t-x)^{(p^{s}-1)/2}.$

The number
$M=\frac{p^{s}-1}{2}=\frac{p-1}{2}+\frac{p-1}{2}p+\dots+\frac{p-1}{2}p^{s-1}$
is the unique positive integer such that $1\leqslant M\leqslant p^{s}$ and
$M\equiv-1/2$ $\pmod{p^{s}}$. Define the $p^{s}$-approximation polynomial
$P_{s}(x)$ as the coefficient of $t^{p^{s}-1}$ in the master polynomial
$\Phi_{p^{s}}(t,x)$. Then

(4.4) $\displaystyle
P_{s}(x)=(-1)^{(p^{s}-1)/2}\sum_{k=0}^{(p^{s}-1)/2}\binom{(p^{s}-1)/2}{k}^{2}x^{k}.$

Define $P_{0}(x)=1$.

Recall the hypergeometric function ${}_{2}F_{1}(a,b;c;x)$. Then

(4.5) $\displaystyle
P_{s}(x)=(-1)^{(p^{s}-1)/2}{}_{2}F_{1}\Big{(}\frac{1-p^{s}}{2},\frac{1-p^{s}}{2};1;x\Big{)}.$

The polynomial $P_{s}(x)$ is a solution of the hypergeometric equation (4.2)
modulo $p^{s}$. This follows from Theorem 3.1 or from formula (4.5).

###  4.2. Baby congruences

Let $\varphi_{s}(x)=(x+1)^{(p^{s}-1)/2}$. Then

(4.6)
$\displaystyle\varphi_{s+1}(x)\varphi_{s-1}(x^{p})\equiv\varphi_{s}(x)\varphi_{s}(x^{p})\pmod{p^{s}}.$

This follows from $(x+1)^{p^{s}}\equiv(x^{p}+1)^{p^{s-1}}\pmod{p^{s}}$.

######  Lemma 4.1.

The master polynomials $\Phi_{s}(t,x)$ satisfy the baby congruence

(4.7)
$\displaystyle\Phi_{s+1}(t,x)\Phi_{s-1}(t^{p},x^{p})\equiv\Phi_{s}(t,x)\Phi_{s}(t^{p},x^{p})\pmod{p^{s}}.$

###### Proof.

The lemma follows from (4.6). ∎

###  4.3. Congruences for $P_{s}(x)$

######  Theorem 4.2.

The approximation polynomials $P_{s}(x)$ satisfy the congruence

(4.8) $\displaystyle P_{s+1}(x)P_{s-1}(x^{p})\equiv
P_{s}(x)P_{s}(x^{p})\pmod{p^{s}}.$

This theorem follows from a more general Theorem 4.4 below.

Using formula (4.5) we may rewrite (4.8) as the congruence

(4.9)
$\displaystyle{}_{2}F_{1}\Big{(}\frac{1-p^{s+1}}{2},\frac{1-p^{s+1}}{2};1;x\Big{)}{}_{2}F_{1}\Big{(}\frac{1-p^{s-1}}{2},\frac{1-p^{s-1}}{2};1;x^{p}\Big{)}\equiv\phantom{aaaaaaaaaa}$
$\displaystyle\equiv{}_{2}F_{1}\Big{(}\frac{1-p^{s}}{2},\frac{1-p^{s}}{2};1;x\Big{)}{}_{2}F_{1}\Big{(}\frac{1-p^{s}}{2},\frac{1-p^{s}}{2};1;x^{p}\Big{)}\pmod{p^{s}}.$

Let $\alpha$ be a rational number which is a $p$-adic unit,
$\alpha=\alpha_{0}+\alpha_{1}p+\alpha_{2}p^{2}+\cdots$. Denote by
$[\alpha]_{s}$ the sum of the first $s$ summands. Then congruence (4.9) takes
the form:

(4.10)
$\displaystyle{}_{2}F_{1}\big{(}[-\frac{1}{2}]_{s+1},[-\frac{1}{2}]_{s+1};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{1}{2}]_{s-1},[-\frac{1}{2}]_{s-1};1;x^{p}\big{)}\equiv\phantom{aaaaaaaaaaaa}$
$\displaystyle\equiv{}_{2}F_{1}\big{(}[-\frac{1}{2}]_{s},[-\frac{1}{2}]_{s};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{1}{2}]_{s},[-\frac{1}{2}]_{s};1;x^{p}\big{)}\pmod{p^{s}}.$

###  4.4. Coefficients of master polynomials

Consider

$\displaystyle\hat{\Phi}_{s}(t,x)$ $\displaystyle:=$ $\displaystyle
t^{-(p^{s}-1)}\Phi_{s}(t,x)=t^{-(p^{s}-1)/2}\big{(}(t-1)(t-x)\big{)}^{(p^{s}-1)/2}$
$\displaystyle=$
$\displaystyle\big{(}(t-1)(1-x/t)\big{)}^{(p^{s}-1)/2}=\sum_{j=-(p^{s}-1)/2}^{(p^{s}-1)/2}C_{s,j}(x)t^{j},$

where

$\displaystyle
C_{s,j}(x)=(-1)^{\frac{p^{s}-1}{2}-j}\sum_{m}\binom{\frac{p^{s}-1}{2}}{m+j}\binom{\frac{p^{s}-1}{2}}{m}x^{m}.$

In particular, $C_{s,0}(x)=P_{s}(x)$. Every coefficient $C_{s,j}(x)$ is a
hypergeometric function:

$\displaystyle
C_{s,j}(x)=(-1)^{\frac{p^{s}-1}{2}-j}\binom{\frac{p^{s}-1}{2}}{j}{}_{2}F_{1}\Big{(}\frac{1-p^{s}}{2},\frac{1-p^{s}}{2}+j;j+1;x\Big{)}\quad\text{for}\;j\geqslant
0,$

while a hypergeometric expression in the case $j<0$ comes out from the
following simple fact.

######  Lemma 4.3.

We have $\hat{\Phi}_{s}(x/t,x)=\hat{\Phi}_{s}(t,x)$ and hence

(4.12) $\displaystyle C_{s,-j}(x)=x^{j}C_{s,j}(x).$

We expand congruence (4.7) into a congruence of polynomials $C_{s,j}(x)$. The
constant term in $t$ gives us

(4.13)
$\displaystyle{\sum}_{k}C_{s+1,kp}(x)C_{s-1,-k}(x^{p})\equiv{\sum}_{k}C_{s,kp}(x)C_{s,-k}(x^{p})\pmod{p^{s}}.$

The following Theorem 4.4 establishes the congruences of individual pairs of
terms in (4.13).

######  Theorem 4.4.

For any $k$ appearing in (4.13) we have

(4.14) $\displaystyle C_{s+1,kp}(x)C_{s-1,-k}(x^{p})\equiv
C_{s,kp}(x)C_{s,-k}(x^{p})\pmod{p^{s}}.$

In particular for $k=0$ we have congruence (4.8).

###### Proof.

Every index $k$ appearing in (4.13) can be written uniquely as

(4.15)
$\displaystyle\phantom{aaa}k=k_{0}+k_{1}p+\dots+k_{s-2}p^{s-2},\quad-(p-1)/2\leqslant
k_{i}\leqslant(p-1)/2.$

Using (4.12) we reformulate (4.14) as

(4.16) $\displaystyle C_{s+1,kp}(x)C_{s-1,k}(x^{p})\equiv
C_{s,kp}(x)C_{s,k}(x^{p})\mod{p^{s}}$

and prove it below. We have

$\displaystyle C_{s+1,kp}(x)$ $\displaystyle=$
$\displaystyle\operatorname{CT}_{t}\Big{[}\hat{\Phi}_{1}(t,x)\Big{(}\prod_{i=0}^{s-2}\big{(}t^{-k_{i}}\hat{\Phi}_{1}(t,x)\big{)}^{p^{i+1}}\Big{)}\hat{\Phi}_{1}(t,x)^{p^{s}}\Big{]},$
$\displaystyle C_{s-1,k}(x)$ $\displaystyle=$
$\displaystyle\operatorname{CT}_{t}\Big{[}\prod_{i=0}^{s-2}\big{(}t^{-k_{i}}\hat{\Phi}_{1}(t,x)\big{)}^{p^{i}}\Big{]},$
$\displaystyle C_{s,kp}(x)$ $\displaystyle=$
$\displaystyle\operatorname{CT}_{t}\Big{[}\hat{\Phi}_{1}(t,x)\prod_{i=0}^{s-2}\big{(}t^{-k_{i}}\hat{\Phi}_{1}(t,x)\big{)}^{p^{i+1}}\Big{]},$
$\displaystyle C_{s,k}(x)$ $\displaystyle=$
$\displaystyle\operatorname{CT}_{t}\Big{[}\Big{(}\prod_{i=0}^{s-2}\big{(}t^{-k_{i}}\hat{\Phi}_{1}(t,x)\big{)}^{p^{i}}\Big{)}\hat{\Phi}_{1}(t,x)^{p^{s-1}}\Big{]}.$

It is easy to see that the $(s+1)$-tuple of Laurent polynomials

$\displaystyle\hat{\Phi}_{1}(t,x),\,t^{-k_{0}}\hat{\Phi}_{1}(t,x),\,\dots,\,t^{-k_{s-2}}\hat{\Phi}_{1}(t,x),\,\hat{\Phi}_{1}(t,x)$

is admissible in the sense of Definition 2.6. Now the application of Theorem
2.9 gives congruence (4.16) and hence congruence (4.14). ∎

######  Remark.

Denote $A(n,x):={}_{2}F_{1}(-n,-n;1;x)=\sum_{k}\binom{n}{k}^{2}x^{k}$. Let

$\displaystyle
n=n_{0}+n_{1}p+\dots+n_{s-1}p^{s-1},\qquad[n/p]=n_{1}+\dots+n_{s-1}p^{s-2},$

where $0\leqslant n_{i}<p$. Then for any $m\in{\mathbb{Z}}_{\geqslant 0}$ we
have

(4.17) $\displaystyle A(n+mp^{s},x)A([n/p],x^{p})\equiv
A(n,x)A([n/p]+mp^{s-1},x^{p})\pmod{p^{s}}.$

The proof follows from Theorem 2.9 and the identity

$\displaystyle
A(n,x)=\operatorname{CT}_{t}\big{[}\big{(}(t+1)(1+x/t)\big{)}^{n}\big{]}.$

###  4.5. Limits of $P_{s}(x)$

For $\alpha\in{\mathbb{Z}}_{p}$ there exists a unique solution
$\omega(\alpha)\in{\mathbb{Z}}_{p}$ of the equation
$\omega(\alpha)^{p}=\omega(\alpha)$ that is congruent to $x$ modulo $p$. The
element $\omega(\alpha)$ is called the Teichmüller representative of $\alpha$.
For $\alpha\in{\mathbb{F}}_{p}$, $r>0$, define the disc

$\displaystyle
D_{\alpha,r}=\\{x\in{\mathbb{Z}}_{p}\mid|x-\omega(\alpha)|_{p}<r\\}.$

Denote

$\displaystyle\bar{P}_{s}(x):=(-1)^{(p^{s}-1)/2}P_{s}(x)={}_{2}F_{1}\Big{(}\Big{[}\\!-\frac{1}{2}\Big{]}_{s},\Big{[}\\!-\frac{1}{2}\Big{]}_{s};1;x\Big{)},$

see (4.5). Denote

(4.18)
$\displaystyle\mathfrak{D}=\\{x\in{\mathbb{Z}}_{p}\mid|\bar{P}_{1}(x)|_{p}=1\\}.$

######  Theorem 4.5.

For $s\geqslant 1$ the rational function
$\frac{\bar{P}_{s+1}(x)}{\bar{P}_{s}(x^{p})}$ is regular on $\mathfrak{D}$.
The sequence
$\big{(}\frac{\bar{P}_{s+1}(x)}{\bar{P}_{s}(x^{p})}\big{)}_{s\geqslant 1}$
uniformly converges on $\mathfrak{D}$. The limiting analytic function $f(x)$
equals the ratio $\frac{F(x)}{F(x^{p})}$ on the disc $D_{0,1}$ where
$F(x):={}_{2}F_{1}(1/2,1/2;1;x)$ is defined by the convergent power series
(4.1). We also have $|f(x)|_{p}=1$ for $x\in\mathfrak{D}$.

###### Proof.

We have ${\mathbb{Z}}_{p}=\bigcup_{\alpha\in{\mathbb{F}}_{p}}D_{\alpha,1}$ and
also
$\mathfrak{D}=\bigcup_{\alpha\in{\mathbb{F}}_{p},\,|\bar{P}_{1}(\omega(\alpha))|_{p}=1}D_{\alpha,1}$
since $\bar{P}_{1}(x)$ has coefficients in ${\mathbb{Z}}_{p}$. In particular,
$D_{0,1}\subset\mathfrak{D}$. We also have

$\\{x\in{\mathbb{Z}}_{p}\mid|\bar{P}_{1}(x^{p})|_{p}=1\\}=\bigcup_{\alpha\in{\mathbb{F}}_{p},\,|\bar{P}_{1}(\omega(\alpha))|_{p}=1}D_{\alpha,1}=\mathfrak{D}$

for the same reason.

By Lucas’ theorem
$\bar{P}_{s}(x)\equiv\bar{P}_{1}(x)\bar{P}_{1}(x^{p})\dots\bar{P}_{1}(x^{p^{s-1}})$
$\pmod{p}$. Hence $|\bar{P}_{s}(x)|_{p}=|\bar{P}_{s}(x^{p})|_{p}=1$ for
$s\geqslant 1$, $x\in\mathfrak{D}$. Hence the rational functions
$\frac{\bar{P}_{s+1}(x)}{\bar{P}_{s}(x^{p})}$ are regular on $\mathfrak{D}$.

Congruence (4.8) implies that

(4.19)
$\displaystyle\Big{|}\frac{\bar{P}_{s+1}(x)}{\bar{P}_{s}(x^{p})}-\frac{\bar{P}_{s}(x)}{\bar{P}_{s-1}(x^{p})}\Big{|}_{p}\leqslant
p^{-s}\quad\operatorname{for}\ x\in\mathfrak{D}.$

This shows the uniform convergence of our sequence of rational functions on
the domain $\mathfrak{D}$. For the limiting function $f(x)$ we have
$|f(x)|_{p}=1$ for $x\in\mathfrak{D}$.

Clearly, for any fixed index $k$ the coefficient $\binom{(p^{s}-1)/2}{k}^{2}$
of $x^{k}$ in $\bar{P}_{s}(x)$ converges $p$-adically to the coefficient
$\binom{-1/2}{k}^{2}$ of $x^{k}$ in $F(x)$. Hence the sequence
$(\bar{P}_{s}(x))_{s\geqslant 1}$ converges to $F(x)$ on $D_{0,1}$, so that
$f(x)=\frac{F(x)}{F(x^{p})}$ on $D_{0,1}$. The theorem is proved. ∎

Dwork gives in [Dw] a different construction of analytic continuation of the
ratio $\frac{F(x)}{F(x^{p})}$ from $D_{0,1}$ to a larger domain. He considers
the sequence of polynomials

$\displaystyle F_{s}(x)=\sum_{k=0}^{p^{s}-1}\binom{-1/2}{k}^{2}x^{k},$

which are truncations of the hypergeometric series $F(x)$, and shows that the
sequence of rational functions
$\big{(}\frac{F_{s+1}(x)}{F_{s}(x^{p})}\big{)}_{s\geqslant 1}$ uniformly
converges on the domain
$\mathfrak{D}^{\operatorname{Dw}}=\\{x\in{\mathbb{Z}}_{p}\ |\
|g(x)|_{p}=1\\}$, where the polynomial

(4.20) $\displaystyle g(x)=\sum_{k=0}^{(p-1)/2}\binom{-1/2}{k}^{2}x^{k}$

is attributed by Dwork to Igusa [Ig]. Clearly his limiting function
$f^{\operatorname{Dw}}(x)$ equals the ratio $\frac{F(x)}{F(x^{p})}$ on
$D_{0,1}$.

It is easy to see that the two sequences of rational functions
$\big{(}\frac{\bar{P}_{s+1}(x)}{\bar{P}_{s}(x^{p})}\big{)}_{s\geqslant 1}$ and
$\big{(}\frac{F_{s+1}(x)}{F_{s}(x^{p})}\big{)}_{s\geqslant 1}$ have the same
limiting functions on the same domain. Indeed, $\bar{P}_{1}(x)\equiv g(x)$
$\pmod{p}$ and hence $\mathfrak{D}=\mathfrak{D}^{\operatorname{Dw}}$. Also
$f(x)=f^{\operatorname{Dw}}(x)$ on $D_{0,1}$ and hence on $\mathfrak{D}$.

Dwork shows in [Dw] interesting properties of the function $f(x)$. For
example, let $\alpha\in{\mathbb{F}}^{\times}_{p}-\\{1\\}$ be such that
$\omega(\alpha)\in{\mathfrak{D}}$. Dwork shows that the zeta function of the
elliptic curve defined over ${\mathbb{F}}_{p}$ by the equation
$y^{2}=x(x-1)(x-\alpha)$ has two zeros, which are
$1/((-1)^{(p-1)/2}f(\omega(\alpha)))$ and $(-1)^{(p-1)/2}f(\omega(\alpha))/p$.
Clearly $|f(\omega(\alpha))|_{p}=1$. The number
$(-1)^{(p-1)/2}f(\omega(\alpha))$ is called the unit root of that elliptic
curve.

According to our discussion this unit root can be calculated as the value at
$x=\omega(\alpha)$ of the limit as $s\to\infty$ of the ratio
$\frac{\bar{P}_{s+1}(x)}{\bar{P}_{s}(x^{p})}$ of approximation polynomials
multiplied by $(-1)^{(p-1)/2}$.

##  5\. Function ${}_{2}F_{1}\Big{(}\frac{2}{3},\frac{1}{3};1;x\Big{)}$

###  5.1. Two hypergeometric integrals

The integral

(5.1) $\displaystyle I^{(C)}(x)=\int_{C}t^{-1/3}(t-1)^{-1/3}(t-x)^{-2/3}dt$

where $C\subset{\mathbb{C}}-\\{0,1,x\\}$ is a contour on which the integrand
takes its initial value when $t$ encircles $C$ satisfies the hypergeometric
differential equation

(5.2) $\displaystyle x(1-x)I^{\prime\prime}+(1-2x)I^{\prime}-\frac{2}{9}I=0.$

For a suitable choice of $C$ the integral $I^{(C)}(x)$ presents the
hypergeometric function

(5.3)
${}_{2}F_{1}\Big{(}\frac{2}{3},\frac{1}{3};1,x\Big{)}=\sum_{k=0}^{\infty}\binom{-1/3}{k}\binom{-2/3}{k}x^{k}\,.$

The integral

(5.4) $\displaystyle J^{(D)}(x)=\int_{D}t^{-2/3}(t-1)^{-2/3}(t-x)^{-1/3}dt$

where $D\subset{\mathbb{C}}-\\{0,1,x\\}$ is a contour on which the integrand
takes its initial value when $t$ encircles $D$ satisfies the same
hypergeometric differential equation. For a suitable choice of $D$ the
integral $J^{(D)}(x)$ presents the same hypergeometric function
${}_{2}F_{1}\Big{(}\frac{2}{3},\frac{1}{3};1,x\Big{)}$.

The differential form $t^{-1/3}(t-1)^{-1/3}(t-z)^{-2/3}dt$ is transformed to
the differential form $-t^{-2/3}(t-1)^{-2/3}(t-z)^{-1/3}dt$ by the change of
the variable $t\mapsto(t-z)/(t-1)$.

In this section we discuss the $p^{s}$-approximations of the integrals
$I^{(C)}(x)$ and $J^{(D)}(x)$.

###  5.2. The case $p=3\ell+1$

The master polynomial for $I^{(C)}(x)$ is given by the formula

$\displaystyle\Phi_{s}(t,x)=t^{(p^{s}-1)/3}(t-1)^{(p^{s}-1)/3}(t-x)^{2(p^{s}-1)/3}.$

The $p^{s}$-approximation polynomial $Q_{s}(x)$ is defined as the coefficient
of $t^{p^{s}-1}$ in $\Phi_{s}(t,x)$,

$\displaystyle
Q_{s}(x)=(-1)^{(p^{s}-1)/3}\sum_{k}\binom{2(p^{s}-1)/3}{k}\binom{(p^{s}-1)/3}{k}x^{k}.$

Define $Q_{0}(x)=1$. We have

(5.5) $\displaystyle
Q_{s}(x)={}_{2}F_{1}\Big{(}\frac{2-2p^{s}}{3},\frac{1-p^{s}}{3};1;x\Big{)},$

since $(-1)^{(p^{s}-1)/3}=1$.

The polynomial $Q_{s}(x)$ is a solution of the hypergeometric equation (5.2)
modulo $p^{s}$. This follows from Theorem 3.1 or from formula (5.5).

The master polynomial for $J^{(D)}(x)$ is given by the formula

$\displaystyle\Psi_{s}(t,x)=t^{2(p^{s}-1)/3}(t-1)^{2(p^{s}-1)/3}(t-x)^{(p^{s}-1)/3}.$

The $p^{s}$-approximation polynomial $R_{s}(x)$ is defined as the coefficient
of $t^{p^{s}-1}$ in $\Psi_{s}(t,x)$,

$\displaystyle
R_{s}(x)=\sum_{k}\binom{2(p^{s}-1)/3}{k}\binom{(p^{s}-1)/3}{k}x^{k}.$

Define $R_{0}(x)=1$. We have

(5.6) $\displaystyle
Q_{s}(x)={}_{2}F_{1}\Big{(}2\frac{1-p^{s}}{3},\frac{1-p^{s}}{3};1;x\Big{)}.$

Master polynomials satisfy baby congruences,

(5.7)
$\displaystyle\Phi_{s+1}(t,x)\Phi_{s-1}(t^{p},x^{p})\equiv\Phi_{s}(t,x)\Phi_{s}(t^{p},x^{p})\pmod{p^{s}},$
$\displaystyle\Psi_{s+1}(t,x)\Psi_{s-1}(t^{p},x^{p})\equiv\Psi_{s}(t,x)\Psi_{s}(t^{p},x^{p})\pmod{p^{s}},$

by formula (4.6).

######  Theorem 5.1.

For $p=3\ell+1$ the approximation polynomials $R_{s}(x)$ and $Q_{s}(x)$
satisfy the congruences

(5.8) $\displaystyle Q_{s+1}(x)Q_{s-1}(x^{p})\equiv
Q_{s}(x)Q_{s}(x^{p})\pmod{p^{s}},$ (5.9) $\displaystyle
R_{s+1}(x)R_{s-1}(x^{p})\equiv R_{s}(x)R_{s}(x^{p})\pmod{p^{s}}.$

###### Proof.

Denote $\hat{\Phi}_{1}(t,x)=(t-1)^{\ell}(1-x/t)^{2\ell}$. Then
$Q_{s+1}(x)=\operatorname{CT}_{t}\big{[}\hat{\Phi}_{1}(t,x)^{1+p+\dots+p^{s}}\big{]}$.
It is easy to see that the $(s+1)$-tuple of Laurent polynomials
$(\hat{\Phi}_{1}(t,x),\hat{\Phi}_{1}(t,x),\,\dots,\hat{\Phi}_{1}(t,x))$ is
admissible in the sense of Definition 2.6. Now the application of Theorem 2.9
gives congruence (5.8). Congruence (5.9) is proved in the same way applied to
the formula $R_{s+1}(x)$ $=$
$\operatorname{CT}_{t}\big{[}\hat{\Psi}_{1}(t,x)^{1+p+\dots+p^{s}}\big{]}$,
where $\hat{\Psi}_{1}(t,x)=(t-1)^{2\ell}(1-x/t)^{\ell}$. Congruence (5.9) also
follows from (5.8) since $R_{s}(x)=Q_{s}(x)$. ∎

Formulas (5.5) and (5.6) imply that for $p=3\ell+1$, $s\geqslant 1$ we have

(5.10)
$\displaystyle{}_{2}F_{1}\Big{(}\frac{2-2p^{s+1}}{3},\frac{1-p^{s+1}}{3};1;x\Big{)}{}_{2}F_{1}\Big{(}\frac{2-2p^{s-1}}{3},\frac{1-p^{s-1}}{3};1;x^{p}\Big{)}\equiv\phantom{aaaaaaaaaaaaaaaaaa}$
$\displaystyle\equiv{}_{2}F_{1}\Big{(}\frac{2-2p^{s}}{3},\frac{1-p^{s}}{3};1;x\Big{)}{}_{2}F_{1}\Big{(}\frac{2-2p^{s}}{3},\frac{1-p^{s}}{3};1;x^{p}\Big{)}\pmod{p^{s}}.$

Using the expansions

$\displaystyle-1/3$ $\displaystyle=\ell+\ell p+\ell p^{2}+\cdots,$
$\displaystyle\qquad-2/3$ $\displaystyle=2\ell+2\ell p+2\ell p^{2}+\cdots,$
$\displaystyle(p^{s}-1)/3$ $\displaystyle=\ell+\ell p+\ell p^{2}+\dots+\ell
p^{s-1},$ $\displaystyle\qquad(p^{s}-2)/3$ $\displaystyle=2\ell+2\ell p+2\ell
p^{2}+\dots+2\ell p^{s-1},$

we conclude that for $p=3\ell+1$ and $s\geqslant 1$ we have

(5.11)
$\displaystyle{}_{2}F_{1}\big{(}[-\frac{2}{3}]_{s+1},[-\frac{1}{3}]_{s+1};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{2}{3}]_{s-1},[-\frac{1}{3}]_{s-1};1;x^{p}\big{)}\equiv\phantom{aaaaaaaaaaaa}$
$\displaystyle\equiv{}_{2}F_{1}\big{(}[-\frac{2}{3}]_{s},[-\frac{1}{3}]_{s};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{2}{3}]_{s},[-\frac{1}{3}]_{s};1;x^{p}\big{)}\pmod{p^{s}}.$

###  5.3. The case $p=3\ell+2>2$

The master polynomial for $I^{(C)}(x)$ is given by the formulas

$\displaystyle\Phi_{s}(t,x)$ $\displaystyle=$ $\displaystyle
t^{(2p^{s}-1)/3}(t-1)^{(2p^{s}-1)/3}(t-x)^{(p^{s}-2)/3},\qquad\\!\\!\operatorname{odd}\,s,$
$\displaystyle\Phi_{s}(t,x)$ $\displaystyle=$ $\displaystyle
t^{(p^{s}-1)/3}(t-1)^{(p^{s}-1)/3}(t-x)^{2(p^{s}-1)/3},\qquad\operatorname{even}\,s.$

The $p^{s}$-approximation polynomial $Q_{s}(x)$ is defined as the coefficient
of $t^{p^{s}-1}$ in $\Phi_{s}(t,x)$,

$\displaystyle Q_{s}(x)$ $\displaystyle=$
$\displaystyle(-1)^{(2p^{s}-1)/3}\sum_{k}\binom{(2p^{s}-1)/3}{k}\binom{(p^{s}-2)/3}{k}x^{k},\qquad\\!\\!\operatorname{odd}\,s,$
$\displaystyle Q_{s}(x)$ $\displaystyle=$
$\displaystyle(-1)^{(p^{s}-1)/3}\sum_{k}\binom{2(p^{s}-1)/3}{k}\binom{(p^{s}-1)/3}{k}x^{k},\qquad\operatorname{even}\,s.$

Define $Q_{0}(x)=1$. We have

(5.12) $\displaystyle Q_{s}(x)$ $\displaystyle=$
$\displaystyle-\,{}_{2}F_{1}\Big{(}\frac{2-p^{s}}{3},\frac{1-2p^{s}}{3};1;x\Big{)},\qquad\\!\\!\\!\operatorname{odd}\,s,$
$\displaystyle Q_{s}(x)$ $\displaystyle=$
$\displaystyle{}_{2}F_{1}\Big{(}\frac{2-2p^{s}}{3},\frac{1-p^{s}}{3};1;x\Big{)},\
\qquad\operatorname{even}\,s.$

Here we use the fact that for $p=3\ell+2>2$ we have $(-1)^{(2p^{s}-1)/3}=-1$
for odd $s$ and $(-1)^{(p^{s}-1)/3}=1$ for even $s$.

The polynomial $Q_{s}(x)$ is a solution of the hypergeometric equation (5.2)
modulo $p^{s}$. This follows from Theorem 3.1 or from formula (5.12).

The master polynomial for $J^{(D)}(x)$ is given by the formulas

$\displaystyle\Psi_{s}(t,x)$ $\displaystyle=$ $\displaystyle
t^{(p^{s}-2)/3}(t-1)^{(p^{s}-2)/3}(t-x)^{(2p^{s}-1)/3},\qquad\,\operatorname{odd}\,s,$
$\displaystyle\Psi_{s}(t,x)$ $\displaystyle=$ $\displaystyle
t^{2(p^{s}-1)/3}(t-1)^{2(p^{s}-1)/3}(t-x)^{(p^{s}-1)/3},\qquad\\!\operatorname{even}\,s.$

The $p^{s}$-approximation polynomial $R_{s}(x)$ is defined as the coefficient
of $t^{p^{s}-1}$ in $\Phi_{s}(t,x)$,

(5.13) $\displaystyle R_{s}(x)$ $\displaystyle=$
$\displaystyle(-1)^{(p^{s}-2)/3}\sum_{k}\binom{(2p^{s}-1)/3}{k}\binom{(p^{s}-2)/3}{k}x^{k},\qquad\operatorname{odd}\,s,$
$\displaystyle R_{s}(x)$ $\displaystyle=$
$\displaystyle(-1)^{2(p^{s}-1)/3}\sum_{k}\binom{2(p^{s}-1)/3}{k}\binom{(p^{s}-1)/3}{k}x^{k},\qquad\\!\\!\operatorname{even}\,s.$

Define $R_{0}(x)=1$. We have

(5.14) $\displaystyle R_{s}(x)$ $\displaystyle=$
$\displaystyle-\,{}_{2}F_{1}\Big{(}\frac{2-p^{s}}{3},\frac{1-2p^{s}}{3};1;x\Big{)},\qquad\operatorname{odd}\,s,$
$\displaystyle R_{s}(x)$ $\displaystyle=$
$\displaystyle{}_{2}F_{1}\Big{(}\frac{2-2p^{s}}{3},\frac{1-p^{s}}{3};1;x\Big{)},\quad\
\qquad\\!\\!\operatorname{even}\,s.$

Here we use the fact that for $p=3\ell+2>2$ we have $(-1)^{(p^{s}-2)/3}=-1$
for odd $s$ and $(-1)^{(2p^{s}-1)/3}=1$ for even $s$.

The polynomial $R_{s}(x)$ is a solution of the hypergeometric equation (5.2)
modulo $p^{s}$. This follows from Theorem 3.1 or from formula (5.14).

######  Lemma 5.2.

Master polynomials satisfy baby congruences,

(5.15)
$\displaystyle\Phi_{s+1}(t,x)\Psi_{s-1}(t^{p},x^{p})\equiv\Phi_{s}(t,x)\Psi_{s}(t^{p},x^{p})\pmod{p^{s}},$
(5.16)
$\displaystyle\Psi_{s+1}(t,x)\Phi_{s-1}(t^{p},x^{p})\equiv\Psi_{s}(t,x)\Phi_{s}(t^{p},x^{p})\pmod{p^{s}}.$

###### Proof.

We prove (5.15) for an odd $s$. The case of an even $s$ and congruence (5.16)
are proved similarly. The left-hand side of (5.15) for an odd $s=2k+1$ equals

$\displaystyle
t^{(p^{s+1}-1)/3}(t-1)^{(p^{s+1}-1)/3}(t-x)^{2(p^{s+1}-1)/3}t^{2p(p^{s-1}-1)/3}(t^{p}-1)^{2(p^{s-1}-1)/3}(t^{p}-x^{p})^{(p^{s-1}-1)/3},$

while the right-hand side equals

$\displaystyle
t^{(2p^{s}-1)/3}(t-1)^{(2p^{s}-1)/3}(t-x)^{(p^{s}-2)/3}t^{p(p^{s}-2)/3}(t^{p}-1)^{(p^{s}-2)/3}(t^{p}-x^{p})^{(2p^{s}-1)/3}.$

Now the congruence (5.15) for an odd $s$ follows from formula (4.6). ∎

######  Theorem 5.3.

For $p=3\ell+2>2$ the approximation polynomials $R_{s}(x)$ and $Q_{s}(x)$
satisfy the congruences

(5.17) $\displaystyle Q_{s+1}(x)R_{s-1}(x^{p})\equiv
Q_{s}(x)R_{s}(x^{p})\pmod{p^{s}},$ (5.18) $\displaystyle
R_{s+1}(x)Q_{s-1}(x^{p})\equiv R_{s}(x)Q_{s}(x^{p})\pmod{p^{s}}.$

###### Proof.

We prove (5.17) for an odd $s$. The case of an even $s$ and congruence (5.18)
are proved similarly. Denote

$\displaystyle f(t,x)$ $\displaystyle=$
$\displaystyle(t-1)^{(2p-1)/3}(1-x/t)^{(p-2)/3}=(t-1)^{2\ell+1}(1-x/t)^{\ell},$
$\displaystyle g(t,x)$ $\displaystyle=$
$\displaystyle(t-1)^{(p-2)/3}(1-x/t)^{(2p-1)/3}=(t-1)^{\ell}(1-x/t)^{2\ell+1}.$

It is easy to see that for an odd $s$ we have

$\displaystyle Q_{s+1}(x)$ $\displaystyle=$
$\displaystyle\operatorname{CT}_{t}\Big{[}(t-1)^{(p^{s+1}-1)/3}(1-x/t)^{2(p^{s+1}-1)/3}\Big{]}$
$\displaystyle=$
$\displaystyle\operatorname{CT}_{t}\Big{[}f(t,x)g(t,x)^{p}\dots
f(t,x)^{p^{s-1}}g(t,x)^{p^{s}}\Big{]},$ $\displaystyle R_{s-1}(x)$
$\displaystyle=$
$\displaystyle\operatorname{CT}_{t}\Big{[}(t-1)^{2(p^{s-1}-1)/3}(1-x/t)^{(p^{s-1}-1)/3}\Big{]}$
$\displaystyle=$
$\displaystyle\operatorname{CT}_{t}\Big{[}g(t,x)f(t,x)^{p}\dots
g(t,x)^{p^{s-3}}f(t,x)^{p^{s-2}}\Big{]},$ $\displaystyle Q_{s}(x)$
$\displaystyle=$
$\displaystyle\operatorname{CT}_{t}\Big{[}(t-1)^{(2p^{s}-1)/3}(1-x/t)^{(p^{s}-2)/3}\Big{]}$
$\displaystyle=$
$\displaystyle\operatorname{CT}_{t}\Big{[}f(t,x)g(t,x)^{p}\dots
g(t,x)^{p^{s-2}}f(t,x)^{p^{s-1}}\Big{]},$ $\displaystyle R_{s}(x)$
$\displaystyle=$
$\displaystyle\operatorname{CT}_{t}\Big{[}(t-1)^{(p^{s}-2)/3}(1-x/t)^{(2p^{s}-1)/3}\Big{]}$
$\displaystyle=$
$\displaystyle\operatorname{CT}_{t}\Big{[}g(t,x)f(t,x)^{p}\dots
f(t,x)^{p^{s-2}}g(t,x)^{p^{s-1}}\Big{]},$

Observe that the $(s+1)$-tuple of Laurent polynomials
$(f(t,x),g(t,x),\dots,f(t,x),g(t,x))$ is admissible in the sense of Definition
2.6. Now the application of Theorem 2.9 gives congruence (5.17) for an odd
$s$. ∎

######  Remark.

In general we may take any admissible tuple of Laurent polynomials and obtain
the coresponding Dwork congruences. For example the tuples
$(f,g,f,f,f,g,g,f,\dots)$ and $(f,f,\dots)$ are admissible.

Using formulas (5.12) and (5.14) we may reformulate congruences (5.17) and
(5.18) as

(5.19)
$\displaystyle{}_{2}F_{1}\Big{(}\frac{2-p^{s+1}}{3},\frac{1-2p^{s+1}}{3};1;x\Big{)}{}_{2}F_{1}\Big{(}\frac{2-p^{s-1}}{3},\frac{1-2p^{s-1}}{3};1;x^{p}\Big{)}\equiv\phantom{aaaaaaaaaaaaaaaaaa}$
$\displaystyle\equiv{}_{2}F_{1}\Big{(}\frac{2-2p^{s}}{3},\frac{1-p^{s}}{3};1;x\Big{)}{}_{2}F_{1}\Big{(}\frac{2-2p^{s}}{3},\frac{1-p^{s}}{3};1;x^{p}\Big{)}\pmod{p^{s}},\
\ \operatorname{odd}\,s,$ (5.20)
$\displaystyle{}_{2}F_{1}\Big{(}\frac{2-2p^{s+1}}{3},\frac{1-p^{s+1}}{3};1;x\Big{)}{}_{2}F_{1}\Big{(}\frac{2-2p^{s-1}}{3},\frac{1-p^{s-1}}{3};1;x^{p}\Big{)}\equiv\phantom{aaaaaaaaaaaaaaaaaa}$
$\displaystyle\equiv{}_{2}F_{1}\Big{(}\frac{2-p^{s}}{3},\frac{1-2p^{s}}{3};1;x\Big{)}{}_{2}F_{1}\Big{(}\frac{2-p^{s}}{3},\frac{1-2p^{s}}{3};1;x^{p}\Big{)}\pmod{p^{s}},\
\ \operatorname{even}\,s.$

Recall that in these congruences we have $p=3\ell+2$.

Consider the $p$-adic presentations

(5.21) $\displaystyle-1/3$ $\displaystyle=$ $\displaystyle 2\ell+1+\ell
p+(2\ell+1)p^{2}+\ell p^{3}+\cdots,$ (5.22) $\displaystyle-2/3$
$\displaystyle=$ $\displaystyle\ell+(2\ell+1)p+\ell
p^{2}+(2\ell+1)p^{3}+\cdots,$

Recall that $[-1/3]_{s}$ (resp., $[-2/3]_{s}$) is the sum of the first $s$
summands in (5.21) (resp., (5.22)). Then congruences (5.19) and (5.20) imply
that for $p=3\ell+2$, $s\geqslant 1$ we have

(5.23)
$\displaystyle{}_{2}F_{1}\big{(}[-\frac{2}{3}]_{s+1},[-\frac{1}{3}]_{s+1};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{2}{3}]_{s-1},[-\frac{1}{3}]_{s-1};1;x^{p}\big{)}\equiv\phantom{aaaaaaaaaaaa}$
$\displaystyle\equiv{}_{2}F_{1}\big{(}[-\frac{2}{3}]_{s},[-\frac{1}{3}]_{s};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{2}{3}]_{s},[-\frac{1}{3}]_{s};1;x^{p}\big{)}\pmod{p^{s}}.$

###  5.4. Limits of $\bar{Q}_{s}(x)$

Define

(5.24)
$\displaystyle\bar{Q}_{s}(x)={}_{2}F_{1}\Big{(}\Big{[}\\!-\frac{2}{3}\Big{]}_{s},\Big{[}\\!-\frac{1}{3}\Big{]}_{s};1;x\Big{)}.$

Then for any prime $p>3$ we have

(5.25)
$\displaystyle\bar{Q}_{s+1}(x)\bar{Q}_{s-1}(x^{p})\equiv\bar{Q}_{s}(x)\bar{Q}_{s}(x^{p})\pmod{p^{s}}$

by (5.11) and (5.23).

######  Theorem 5.4.

For any prime $p>3$ and integer $s\geqslant 1$ the rational function
$\frac{\bar{Q}_{s+1}(x)}{\bar{Q}_{s}(x^{p})}$ is regular on the domain

(5.26) $\displaystyle\mathfrak{D}=\\{x\in{\mathbb{Z}}_{p}\ |\
|\bar{Q}_{1}(x)|_{p}=1\\}.$

The sequence
$\big{(}\frac{\bar{Q}_{s+1}(x)}{\bar{Q}_{s}(x^{p})}\big{)}_{s\geqslant 1}$
uniformly converges on $\mathfrak{D}$. The limiting analytic function $f(x)$
equals the ratio $\frac{F(x)}{F(x^{p})}$ on the disc $D_{0,1}$ where
$F(x):={}_{2}F_{1}(2/3,1/3;1;x)$ is defined by the corresponding convergent
power series.

###### Proof.

The proof is the same as the proof of Theorem 4.5. ∎

###  5.5. Remark

Although the congruences (5.11) and (5.23) look the same for $p=3\ell+1$ and
$p=3\ell+2$ the proofs of them are different as presented above. The proof of
(5.11) for $p=3\ell+1$ uses just any one of the two master polynomials:
$\Phi_{s}(t,x)$ or $\Psi_{s}(t,x)$, while the proof of (5.23) for $p=3\ell+2$
uses the interaction of the two master polynomials $\Phi_{s}(t,x)$ and
$\Psi_{s}(t,x)$. Cf. the baby congruences (5.7), (5.15), (5.16).

###  5.6. Congruences related to $-\frac{1}{5}$, $-\frac{2}{5}$,
$-\frac{3}{5}$, $-\frac{4}{5}$

In this section we formulate the congruences related to the above rational
numbers. The proof of these congruences is similar to the corresponding proofs
in Sections 4 and 5.

For $p=5\ell\pm 2$ and any $s\geqslant 1$ we have

(5.27)
$\displaystyle{}_{2}F_{1}\big{(}[-\frac{4}{5}]_{s+1},[-\frac{1}{5}]_{s+1};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{3}{5}]_{s-1},[-\frac{2}{5}]_{s-1};1;x^{p}\big{)}\equiv\phantom{aaaaaaaaaaaa}$
$\displaystyle\equiv{}_{2}F_{1}\big{(}[-\frac{4}{5}]_{s},[-\frac{1}{5}]_{s};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{3}{5}]_{s},[-\frac{2}{5}]_{s};1;x^{p}\big{)}\pmod{p^{s}},$
(5.28)
$\displaystyle{}_{2}F_{1}\big{(}[-\frac{3}{5}]_{s+1},[-\frac{2}{5}]_{s+1};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{4}{5}]_{s-1},[-\frac{1}{5}]_{s-1};1;x^{p}\big{)}\equiv\phantom{aaaaaaaaaaaa}$
$\displaystyle\equiv{}_{2}F_{1}\big{(}[-\frac{3}{5}]_{s},[-\frac{2}{5}]_{s};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{4}{5}]_{s},[-\frac{1}{5}]_{s};1;x^{p}\big{)}\pmod{p^{s}}.$

For $p=5\ell\pm 1$ and any $s\geqslant 1$ we have

(5.29)
$\displaystyle{}_{2}F_{1}\big{(}[-\frac{4}{5}]_{s+1},[-\frac{1}{5}]_{s+1};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{4}{5}]_{s-1},[-\frac{1}{5}]_{s-1};1;x^{p}\big{)}\equiv\phantom{aaaaaaaaaaaa}$
$\displaystyle\equiv{}_{2}F_{1}\big{(}[-\frac{4}{5}]_{s},[-\frac{1}{5}]_{s};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{4}{5}]_{s},[-\frac{1}{5}]_{s};1;x^{p}\big{)}\pmod{p^{s}},$
(5.30)
$\displaystyle{}_{2}F_{1}\big{(}[-\frac{3}{5}]_{s+1},[-\frac{2}{5}]_{s+1};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{3}{5}]_{s-1},[-\frac{2}{5}]_{s-1};1;x^{p}\big{)}\equiv\phantom{aaaaaaaaaaaa}$
$\displaystyle\equiv{}_{2}F_{1}\big{(}[-\frac{3}{5}]_{s},[-\frac{2}{5}]_{s};1;x\big{)}\,{}_{2}F_{1}\big{(}[-\frac{3}{5}]_{s},[-\frac{2}{5}]_{s};1;x^{p}\big{)}\pmod{p^{s}}.$

Similar congruences hold for rational numbers of the form $a/b$, where $b$ is
a prime and $1-b\leqslant a\leqslant-1$. These congruences will be described
somewhere else.

##  6\. KZ equations

###  6.1. KZ equations

Let ${\mathfrak{g}}$ be a simple Lie algebra with an invariant scalar product.
The Casimir element is

$\displaystyle\Omega={\sum}_{i}\,h_{i}\otimes h_{i}\ \ \in\
{\mathfrak{g}}\otimes{\mathfrak{g}},$

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. We consider the
following system of differential and algebraic equations for a column
$3$-vector $I=(I_{1},I_{2},I_{3})$ depending on variables
$z=(z_{1},z_{2},z_{3})$ :

(6.1) $\displaystyle\frac{\partial I}{\partial z_{1}}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}\Big{(}\frac{\Omega_{12}}{z_{1}-z_{2}}+\frac{\Omega_{13}}{z_{1}-z_{3}}\Big{)}I,\qquad\frac{\partial
I}{\partial
z_{2}}={\frac{1}{2}}\Big{(}\frac{\Omega_{21}}{z_{2}-z_{1}}+\frac{\Omega_{23}}{z_{2}-z_{3}}\Big{)}I,$
$\displaystyle\frac{\partial I}{\partial z_{3}}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}\Big{(}\frac{\Omega_{31}}{z_{3}-z_{1}}+\frac{\Omega_{32}}{z_{3}-z_{2}}\Big{)}I,\qquad\quad
0=I_{1}+I_{2}+I_{3},$

where $\Omega_{ij}=\Omega_{ji}$ and

$\displaystyle\Omega_{12}=\begin{pmatrix}-1&1&0\\\ 1&-1&0\\\
0&0&0\end{pmatrix},\quad\Omega_{13}=\begin{pmatrix}-1&0&1\\\ 0&0&0\\\
1&0&-1\end{pmatrix},\quad\Omega_{23}=\begin{pmatrix}0&0&0\\\ 0&-1&1\\\
0&1&-1\end{pmatrix}.$

Denote

$\displaystyle
H_{1}(z)={\frac{1}{2}}\Big{(}\frac{\Omega_{12}}{z_{1}-z_{2}}+\frac{\Omega_{13}}{z_{1}-z_{3}}\Big{)},\quad
H_{2}(z)={\frac{1}{2}}\Big{(}\frac{\Omega_{21}}{z_{2}-z_{1}}+\frac{\Omega_{23}}{z_{2}-z_{3}}\Big{)},\quad
H_{3}(z)={\frac{1}{2}}\Big{(}\frac{\Omega_{31}}{z_{3}-z_{1}}+\frac{\Omega_{32}}{z_{3}-z_{2}}\Big{)},$
$\displaystyle\nabla_{i}^{\operatorname{KZ}}=\frac{\partial}{\partial
z_{i}}-H_{i}(z),\qquad i=1,2,3.$

Then the KZ equations can be written as the system of equations,

$\displaystyle\nabla_{i}^{\operatorname{KZ}}I=0,\quad i=1,2,3,\qquad
I_{1}+I_{2}+I_{3}=0.$

System (6.1) is the system of KZ equations with parameter $\kappa=2$
associated with the Lie algebra ${\mathfrak{sl}}_{2}$ and the subspace of
singular vectors of weight $1$ of the tensor power
$({\mathbb{C}}^{2})^{\otimes 3}$ of two-dimensional irreducible
${\mathfrak{sl}}_{2}$-modules, up to a gauge transformation, see this example
in [V2, Section 1.1].

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

Define the master function

$\displaystyle\Phi(t,z)=(t-z_{1})^{-1/2}(t-z_{2})^{-1/2}(t-z_{3})^{-1/2}$

and the column 3-vector

(6.2) $\displaystyle
I^{(C)}(z)=(I_{1}(z),I_{2}(z),I_{3}(z)):=\int_{C}\Big{(}\frac{\Phi(t,z)}{t-z_{1}},\frac{\Phi(t,z)}{t-z_{2}},\frac{\Phi(t,z)}{t-z_{3}}\Big{)}dt\,,$

where $C\subset{\mathbb{C}}-\\{z_{1},z_{2},z_{3}\\}$ is a contour on which the
integrand takes its initial value when $t$ encircles $C$.

######  Theorem 6.1 (cf.[V4]).

The function $I^{(C)}(z)$ is a solution of system (6.1).

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

###### Proof.

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

(6.3)
$\displaystyle-\frac{1}{2}\,\Big{(}\frac{\Phi(t,z)}{t-z_{1}}+\frac{\Phi(t,z)}{t-z_{2}}+\frac{\Phi(t,z)}{t-z_{3}}\Big{)}\,=\,\frac{\partial\Phi}{\partial
t}(t,z)\,,$ (6.4) $\displaystyle\Big{(}\frac{\partial}{\partial
z_{i}}-\frac{1}{2}\sum_{j\neq
i}\frac{\Omega_{i,j}}{z_{i}-z_{j}}\Big{)}\Big{(}\frac{\Phi(t,z)}{t-z_{1}},\frac{\Phi(t,z)}{t-z_{2}},\frac{\Phi(t,z)}{t-z_{3}}\Big{)}\,=\frac{\partial\Psi^{i}}{\partial
t}(t,z),$

where $\Psi^{i}(t,z)$ is the column $3$-vector
$(0,\dots,0,-\frac{\Phi(t,z)}{t-z_{i}},0,\dots,0)$ with the nonzero element at
the $i$-th place. ∎

######  Theorem 6.2 (cf. [V1, Formula (1.3)]).

All solutions of system (6.1) have this form. Namely, the complex vector space
of solutions of the form (6.2) is $2$-dimensional.

###  6.3. Solutions as vectors of first derivatives

Consider the elliptic integral

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

Then

(6.6) $\displaystyle I^{(C)}(z)=\,2\,\Big{(}\frac{\partial T^{(C)}}{\partial
z_{1}},\frac{\partial T^{(C)}}{\partial z_{2}},\frac{\partial
T^{(C)}}{\partial z_{3}}\Big{)}.$

Denote $\nabla T=\Big{(}\frac{\partial T}{\partial z_{1}},\frac{\partial
T}{\partial z_{2}},\frac{\partial T}{\partial z_{3}}\Big{)}$. Then the column
gradient vector of the function $T(z)$ satisfies the following system of (KZ)
equations

(6.7) $\displaystyle\nabla_{i}^{\operatorname{KZ}}\nabla T=0,\quad
i=1,2,3,\qquad\frac{\partial T}{\partial z_{1}}+\frac{\partial T}{\partial
z_{2}}+\frac{\partial T}{\partial z_{3}}=0.$

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

###  6.4. Solutions modulo $p^{s}$

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

(6.8)
$\displaystyle\Phi_{s}(t,z)=\big{(}(t-z_{1})(t-z_{1})(t-z_{1})\big{)}^{(p^{s}-1)/2}.$

Define the column 3-vector

(6.9) $\displaystyle I_{s}(z)=(I_{s,1}(z),I_{s,2}(z),I_{s,3}(z))$

as the coefficient of $t^{p^{s}-1}$ in the polynomial

(6.10)
$\displaystyle\Big{(}\frac{\Phi_{s}(t,z)}{t-z_{1}},\frac{\Phi_{s}(t,z)}{t-z_{2}},\frac{\Phi_{s}(t,z)}{t-z_{3}}\Big{)}.$

######  Theorem 6.3 ([V4]).

The polynomial $I_{s}(z)$ is a solution of system (6.1) modulo $p^{s}$.

###### Proof.

We have the following modifications of identities (6.3), (6.4) :

(6.11)
$\displaystyle\frac{p^{s}-1}{2}\,\Big{(}\frac{\Phi_{s}(t,z)}{t-z_{1}}+\frac{\Phi_{s}(t,z)}{t-z_{2}}+\frac{\Phi_{s}(t,z)}{t-z_{3}}\Big{)}\,=\,\frac{\partial\Phi_{s}}{\partial
t}(t,z)\,,$ (6.12) $\displaystyle\Big{(}\frac{\partial}{\partial
z_{i}}+\frac{p^{s}-1}{2}\sum_{j\neq
i}\frac{\Omega_{i,j}}{z_{i}-z_{j}}\Big{)}\Big{(}\frac{\Phi_{s}(t,z)}{t-z_{1}},\frac{\Phi_{s}(t,z)}{t-z_{2}},\frac{\Phi_{s}(t,z)}{t-z_{3}}\Big{)}\,=\frac{\partial\Psi_{s}^{i}}{\partial
t}(t,z),$

where $\Psi_{s}^{i}(t,z)$ is the column $3$-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 6.3 follows from these identities. ∎

###  6.5. $p^{s}$-Approximation polynomials of $T(z)$

Define the $p^{s}$-approximation polynomial $T_{s}(z)$ of the elliptic
integral $T(z)$ as the coefficient of $t^{p^{s}-1}$ in the master polynomial
$\Phi_{s}(t,z)$,

(6.13)
$\displaystyle\\!\\!\\!\\!\\!T_{s}(z)=(-1)^{(p^{s}-1)/2}\sum_{k_{1}+k_{2}+k_{3}=(p^{s}-1)/2}\binom{(p^{s}-1)/2}{k_{1}}\binom{(p^{s}-1)/2}{k_{2}}\binom{(p^{s}-1)/2}{k_{3}}z_{1}^{k_{1}}z_{2}^{k_{2}}z_{3}^{k_{3}}.$

We put $T_{0}(x)=1$.

The polynomial $T_{s}(z_{1},z_{2},z_{3})$ is symmetric with respect to
permutations of $z_{1},z_{2},z_{3}$ and

(6.14) $\displaystyle T_{s}(1,z_{2},0)=P_{s}(z_{2}),$

where $P_{s}(x)$ is defined in (4.4). The gradient vector $\nabla
T_{s}:=\Big{(}\frac{\partial T_{s}}{\partial z_{1}},\frac{\partial
T_{s}}{\partial z_{2}},\frac{\partial T_{s}}{\partial z_{3}}\Big{)}$ of the
$p^{s}$-approximation polynomial $T_{s}(z)$ is a solution modulo $p^{s}$ of
system (6.1) since

(6.15) $\displaystyle\nabla
T_{s}=\frac{1-p^{s}}{2}\,(I_{s,1}(z),I_{s,2}(z),I_{s,3}(z)).$

######  Lemma 6.4.

For $s\geqslant 1$ the master polynomials satisfy baby congruences,

(6.16)
$\displaystyle\Phi_{s+1}(t,z)\Phi_{s-1}(t^{p},z_{1}^{p},z_{2}^{p},z_{3}^{p})\equiv\Phi_{s}(t,z)\Phi_{s}(t^{p},z_{1}^{p},z_{2}^{p},z_{3}^{p})\pmod{p^{s}}.$

∎

######  Theorem 6.5.

For $s\geqslant 1$ we have

(6.17) $\displaystyle
T_{s+1}(z_{1},z_{2},z_{3})\,T_{s-1}(z_{1}^{p},z_{2}^{p},z_{3}^{p})\,\equiv\,T_{s}(z_{1},z_{2},z_{3})\,T_{s}(z_{1}^{p},z_{2}^{p},z_{3}^{p})\pmod{p^{s}}.$

###### Proof.

Let $h(t,z)=t^{1-p}\big{(}(t-z_{1})(t-z_{2})(t-z_{3})\big{)}^{(p-1)/2}$. Then

$\displaystyle T_{s}(z)=\operatorname{CT}_{t}\big{[}h(t,z)h(t,z)^{p}\dots
h(t,z)^{p^{s-1}}\big{]}.$

The tuple of Laurent polynomials $(h(t,z),h(t,z),\dots)$ is admissible in the
sense of Definition 2.6. Now the application of Theorem 2.9 gives congruence
(6.17). ∎

###  6.6. Limits of $T_{s}(z)$

Denote $\bar{T}_{s}(z):=(-1)^{(p^{s}-1)/2}T_{s}(z)$, see (6.13),

(6.18)
$\displaystyle\mathfrak{D}=\\{(z_{1},z_{2},z_{3})\in{\mathbb{Z}}_{p}^{3}\,\mid|\bar{T}_{1}(z_{1},z_{2},z_{3})|_{p}=1\\}.$

Notice that
${\mathbb{Z}}_{p}^{3}=\bigcup_{\alpha,\beta,\gamma\in{\mathbb{F}}_{p}}D_{\alpha,1}\times
D_{\beta,1}\times D_{\gamma,1}$. Since $\bar{T}_{1}(z)$ has coefficients in
${\mathbb{Z}}_{p}$,

$\displaystyle\mathfrak{D}={\bigcup}^{o}D_{\alpha,1}\times D_{\beta,1}\times
D_{\gamma,1}$

where the summation ${\bigcup}^{o}$ is over all
$\alpha,\beta,\gamma\in{\mathbb{F}}_{p}$ such that
$|T(\omega(\alpha),\omega(\beta),\omega(\gamma))|_{p}=1$. For the same reason
we have

(6.19)
$\displaystyle\mathfrak{D}=\\{(z_{1},z_{2},z_{3})\in{\mathbb{Z}}_{p}^{3}\,\mid|\bar{T}_{1}(z_{1}^{p},z_{2}^{p},z_{3}^{p})|_{p}=1\\}.$

Denote

$\displaystyle\mathfrak{E}=\\{(1,z_{2},0)\in{\mathbb{Z}}_{p}^{3}\ |\
|z_{2}|_{p}<1\\}.$

######  Lemma 6.6.

We have $\mathfrak{E}\subset D_{1,1}\times D_{0,1}\times
D_{0,1}\subset\mathfrak{D}$.

###### Proof.

The first inclusion is clear. The second inclusion follows from the equality
$\bar{T}_{s}(1,0,0)=1$. ∎

######  Theorem 6.7.

For $s\geqslant 1$ the rational function
$\frac{\bar{T}_{s+1}(z)}{\bar{T}_{s}(z^{p})}$ is regular on $\mathfrak{D}$.
The sequence
$\big{(}\frac{\bar{T}_{s+1}(z)}{\bar{T}_{s}(z^{p})}\big{)}_{s\geqslant 1}$
uniformly converges on $\mathfrak{D}$. The limiting analytic function $f(z)$,
restricted to $\mathfrak{E}$, equals the ratio $\frac{F(z_{2})}{F(z_{2}^{p})}$
where $F(z_{2}):={}_{2}F_{1}(1/2,1/2;1;z_{2})$ is defined by the convergent
power series (4.1). We also have $|f(z)|_{p}=1$ for every $z\in\mathfrak{D}$.

###### Proof.

By Lucas’ theorem
$\bar{T}_{s}(z)\equiv\bar{T}_{1}(z)\bar{T}_{1}(z^{p})\dots\bar{T}_{1}(x^{p^{s-1}})$
$\pmod{p}$. Hence $|\bar{T}_{s}(z)|_{p}=|\bar{T}_{s}(z^{p})|_{p}=1$ for
$s\geqslant 1$, $z\in\mathfrak{D}$. Hence the rational functions
$\frac{\bar{T}_{s+1}(z)}{\bar{T}_{s}(z^{p})}$ are regular on $\mathfrak{D}$

Congruence (6.17) implies that

(6.20)
$\displaystyle\Big{|}\frac{\bar{T}_{s+1}(z)}{\bar{T}_{s}(z^{p})}-\frac{\bar{T}_{s}(z)}{\bar{T}_{s-1}(z^{p})}\Big{|}_{p}\leqslant
p^{-s}\quad\operatorname{for}\ z\in\mathfrak{D}.$

This shows the uniform convergence of
$\big{(}\frac{\bar{T}_{s+1}(z)}{\bar{T}_{s}(z^{p})}\big{)}_{s\geqslant 1}$ on
$\mathfrak{D}$. For the limiting function $f(z)$ we have $|f(z)|_{p}=1$ for
$z\in\mathfrak{D}$.

We have
$\bar{T}_{s}(1,z_{2},0)=\bar{P}_{s}(z_{2})=\sum_{k}\binom{(p^{s}-1)/2}{k}^{2}z_{2}^{k}$.
Clearly, for any fixed index $k$ the coefficient $\binom{(p^{s}-1)/2}{k}^{2}$
of $z_{2}^{k}$ in $\bar{T}_{s}(1,z_{2},0)$ converges $p$-adically to the
coefficient $\binom{-1/2}{k}^{2}$ of $z_{2}^{k}$ in $F(z_{2})$. Hence the
sequence $(\bar{T}_{s}(1,z_{2},0))_{s\geqslant 1}$ converges to $F(z_{2})$ on
$\mathfrak{E}$, so that $f(1,z_{2},0)=\frac{F(z_{2})}{F(z^{p})}$ on
$\mathfrak{E}$. The theorem is proved. ∎

######  Remark.

The analytic function $f(z)$ of Theorem 6.9 exhibits behavior very different
from the behavior of the corresponding ratio $T^{(C)}(z)/T^{(C)}(z^{p})$ of
complex elliptic integrals.

By Theorem 6.9 the function $f(z)$ restricted to the one-dimensional discs
$\\{(z_{1},0,1)\in{\mathbb{Z}}_{p}^{3}\ |\ |z_{1}|_{p}<1\\}$,
$\\{(1,z_{2},0)\in{\mathbb{Z}}_{p}^{3}\ |\ |z_{2}|_{p}<1\\}$,
$\\{(0,1,z_{3})\in{\mathbb{Z}}_{p}^{3}\ |\ |z_{3}|_{p}<1\\}$ equals
$\frac{F(z_{1})}{F(z_{1}^{p})}$ , $\frac{F(z_{2})}{F(z_{2}^{p})}$ ,
$\frac{F(z_{3})}{F(z_{3}^{p})}$, respectively.

In the complex case, for the ratio $T^{(C_{1})}(z)/T^{(C_{1})}(z^{p})$ to be
equal to $\frac{F(z_{1})}{F(z_{1}^{p})}$ on
$\\{(z_{1},0,1)\in{\mathbb{C}}^{3}\ |\ |z_{1}|<1\\}$, the contour $C_{1}$ must
be the cycle on the elliptic curve $y^{2}=(t-z_{1})t(t-1)$ vanishing at
$z_{1}=0$. Similarly for $T^{(C_{2})}(z)/T^{(C_{2})}(z^{p})$ to be equal to
$\frac{F(z_{2})}{F(z_{2}^{p})}$ on $\\{(1,z_{2},0)\in{\mathbb{C}}^{3}\ |\
|z_{2}|<1\\}$, the contour $C_{2}$ must be the cycle on the elliptic curve
$y^{2}=(t-1)(t-z_{2})t$ vanishing at $z_{2}=0$, and for
$T^{(C_{3})}(z)/T^{(C_{3})}(z^{p})$ to be equal to
$\frac{F(z_{3})}{F(z_{3}^{p})}$ on $\\{(0,1,z_{3})\in{\mathbb{C}}^{3}\ |\
|z_{3}|<1\\}$, the contour $C_{3}$ must be the cycle on the elliptic curve
$y^{2}=t(t-1)(t-z_{3})$ vanishing at $z_{3}=0$. But these three local complex
analytic functions are not restrictions of a single univalued complex analytic
function due to the irreducibility of the monodromy representation of the
Gauss–Manin connection associated with the family of elliptic curves
$y^{2}=(t-z_{1})(t-z_{2})(t-z_{3})$.

For $i,j\in\\{1,2,3\\}$ and $s\geqslant 1$ denote

$\displaystyle
f_{s}(z)=T_{s}(z)\Big{/}T_{s-1}(z^{p}),\qquad\eta^{(i)}_{s}(z)=\frac{\partial
T_{s}}{\partial
z_{i}}(z)\Big{/}T_{s}(z),\qquad\eta^{(ij)}_{s}(z)=\frac{\partial^{2}T_{s}}{\partial
z_{i}\partial z_{j}}(z)\Big{/}T_{s}(z).$

######  Theorem 6.8.

For $s\geqslant 1$ the rational functions $\eta^{(i)}_{s}(z)$ and
$\eta^{(ij)}_{s}(z)$ are regular on $\mathfrak{D}$. The sequences of rational
functions $(\eta^{(i)}_{s}(z))_{s\geqslant 1}$ and
$(\eta^{(ij)}_{s}(z))_{\geqslant 1}$ converge uniformly on $\mathfrak{D}$ to
analytic functions. If $\eta^{(j)}$ and $\eta^{(ij)}$ denote the corresponding
limits, then

(6.21) $\displaystyle\eta^{(1)}+\eta^{(2)}+\eta^{(3)}=0,$ (6.22)
$\displaystyle\eta^{(j1)}+\eta^{(j2)}+\eta^{(j3)}=0,\qquad j=1,2,3,$ (6.23)
$\displaystyle\frac{\partial}{\partial
z_{j}}\eta^{(i)}=\eta^{(ji)}-\eta^{(i)}\eta^{(j)}.$

###### Proof.

Denote $\delta_{i}=z_{i}\frac{\partial}{\partial z_{i}}$. By Theorem 6.7 the
sequence $(f_{s})$ uniformly converges to the analytic function $f$ on
$\mathfrak{D}$. Therefore, the sequence of the derivatives
$(\frac{\partial}{\partial z_{i}}f_{s})$ uniformly converges on $\mathfrak{D}$
to $\frac{\partial}{\partial z_{i}}f$. Hence the sequence
$\big{(}(\delta_{i}f_{s})/f_{s}\big{)}$ uniformly converges on $\mathfrak{D}$
to the function $(\delta_{i}f)/f$. At the same time

$\displaystyle\frac{\delta_{i}f_{s}}{f_{s}}(z)=\frac{\delta_{i}T_{s}}{T_{s}}(z)-p\,\frac{\delta_{i}T_{s-1}}{T_{s-1}}(z^{p})$

and, more generally,

$\displaystyle\frac{\delta_{i}f_{s-k}}{f_{s-k}}(z^{p^{k}})=\frac{\delta_{i}T_{s-k}}{T_{s-k}}(z^{p^{k}})-p\,\frac{\delta_{i}T_{s-k-1}}{T_{s-k-1}}(z^{p^{k+1}})\qquad\text{for}\quad
k=0,1,\dots,s.$

Summing the relations up with suitable weights to get telescoping we obtain,
for any $r\leqslant s$,

$\displaystyle\sum_{k=0}^{r-1}\,p^{k}\,\frac{\delta_{i}f_{s-k}}{f_{s-k}}(z^{p^{k}})\,=\,\frac{\delta_{i}T_{s}}{T_{s}}(z)\,-\,p^{r}\,\frac{\delta_{i}T_{s-r}}{T_{s-r}}(z^{p^{r}}).$

Choosing $r=[s/2]$ and taking the limit as $s\to\infty$ on both sides we
arrive at

$\displaystyle\sum_{k=0}^{\infty}\,p^{k}\,\frac{\delta_{i}f}{f}(z^{p^{k}})=\lim_{s\to\infty}\frac{\delta_{i}T_{s}}{T_{s}}(z).$

The series on the left uniformly converges on $\mathfrak{D}$. Hence there
exists the limit on the right-hand side. This means that

$\displaystyle\eta^{(i)}(x)=\lim_{s\to\infty}\frac{\frac{\partial}{\partial
z_{i}}T_{s}}{T_{s}}(z)=\frac{1}{z_{i}}\sum_{k=0}^{\infty}\,p^{k}\,\frac{\delta_{i}f}{f}(z^{p^{k}}).$

One can further differentiate the resulting equality with respect to any of
the variables $z_{1},z_{2},z_{3}$ to get by induction formulas for
$\eta^{(ij)}$ and more generally for $\eta^{(ijk\dots)}$. Note that (6.23)
comes out from differentiating logarithmic derivatives.

Formulas (6.21) and (6.22) follow from (6.11). The theorem is proved. ∎

######  Theorem 6.9.

We have the following system of equations on $\mathfrak{D}$ :

(6.24) $\displaystyle\phantom{aaaa}\begin{pmatrix}\eta^{(11)}\\\
\eta^{(12)}\\\
\eta^{(13)}\end{pmatrix}={\frac{1}{2}}\Big{(}\frac{\Omega_{12}}{z_{1}-z_{2}}+\frac{\Omega_{13}}{z_{1}-z_{3}}\Big{)}\begin{pmatrix}\eta^{(1)}\\\
\eta^{(2)}\\\ \eta^{(3)}\end{pmatrix},\qquad\begin{pmatrix}\eta^{(21)}\\\
\eta^{(22)}\\\
\eta^{(23)}\end{pmatrix}={\frac{1}{2}}\Big{(}\frac{\Omega_{21}}{z_{2}-z_{1}}+\frac{\Omega_{23}}{z_{2}-z_{3}}\Big{)}\begin{pmatrix}\eta^{(1)}\\\
\eta^{(2)}\\\ \eta^{(3)}\end{pmatrix},$
$\displaystyle\begin{pmatrix}\eta^{(31)}\\\ \eta^{(32)}\\\
\eta^{(33)}\end{pmatrix}={\frac{1}{2}}\Big{(}\frac{\Omega_{31}}{z_{3}-z_{1}}+\frac{\Omega_{33}}{z_{3}-z_{2}}\Big{)}\begin{pmatrix}\eta^{(1)}\\\
\eta^{(2)}\\\
\eta^{(3)}\end{pmatrix},\qquad\eta^{(1)}+\eta^{(2)}+\eta^{(3)}=0.$

###### Proof.

The theorem follows from Theorems 6.3 and 6.9. ∎

######  Theorem 6.10.

The column vector

(6.25)
$\displaystyle\vec{\eta}(z):=(\eta^{(1)}(z),\eta^{(2)}(z),\eta^{(3)}(z))$

is nonzero at every point $z\in\mathfrak{D}$.

###### Proof.

On the one hand, if $\vec{\eta}(a)=0$ for some $a\in\mathfrak{D}$, then all
derivatives of $\vec{\eta}(z)$ at $a$ are equal to zero. This follows from the
first three equations in (6.24) written as

(6.26) $\displaystyle\frac{\partial}{\partial
z_{i}}\vec{\eta}=(H_{i}-\eta^{(i)})\,\vec{\eta},\qquad i=1,2,3.$

Hence $\vec{\eta}(z)$ equals zero identically on $\mathfrak{D}$. On the other
hand, $\eta^{(2)}(1,0,0)=F^{\prime}(0)/F(0)=1/4$ by Theorem 6.7. This
contradiction implies the theorem. ∎

###  6.7. Subbundle $\mathcal{L}\,\to\,\mathfrak{D}$

Denote $W=\\{(I_{1},I_{2},I_{3})\in{\mathbb{Q}}_{p}^{3}\ |\
I_{1}+I_{2}+I_{3}=0\\}$. The differential operators
$\nabla^{\operatorname{KZ}}_{i}$, $i=1,2,3$, define a connection on the
trivial bundle $W\times\mathfrak{D}\to\mathfrak{D}$, called the KZ connection.
The KZ connection is flat,

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

The flat sections of the KZ connection are solutions of system (6.1) of KZ
equations.

For any $a\in\mathfrak{D}$ let $\mathcal{L}_{a}\subset W$ be the one-
dimensional vector subspace generated by $\vec{\eta}(a)$. Then

$\displaystyle\mathcal{L}:=\bigcup\nolimits_{a\in\mathfrak{D}}\,\mathcal{L}_{a}\,\to\,\mathfrak{D}$

is an analytic line subbundle of the trivial bundle
$W\times\mathfrak{D}\to\mathfrak{D}$.

######  Theorem 6.11.

The subbundle $\mathcal{L}\,\to\,\mathfrak{D}$ is invariant with respect to
the KZ connection. In other words, if $s(z)$ is any section of
$\mathcal{L}\,\to\,\mathfrak{D}$, then sections $\nabla_{i}s(z)$, $i=1,2,3$,
also are sections of $\mathcal{L}\,\to\,\mathfrak{D}$.

###### Proof.

The theorem follows from equations (6.26). ∎

######  Remark.

For any $a\in\mathfrak{D}$ we may find locally a scalar analytic function
$u(z)$ such that $u(z)\cdot\vec{\eta}(z)$ is a solution of the KZ equations
(6.1). Such a function is a solution of the system of equations
$\frac{\partial u}{\partial z_{i}}=-\eta^{(i)}u$, $i=1,2,3$. This system is
compatible since $\frac{\partial\eta^{(j)}}{\partial
z_{i}}=\eta^{(ij)}-\eta^{(i)}\eta^{(j)}=\frac{\partial\eta^{(i)}}{\partial
z_{j}}$.

######  Remark.

The corresponding complex KZ connection does not have invariant line
subbundles due to irreducibility of the monodromy of the KZ connection, which
in our case is the Gauss–Manin connection of the family
$y^{2}=(t-z_{1})(t-z_{2})(t-z_{3})$. Thus the existence of the KZ invariant
line subbundle $\mathcal{L}\to\mathfrak{D}$ is a pure $p$-adic feature.

######  Remark.

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

######  Remark.

Following Dwork we may expect that locally at any point $a\in\mathfrak{D}$,
the solutions of the KZ equations of the form $u(z)\cdot\vec{\eta}(z),$ where
$u(z)$ is a scalar function, are given at $a$ by power series in
$z_{i}-a_{i}$, $i=1,2,3$, 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 [V4, Theorem A.4].

###  6.8. Other definitions of subbundle $\mathcal{L}\to\mathfrak{D}$

####  6.8.1. Line subbundle $\mathcal{M}\to\mathfrak{D}$

Define a polynomial $U_{s}(z)$ as the coefficient of $t^{p^{s}-1}$ in the
master polynomial
$\Phi_{s}(t+z_{3},z)=\big{(}(t-(z_{1}-z_{3}))(t-(z_{2}-z_{3}))t\big{)}^{(p^{s}-1)/2}$.
We have $U_{1}(z)=T_{1}(z)$ $\pmod{p}$, by Lucas’ theorem. Similarly to
Theorem 6.5 we conclude that

(6.27)
$\displaystyle\phantom{aaa}U_{s+1}(z_{1},z_{2},z_{3})\,U_{s-1}(z_{1}^{p},z_{2}^{p},z_{3}^{p})\,\equiv\,U_{s}(z_{1},z_{2},z_{3})\,U_{s}(z_{1}^{p},z_{2}^{p},z_{3}^{p})\pmod{p^{s}}.$

Hence the sequence $\big{(}\frac{U_{s+1}(z)}{U_{s}(z^{p})}\big{)}_{s\geqslant
1}$ uniformly converges to an analytic function on the domain $\mathfrak{D}$
defined in (6.18). The vector-valued polynomial $\nabla
U_{s}(z)=\big{(}\frac{\partial U_{s}}{\partial z_{1}},\frac{\partial
U_{s}}{\partial z_{2}},\frac{\partial U_{s}}{\partial z_{3}}\big{)}$ is a
solution modulo $p^{s}$ of the KZ equations (6.1); see [V4, Theorem 9.1], cf.
the proof of Theorem 6.3. Consider the function

$\displaystyle\vec{\mu}=(\mu^{(1)},\mu^{(2)},\mu^{(3)}):=\lim_{s\to\infty}\frac{\nabla
U_{s}}{U_{s}}$

defined on the same domain $\mathfrak{D}$. Similarly to the proofs of Theorems
6.8–6.11 we conclude that the function $\vec{\mu}(z)$ is nonzero on
$\mathfrak{D}$ and its values span an analytic line subbundle

$\displaystyle\mathcal{M}:=\bigcup\nolimits_{a\in\mathfrak{D}}\,\mathcal{M}_{a}\,\to\,\mathfrak{D}$

of the trivial bundle $W\times\mathfrak{D}\to\mathfrak{D}$; here
$\mathcal{M}_{a}\subset W$ is the one-dimensional subspace generated by
$\vec{\mu}(a)$. The line subbundle $\mathcal{M}\,\to\,\mathfrak{D}$ is
invariant with respect to the KZ connection.

######  Theorem 6.12.

The line bundles $\mathcal{M}\,\to\,\mathfrak{D}$ and
$\mathcal{L}\,\to\,\mathfrak{D}$ coincide.

###### Proof.

The proof rests on the two lemmas.

######  Lemma 6.13.

The line bundles $\mathcal{M}\,\to\,\mathfrak{D}$ and
$\mathcal{L}\,\to\,\mathfrak{D}$ coincide, if there is $a\in\mathfrak{D}$ such
that $\mathcal{M}_{a}=\mathcal{L}_{a}$.

###### Proof.

Let $\mathcal{M}_{a}=\mathcal{L}_{a}$ for some $a\in\mathfrak{D}$. Then
$\mathcal{M}_{z}=\mathcal{L}_{s}$ in some neighborhood of $a$, since locally
the subbundles are generated by the values of the solutions with the same
initial condition at $z=a$. Hence $\mathcal{M}_{z}=\mathcal{L}_{z}$ on
$\mathfrak{D}$. ∎

######  Lemma 6.14.

For $i=1,2,3$ the functions $\frac{\partial T_{s}}{\partial
z_{i}}(z)\big{/}T_{s}(z)$ and $\frac{\partial U_{s}}{\partial
z_{i}}(z)\big{/}U_{s}(z)$ are equal on the line $z_{1}=1$, $z_{3}=0$.

Hence $\mathcal{M}=\mathcal{L}$ over the points of that line and, therefore,
$\mathcal{M}_{z}=\mathcal{L}_{z}$ for $z\in\mathfrak{D}$. ∎

####  6.8.2. Line subbundle $\mathcal{N}\to\hat{\mathfrak{D}}$

Let $\omega(x)=F^{\prime}(x)/F(x)$, where $F(x)={}_{2}F_{1}(1/2,1/2;1;x)$. We
have $\omega(x)=\lim_{s\to\infty}\frac{P_{s}^{\prime}(x)}{P_{s}(x)}$ on
$D_{0,1}$. Introduce new variables

(6.28) $\displaystyle u_{1}=z_{1}-z_{3},\qquad
u_{2}=\frac{z_{2}-z_{3}}{z_{1}-z_{3}},\qquad u_{3}=z_{1}+z_{2}+z_{3},$

and a vector-valued function

(6.29)
$\displaystyle\vec{\omega}(u)=\frac{1}{u_{1}}\big{(}\\!-1/2-\omega(u_{2})u_{2},\,\omega(u_{2}),\,1/2+\omega(u_{2})(u_{2}-1)\big{)}\,.$

Define

$\displaystyle\hat{\mathfrak{D}}_{0}=\\{(z_{1},z_{2},z_{3})\in{\mathbb{Q}}_{p}^{3}\
|\ z_{i}\neq z_{j}\,\ \forall i\neq j\\}.$

For any $\sigma=(i,j,k)\in S_{3}$ define

$\displaystyle\phantom{aaaaa}\hat{\mathfrak{D}}_{1}^{\sigma}=\Big{\\{}(z_{1},z_{2},z_{3})\in\hat{\mathfrak{D}}_{0}\
\Big{|}\ \ \frac{z_{j}-z_{k}}{z_{i}-z_{k}}\in{\mathbb{Z}}_{p},\
\Big{|}g\Big{(}\frac{z_{j}-z_{k}}{z_{i}-z_{k}}\Big{)}\Big{|}_{p}=1\Big{\\}},$
$\displaystyle\hat{\mathfrak{D}}_{2}^{\sigma}=\Big{\\{}(z_{1},z_{2},z_{3})\in\hat{\mathfrak{D}}_{0}\
\Big{|}\
\frac{z_{i}-z_{k}}{z_{j}-z_{k}}\in\hat{\mathfrak{D}}_{1}^{\sigma}\Big{\\}},\qquad\hat{\mathfrak{D}}^{\sigma}=\hat{\mathfrak{D}}_{1}^{\sigma}\cup\hat{\mathfrak{D}}_{2}^{\sigma},\qquad\hat{\mathfrak{D}}=\sum_{\sigma\in
S_{3}}\hat{\mathfrak{D}}^{\sigma},$

where $g(\lambda)$ is the Igusa polynomial in (4.20).

Using Dwork’s results in [Dw], it is shown in [V4, Appendix] that the values
of the analytic continuation of the function $\vec{\omega}(u)$ generate a line
bundle $\mathcal{N}\to\hat{\mathfrak{D}}$ invariant with respect to the KZ
connection.

######  Theorem 6.15.

The line bundles $\mathcal{M}\to\mathfrak{D}$ and
$\mathcal{N}\to\hat{\mathfrak{D}}$ coincide on
$\mathfrak{D}\cap\hat{\mathfrak{D}}$.

Thus we identified the line bundles $\mathcal{L}\to\mathfrak{D}$,
$\mathcal{M}\to\mathfrak{D}$, and $\mathcal{N}\to\hat{\mathfrak{D}}$ over
$\mathfrak{D}\cap\hat{\mathfrak{D}}$.

###### Proof.

We have

$\displaystyle
U_{s}(z)=(z_{1}-z_{3})^{(p^{s}-1)/2}P_{s}\big{(}\frac{z_{2}-z_{3}}{z_{1}-z_{3}}\big{)},\qquad\frac{\partial
U_{s}}{\partial
z_{2}}=(z_{1}-z_{3})^{(p^{s}-1)/2-1}P_{s}^{\prime}\big{(}\frac{z_{2}-z_{3}}{z_{1}-z_{3}}\big{)}$
$\displaystyle\frac{\partial U_{s}}{\partial
z_{1}}=\frac{p^{s}-1}{2}\,\operatorname{const}\,(z_{1}-z_{3})^{(p^{s}-1)/2-1}P_{s}\big{(}\frac{z_{2}-z_{3}}{z_{1}-z_{3}}\big{)}-\operatorname{const}\,(z_{1}-z_{3})^{(p^{s}-1)/2-1}\frac{z_{2}-z_{3}}{z_{1}-z_{3}}P_{s}^{\prime}\big{(}\frac{z_{2}-z_{3}}{z_{1}-z_{3}}\big{)}.$

Hence

$\displaystyle\frac{1}{U_{s}(z)}\Big{(}\frac{\partial U_{s}}{\partial
z_{1}},\frac{\partial U_{s}}{\partial z_{2}},\frac{\partial U_{s}}{\partial
z_{3}}\Big{)}=\frac{1}{u_{1}}\Big{(}\frac{p^{s}-1}{2}-u_{2}\frac{P_{s}^{\prime}(u_{2})}{P_{s}(u_{2})},\,\frac{P_{s}^{\prime}(u_{2})}{P_{s}(u_{2})},\,-\frac{p^{s}-1}{2}+(u_{2}-1)\frac{P_{s}^{\prime}(u_{2})}{P_{s}(u_{2})}\Big{)}.$

Clearly the limit of this vector equals $\vec{\omega}(u)$ as $s\to\infty$. The
theorem is proved. ∎

##  7\. Concluding remarks

###  7.1. Conjectural stronger congruences for $\bar{P}_{s}(x)$

By Theorem 4.2 we have for polynomials
$\bar{P}_{s}(x):=(-1)^{(p^{s}-1)/2}P_{s}(x)$:

$\displaystyle\bar{P}_{4}(x)\bar{P}_{2}(x^{p})-\bar{P}_{3}(x)\bar{P}_{3}(x^{p})\equiv
0\pmod{p^{3}}.$

In particular, for the coefficient of $x^{N_{0}+N_{1}p+N_{2}p^{2}+N_{3}p^{3}}$
in $\bar{P}_{4}(x)\bar{P}_{2}(x^{p})-\bar{P}_{3}(x)\bar{P}_{3}(x^{p})$ we have

(7.1) $\displaystyle\sum_{\genfrac{}{}{0.0pt}{1}{\scriptstyle
k_{1}+l_{1}=N_{1}}{\scriptstyle
k_{2}+l_{2}=N_{2}}}\bigg{(}\binom{(p^{4}-1)/2}{N_{0}+k_{1}p+k_{2}p^{2}+N_{3}p^{3}}^{2}\binom{(p^{2}-1)/2}{l_{1}+l_{2}p}^{2}-$
$\displaystyle\phantom{aaaaaaaa}-\binom{(p^{3}-1)/2}{N_{0}+k_{1}p+k_{2}p^{2}}^{2}\binom{(p^{3}-1)/2}{l_{1}+l_{2}p+N_{3}p^{2}}^{2}\,\bigg{)}\equiv
0\pmod{p^{3}}.$

Computer experiments show that this sum can be split into subsums with at most
four terms so that each a subsum is divisible by $p^{3}$. More precisely, let
$0\leqslant a,b,\,c,c^{\prime},d,d^{\prime}\leqslant p-1$ be integers. Define

$\displaystyle
A(a,b;c,c^{\prime};d,d^{\prime})=\binom{(p^{4}-1)/2}{a+cp+dp^{2}+bp^{3}}^{2}\binom{(p^{2}-1)/2}{c^{\prime}+d^{\prime}p}^{2}-\binom{(p^{3}-1)/2}{a+cp+dp^{2}}^{2}\binom{(p^{3}-1)/2}{c^{\prime}+d^{\prime}p+bp^{2}}^{2}$

and

(7.2) $\displaystyle
B(a,b;c,c^{\prime};d,d^{\prime})=\operatorname{Sym}A(a,b;c,c^{\prime};d,d^{\prime}):=$
$\displaystyle\phantom{aaa}:=A(a,b;c,c^{\prime};d,d^{\prime})+A(a,b;c^{\prime},c;d,d^{\prime})+A(a,b;c,c^{\prime};d^{\prime},d)+A(a,b;c^{\prime},c;d^{\prime},d).$

We expect that the integer $B(a,b;c,c^{\prime};d,d^{\prime})$ is divisible by
$p^{3}$.

More generally, define

$\displaystyle k=(k^{(1)},\dots,k^{(s)}),\qquad
k^{(i)}=(k^{(i)}_{1},k^{(i)}_{2}),$ $\displaystyle
A(a,b;k)=\binom{(p^{s+2}-1)/2}{a+\sum_{i=1}^{s}k^{(i)}_{1}p^{i}+bp^{s+1}}^{2}\binom{(p^{s}-1)/2}{\sum_{i=1}^{s}k^{(i)}_{2}p^{i-1}}^{2}-\binom{(p^{s+1}-1)/2}{a+\sum_{i=1}^{s}k^{(i)}_{1}p^{i}}^{2}\binom{(p^{s+1}-1)/2}{\sum_{i=1}^{s}k^{(i)}_{2}p^{i-1}+bp^{s}}^{2}.$

Set

$\displaystyle B(a,b;k)=\operatorname{Sym}A(a,b;k),$

where $\operatorname{Sym}$ denotes the symmetrization with respect to the
index $j$ in $k^{(i)}_{j}$ in each group $k^{(i)}=(k^{(i)}_{1},k^{(i)}_{2})$.
Thus the symmetrization has $2^{s}$ summands; the case $s=2$ of this
symmetrization is displayed in (7.2).

######  Conjecture 7.1.

The integer $B(a,b;k)$ is divisible by $p^{s+1}$.

This conjecture is supported by computer experiments and is checked for $s=1$
using [Gr].

###  7.2. Dwork crystals

After the first draft of this preprint was posted, Beukers and Vlasenko kindly
pointed out that some of the congruences in Sections 4 and 5 can be deduced
from the general principles in their papers [BV] and [Vl]. For example, the
coefficient $A_{s}(t)$ of $(xy)^{p^{s}-1}$ in the polynomial
$(y^{2}-x(x-1)(x-t))^{p^{s}-1}$ is equal to

$\displaystyle(-1)^{(p^{s}-1)/2}\binom{p^{s}-1}{\frac{p^{s}-1}{2}}P_{s}(x),$

where the polynomial $P_{s}(x)$ is defined in (4.4), hence the congruence

$\displaystyle A_{s+1}(t)A_{s-1}(t^{p})\equiv
A_{s}(t)A_{s}(t^{p})\pmod{p^{s}}$

which follows from [Vl, Theorem I (ii)] is equivalent to the one in Theorem
4.2. A similar analysis of the coefficient of $(xy^{2})^{p^{s}-1}$ in
$(y^{3}-x(x-1)(x-t)^{2})^{p^{s}-1}$ allows one to produce a different proof of
Theorem 5.1 using the main results in [BV]. Notice that these arguments
require introducing an extra variable, $y$, into the story and make it
difficult (if possible) to establish results like Theorems 4.4, 5.3 and
congruences (5.27)–(5.30). However, there is more than these applications in
the methodology of [BV]: one notable example is the connection of the unit
root, which is read off as the $p$-adic limit, with the local zeta function —
see [BV, Part I, Appendix A] for details.

## References

  * [1]
  * [AS] A. Adolphson, S. Sperber, $A$-hypergeometric series and a $p$-adic refinement of the Hasse–Witt matrix, arXiv:2001.07280, 1–28
  * [2]
  * [BV] F. Beukers, M. Vlasenko, Dwork crystals, I, Intern. Math. Res. Not. 2021 (2021), no. 12, 8807–8844; II, Intern. Math. Res. Not. 2021 (2021), no. 6, 4427–4444
  * [3]
  * [Dw] B. Dwork, $p$-adic cycles, Publ. Math. de lHÉS 37 (1969), 27–115
  * [4]
  * [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
  * [5]
  * [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
  * [6]
  * [Gr] A. Granville, Binomial coefficients modulo prime powers, Preprint, 1–24;
  * [7] https://dms.umontreal.ca/~andrew/Binomial/
  * [8]
  * [Ig] J. Igusa, Class number of a definite quaternion with prime discriminant, Proc. Natl. Acad. Sci. USA 44 (1958), no. 4, 312–314
  * [9]
  * [KZ] V. Knizhnik, A. Zamolodchikov, Current algebra and the Wess–Zumino model in two dimensions, Nucl. Phys. B247 (1984), 83–103
  * [10]
  * [LTYZ] L. Long, F.-T. Tu, N. Yui, W. Zudilin, Supercongruences for rigid hypergeometric Calabi–Yau threefolds, Adv. in Math. 393 (2021), Art. 108058, 49 pages, https://doi.org/10.1016/j.aim.2021.108058
  * [11]
  * [MO] D. Maulik, A. Okounkov, Quantum groups and quantum cohomology, Astérisque, t. 408 (Société Mathématique de France, 2019), 1–277;
  * [12] https://doi.org/10.24033/ast.1074
  * [13]
  * [Me] A. Mellit, A proof of Dwork’s congruences, unpublished (October 20, 2009), 1–3.
  * [14]
  * [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.
  * [15]
  * [RV1] R. Rimányi, A. Varchenko, The ${\mathbb{F}}_{p}$-Selberg integral, arXiv:2011.14248, 1–19
  * [16]
  * [RV2] R. Rimányi, A. Varchenko, The $\mathbb{F}_{p}$-Selberg integral of type $A_{n}$, arXiv:2012.01391, 1–21
  * [17]
  * [SvS] K. Samol, D. van Straten, Dwork congruences and reflexive polytopes, Ann. Math. Qué. 39 (2015), no. 2, 185–203.
  * [18]
  * [SV1] V. Schechtman, A. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139–194
  * [19]
  * [SV2] V. Schechtman, A. Varchenko, Solutions of KZ differential equations modulo $p$, Ramanujan J. 48 (2019), no. 3, 655–683;
  * [20] https://doi.org/10.1007/s11139-018-0068-x, arXiv:1707.02615
  * [21]
  * [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
  * [22]
  * [V2] 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
  * [23]
  * [V3] 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
  * [24]
  * [V4] 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
  * [25]
  * [Vl] M. Vlasenko, Higher Hasse–Witt matrices, Indag. Math. 29 (2018), 1411–1424
  * [26]