#  Calogero-Moser eigenfunctions modulo $p^{s}$

Alexander Gorsky$\mskip 2.00002mu{}^{\circ}$ and Alexander Varchenko$\mskip
2.00002mu{}^{\star}$

###### Abstract.

In this note we use the Matsuo-Cherednik duality between the solutions to KZ
equations and eigenfunctions of Calogero-Moser Hamiltonians to get the
polynomial $p^{s}$-truncation of the Calogero-Moser eigenfunctions at a
rational coupling constant. The truncation procedure uses the integral
representation for the hypergeometric solutions to KZ equations. The
$s\rightarrow\infty$ limit to the pure $p$-adic case has been analyzed in the
$n=2$ case.

∘ Institute for Information Transmission Problems, Moscow, Russia
Bolshoy Karetny per. 19, Moscow, 127051, Russia

${}^{\star}\mskip-0.99998mu$Department of Mathematics, University of North
Carolina at Chapel Hill
Chapel Hill, NC 27599-3250, USA

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

###### Contents

  1. 1 Introduction
    1. 1.1 KZ equations and Calogero-Moser systems
    2. 1.2 Exposition of material
  2. 2 Hypergeometric integrals
    1. 2.1 Master function
    2. 2.2 Symmetrization
    3. 2.3 Weight functions
    4. 2.4 Hypergeometric solutions
    5. 2.5 An integral formula for $\operatorname{ch}(I^{(\gamma)})$
    6. 2.6 Reformulation of the equations
    7. 2.7 Truncated exponential function
    8. 2.8 The $p^{s}$-approximations
    9. 2.9 The $p^{s}$-approximations for $\kappa=\frac{r}{q}+m$, $m\in{\mathbb{Z}}$
  3. 3 The case $n=2$
    1. 3.1 Hypergeometric solutions
    2. 3.2 The $p^{s}$-approximations
    3. 3.3 Polynomials $\varphi_{s,d}(z,\lambda,\kappa)$ and $\psi_{s,d}(z,\lambda,\kappa-1)$
    4. 3.4 Shift operator
  4. 4 The $p$-adic limit as $s\to\infty$ for $n=2$
    1. 4.1 Change of variables
    2. 4.2 The $p$-adic limit
  5. 5 Some properties of $p^{s}$-truncated Calogero-Moser model
    1. 5.1 Arithmetic with the Planck constant
    2. 5.2 Semiclassical limit modulo $p^{s}$
  6. 6 Discussion

##  1\. Introduction

###  1.1. KZ equations and Calogero-Moser systems

There exists a mapping between solutions to the KZ equations and
eigenfunctions of the Calogero-Moser systems [Ma, Ch]. The symmetrizations and
antisymmetrizations of solutions to the KZ equation become the eigenfunctions
of the Calogero-Moser operators with the corresponding coupling constants.
This correspondence can be generalized to the relations between the
eigenfunctions of the Riujsenaars models and solutions to qKZ equations. One
can also consider the limiting correspondence when the Calogero-Moser and
Ruijsenaars systems get degenerated to the open Toda and relativistic open
Toda systems and simultaneously the corresponding limit is taken at the KZ and
qKZ side [ZZ1, KPSZ, KZ, GVZ]. This Matsuo-Cherednik duality is referred to as
quantum-quantum duality within the integrability framework.

The semiclassical limit of this duality can also be examined. In this
scenario, the saddle point approximation is employed in the integral
representation of the solutions to the KZ equations, leading to the derivation
of the Bethe anzatz equations and the corresponding Bethe eigenvectors. This
approach is applicable to quantum spin models such as Gaudin, XXX, or XXZ spin
chains. The Bethe Ansatz equations on the quantum spin chain side are
equivalent to the equations for critical points of the master function [GiK,
MTV1, MTV2]. On the Calogero-Moser side, instead of focusing on wave
functions, we examine the equations for the intersection of Lagrangian
submanifolds, which correspond to the Bethe Ansatz equations on the spin chain
side [GiK, MTV2, GZZ]. Within the integrability framework, this duality is
commonly known as classical-quantum duality, as it establishes a connection
between the classical Calogero-Moser system and the quantum spin chain.

There are also dualities within the families, known as bispectral dualities.
On the Calogero-Moser side, these dualities interchange the coordinates and
the spectral variables, providing a symmetry of the eigenfunction
$\Psi(x,\lambda)$ [R, FGNR]. The eigenfunction $\Psi(x,\lambda)$ turns out to
be an eigenfunction for two different Hamiltonians that are related through
bispectral transformation. On the spin chain side, there is an interesting
interchange between the inhomogeneities and the eigenvalues of the twist
operator. Specifically, the solution to the KZ equations for the
inhomogeneities serves as a simultaneous solution to the bispectrally dual,
also known as the dynamical or dual KZ equations, for the twist variables
[FMTV, OS, TV, EV1, MMPZZ]. Notably, the bispectral dualities exhibit
commutativity with both the Matsuo-Cherednik quantum-quantum duality and the
classical-quantum duality [GVZ].

The classical-quantum and Matsuo-Cherednik dualities can also be understood in
terms of the quantum geometry for some quiver varieties, for example
$T^{*}\operatorname{Gr}(k,n)$ for the trigonometric Ruijsenaars-Schneider
model and the dual qKZ equations or some of its degenerations. For the quantum
integrable system the relation with quantum geometry has been uncovered for
the first time for the non-relativistic Toda [GiK] and for relativistic Toda
system in [GL]. The $J$-functions for the partial flags, which are the
counting functions for quantum cohomology and quantum K-theory
correspondingly, turn out to be eigenfunctions of the quantum Toda chain
Hamiltonians[GiK, GL]. The similar relation has been found for Calogero-Moser
system [MTV2] and for different versions of the Ruijsenaar-Schneider system
[KPSZ, KZ]. Semiclassically the quantum K-theory rings of cotangent bundles to
flag varieties or flag varieties themselves, can be regarded as algebras of
functions on the Lagrangian subvarieties in the phase spaces of classical
models.

On the opposite spin chain side the first step towards the relation with the
quantum geometry has been done in [NS1, NS2] where it was shown explicitly
that the extremization equation for the twisted superpotential for
$T^{*}\operatorname{Gr}(k,n)$ yields the Bethe Anzatz equation for the
inhomogeneous XXZ spin chain. This observation has been generalized to the
statement that the vertex function for $T^{*}\operatorname{Gr}(k,n)$ counting
the quasimaps to the quiver manifold introduced and discussed in [O, AO] obeys
the qKZ equations. On the other hand the master function is identified with
the twisted superpotential in the integral solutions to qKZ. The similar
relations were found for different degenerations of
$T^{*}\operatorname{Gr}(k,n)$ [GiK, GL, MTV2, KPSZ, KZ, GLO].

Hence we have two related counting functions for the quiver variables - the
$J$-function and vertex function formulated in the different ways. They are
eigenfunctions of the Calogero-Moser, Ruijsenaars-Schneyder or Toda
differential operators and simultaneously can be constructed from solutions to
KZ or qKZ equations [OS, KPSZ, KZ, SmV2]. To some extend this situation can be
considered as a manifestation of the Matsuo-Cherednik duality in terms of the
quantum geometry. Note also that Matsuo-Cherednik duality can be considered as
the realization of the 3D mirror symmetry for the gauge theories on the brane
worldvolumes. It is formulated in terms of the proper motion of the
corresponding brane configurations taking into account the rules of brane
intersections [GK].

In this study, we are motivated by the recent derivation of the
$p^{s}$-truncated hypergeometric solutions for KZ equations [SV2, EV2]. We
employ the Matsuo-Cherednik duality, which connects the solutions to KZ
equations with the eigenfunctions of Calogero-Moser Hamiltonians. Our aim is
to obtain the polynomial $p^{s}$-truncation of the Calogero-Moser
eigenfunctions at a rational coupling constant. To accomplish this, we utilize
the integral representation for the hypergeometric solutions to the KZ
equation. Furthermore, we analyze the $s\to\infty$ limit, specifically in the
case where $n=2$, to examine the pure $p$-adic scenario.

###  1.2. Exposition of material

Let $p$ be an odd prime number and $n,s$ positive integers, $n\geqslant 2$. In
this paper we construct polynomials $\varphi_{s,d}(z,\lambda,\kappa)$ and
$\psi_{s,d}(z,\lambda,\kappa)$ in variables $z=(z_{1},\dots,z_{n})$,
$\lambda=(\lambda_{1},\dots,\lambda_{n})$, depending on a rational number
$\kappa$ and a discrete parameter $d$. The polynomials are eigenfunctions
modulo $p^{s}$ of the Calogero-Moser operators,

$\displaystyle\Big{(}-\Delta+\kappa(\kappa-1)\sum_{i\neq
j}\frac{1}{(z_{i}-z_{j})^{2}}\Big{)}\varphi\equiv
p^{2}E\varphi\,\pmod{p^{s}},$
$\displaystyle\Big{(}-\Delta+\kappa(\kappa+1)\sum_{i\neq
j}\frac{1}{(z_{i}-z_{j})^{2}}\Big{)}\psi\equiv p^{2}E\psi\,\pmod{p^{s}},$

where $\Delta=\partial_{1}^{2}+\dots+\partial_{n}^{2}$,
$\partial_{i}=\frac{\partial}{\partial z_{i}}$, $i=1,\dots,n$,
$E=-(\lambda_{1}^{2}+\dots+\lambda_{n}^{2})$, see Corollary 2.6.

The polynomials $\psi_{s,d}$ and $\psi_{s,d}$ are derived using the Matsuo-
Cherednik procedure applied to polynomial solutions modulo $p^{s}$ of the KZ
equations. Such solutions modulo $p^{s}$ have been constructed recently in
[EV2, SV2]. The original Matsuo-Cherednik procedure in [Ma, Ch] begins with a
solution $I(z,\lambda,\kappa)$ of appropriate KZ equations and generates two
eigenfunctions $\varphi(z,\lambda,\kappa)$ and $\psi(z,\lambda,\kappa)$ of the
Calogero-Moser operator shown below:

$\displaystyle\Big{(}-\Delta+\kappa(\kappa-1)\sum_{i\neq
j}\frac{1}{(z_{i}-z_{j})^{2}}\Big{)}\varphi$ $\displaystyle=$ $\displaystyle
E\varphi\,,$ $\displaystyle\Big{(}-\Delta+\kappa(\kappa+1)\sum_{i\neq
j}\frac{1}{(z_{i}-z_{j})^{2}}\Big{)}\psi$ $\displaystyle=$ $\displaystyle
E\psi\,.$

The process for constructing the polynomials $\varphi_{s,d}$ and $\psi_{s,d}$
is detailed in Section 2. For a given tuple $(p,n,\kappa,s)$, this
construction yields a finite set of polynomials $\varphi_{s,d}$ and
$\psi_{s,d}$.

In Theorem 2.3, an important new integral formula is introduced for the
eigenfunction $\psi$ in the Matsuo-Cherednik construction when applied to a
hypergeometric integral solution of the KZ equations. The original Matsuo-
Cherednik formula expresses $\psi$ as a summation of $n!$ terms, while the
formula in (2.5) represents $\psi$ as a single integral involving a product of
elementary functions. This formula bears resemblance to the vertex integral
formulas found in [FSTV, SmV1, SmV2], which are associated with 3D mirror
symmetry. A $p^{s}$-analog of this formula can also be found in Corollary 2.8.

We closely examine the case $n=2$ in Section 3. In this situation,
$z=(z_{1},z_{2})$, $\lambda=(\lambda_{1},\lambda_{2})$. The parameter $d$ in
$\varphi_{s,d}(z,\lambda,\kappa)$ and $\psi_{s,d}(z,\lambda,\kappa)$ is a
positive integer. We assume that

(1.1) $\displaystyle
p=kq+1,\qquad\kappa=\frac{r}{q}+m\,,\qquad\frac{1}{2}<\frac{r}{q}<1,$

where $k,q,r,m$ are integers, $k,r,q$ are positive and $r,q$ are relatively
prime. We demonstrate that if $s$ is sufficiently large, then the polynomials
$\varphi_{s,d}$ and $\psi_{s,d}$ become zero for $d>1$ and the polynomials
$\psi_{s,1}$ and $\psi_{s,1}$ are nonzero, see Theorem 3.2.

The polynomials $\varphi_{s,1}(z,\lambda,\kappa)$ and
$\psi_{s,1}(z,\lambda,\kappa-1)$ are eigenfunctions modulo $p^{s}$ of the same
Calogero-Moser operator with the same eigenvalue. In Theorem 3.4, we show that

(1.2)
$\displaystyle(\lambda_{2}-\lambda_{1})\,p\,\psi_{s,1}(z,\lambda,\kappa-1)\,\equiv\,(\kappa+m)\,\varphi_{s,1}(z,\lambda,\kappa)\pmod{p^{s}},$

in other words, these eigenfunctions modulo $p^{s}$ are proportional. Thus,
for sufficiently large $s$, our construction give exactly one nonzero
eigenfunction modulo $p^{s}$ up to a scalar factor.

In Theorem 3.4, we show that our eigenfunctions agree with the shift operator
$D_{\kappa}=\partial_{1}-\partial_{2}-\frac{2\kappa}{z_{1}-z_{2}}$, namely,

$\displaystyle
D_{\kappa}\psi_{s,d}(z,\lambda,\kappa-1)\,\equiv\,\kappa\,\psi_{s,d}(z,\lambda,\kappa)\pmod{p^{s}}.$

In Section 4, we consider the polynomials $\varphi_{s,1}(z,\lambda,\kappa)$
for $\kappa=\frac{r}{q}$ as in (1.1) and study the $p$-adic limit of these
polynomials as $s\to\infty$. After a suitable normalization, we present the
polynomials $\varphi_{s,1}(z,\lambda,\kappa)$ as polynomials $M^{0}_{s}(x)$ in
one variable $x=(\lambda_{2}-\lambda_{1})(z_{1}-z_{2})$. In Theorem 4.2, we
show that the sequence of polynomials $\\{M^{0}_{s}(x)\\}$ uniformly
$p$-adically converges on ${\mathbb{Z}}_{p}$ as $s\to\infty$ to the function

$\displaystyle M(x)\ =\
\sum_{{n=0}}^{\infty}\frac{\big{(}-\frac{r}{q}\big{)}_{n}}{\big{(}-\frac{2r}{q}\big{)}_{n}\,n!}\,p^{n}x^{n}$

analytic on ${\mathbb{Z}}_{p}$. This function $M(x)$ is a special case of
Kummer’s (confluent hypergeometric) function $M(a,b;x)$,

$\displaystyle
M(a,b;x)={\frac{\Gamma(b)}{\Gamma(a)\Gamma(b-a)}}\sum_{{n=0}}^{\infty}\frac{\Gamma(a+n)x^{n}}{\Gamma(b+n)n!}=\sum_{{n=0}}^{\infty}\frac{(a)_{n}x^{n}}{(b)_{n}n!}={}_{1}F_{1}(a;b;x),$

where $(a)_{n}=a(a+1)\dots(a+n-1)$.

In Section 5.1 we discuss a parallel between the roles of the reciprocal
$1/\hbar$ of the Planck constant in the conventional quantum mechanics and the
prime number $p$ in our $p^{s}$-approximations of the eigenfunctions of the
Calogero-Moser operator. In Section 5.2 we consider the quasiclassical limit
modulo $p^{s}$ of the Matsuo-Cherednik duality for $n=2$. In Section 6 we
discuss some open problems.

### Acknowledgments

The authors are grateful to A. Adolphson and S. Sperber for their consultation
on the inequality (4.10). The authors thank IHES for hospitality in summer
2023.

##  2\. Hypergeometric integrals

###  2.1. Master function

Fix a positive integer $n$. Let $\lambda=(\lambda_{1},\dots,\lambda_{n})$,
$z=(z_{1},\dots,z_{n})$,

$\displaystyle t=(t_{i,j}),\qquad i=1,\dots,n-1,\quad j=1,\dots,n-i.$

The total number of variables $t_{i,j}$ equals $\frac{n(n-1)}{2}$. For
example, for $n=2$, we have $t=(t_{1,1})$ and for $n=3$,
$t=(t_{1,1},t_{1,2},t_{2,1})$.

Define the master function by the formula

$\displaystyle\Phi(t,z,\lambda,\kappa)$ $\displaystyle=$ $\displaystyle
e^{\lambda_{1}\sum_{a=1}^{n}z_{a}+\sum_{i=1}^{n-1}(\lambda_{i+1}-\lambda_{i})\sum_{j=1}^{n-i}t_{i,j}}\prod_{1\leqslant
a<b\leqslant
n}(z_{a}-z_{b})^{\kappa}\prod_{j=1}^{n-1}\prod_{a=1}^{n}(t_{1,j}-z_{a})^{-\kappa}$
$\displaystyle\times$ $\displaystyle\prod_{i=1}^{n-2}\prod_{1\leqslant
j<j^{\prime}\leqslant
n-i}(t_{i,j}-t_{i,j^{\prime}})^{2\kappa}\prod_{i=1}^{n-2}\prod_{j=1}^{n-i}\prod_{j^{\prime}=1}^{n-i-1}(t_{i,j}-t_{i+1,j^{\prime}})^{-\kappa}.$

###  2.2. Symmetrization

For a function $f(x_{1},\dots,x_{k})$ define

$\displaystyle\operatorname{Sym}_{x_{1},\dots,x_{k}}f=\sum_{\sigma\in
S_{k}}f(x_{\sigma(1)},\dots,x_{\sigma(k)}).$

###  2.3. Weight functions

Consider ${\mathbb{C}}^{n}$ as the $\mathfrak{gl}_{n}$-module of highest
weight $(1,0,\dots,0)$ with the standard weight basis $v_{1},\dots,v_{n}$. We
have $e_{i,i}v_{j}=\delta_{i,j}v_{j}$. Let

$\displaystyle({\mathbb{C}}^{n})^{\otimes n}[1,\dots,1]\ \subset\
({\mathbb{C}}^{n})^{\otimes n}$

be the weight subspace of $({\mathbb{C}}^{n})^{\otimes n}$ of weight
$(1,\dots,1)$. A basis of $({\mathbb{C}}^{n})^{\otimes n}[1,\dots,1]$ is given
by the vectors $v_{\sigma}$, $\sigma\in S_{n}$, where

$\displaystyle v_{\sigma}=v_{\sigma(1)}\otimes\dots\otimes v_{\sigma(n)}\,.$

For every $\sigma\in S_{n}$ we construct a scalar weight function
$V_{\sigma}(t,z)$ as follows. Define

$\displaystyle
V^{\circ}(t,z)=\prod_{a=1}^{n-1}\frac{1}{(t_{1,n-a}-z_{a+1})(t_{2,n-a}-t_{1,n-a})(t_{3,n-a}-t_{2,n-a})\dots(t_{a,n-a}-t_{a-1,n-a})}\,,$
$\displaystyle
V(t,z)=\operatorname{Sym}_{t_{1,1},\dots,t_{1,n-1}}\operatorname{Sym}_{t_{2,1},\dots,t_{2,n-2}}\dots\operatorname{Sym}_{t_{n-2,1},t_{n-2,2}}V^{\circ}(t,z).$

For example, for $n=2$,

$\displaystyle V(t_{1,1},z_{1},z_{2})=\frac{1}{t_{1,1}-z_{2}}\,,$

for $n=3$,

$\displaystyle V^{\circ}(t_{1,1},t_{1,2},t_{2,1},z_{1},z_{2},z_{3})\,$
$\displaystyle=$
$\displaystyle\,\frac{1}{t_{1,2}-z_{2}}\,\frac{1}{(t_{1,1}-z_{3})(t_{2,1}-t_{1,1})}\,.$
$\displaystyle V(t_{1,1},t_{1,2},t_{2,1},z_{1},z_{2},z_{3})\,$
$\displaystyle=$
$\displaystyle\,\frac{1}{t_{1,2}-z_{2}}\,\frac{1}{(t_{1,1}-z_{3})(t_{2,1}-t_{1,1})}\,$
$\displaystyle+$
$\displaystyle\,\frac{1}{t_{1,1}-z_{2}}\,\frac{1}{(t_{1,2}-z_{3})(t_{2,1}-t_{1,2})}\,.$

For $\sigma\in S_{n}$, we define

$\displaystyle
V_{\sigma}(t,z_{1},\dots,z_{n})=V(t,z_{\sigma^{-1}(1)},\dots,z_{\sigma^{-1}(n)}),$

and

(2.1) $\displaystyle
I^{(\gamma)}_{\sigma}(z,\lambda,\kappa)=\int_{\gamma}\Phi(t,z,\lambda,\kappa)V_{\sigma}(t,z)dt\,$

where $\gamma$ is an $\frac{n(n-1)}{2}$-dimensional cycle and

$\displaystyle dt=\wedge_{i,j}dt_{i,j}$

is the product in the lexicographical order of all differentials $dt_{i,j}$.

###  2.4. Hypergeometric solutions

Denote

$\displaystyle\partial_{i}=\frac{\partial}{\partial z_{i}},\quad
i=1,\dots,n,\quad\quad\Delta$ $\displaystyle=$
$\displaystyle\partial_{1}^{2}+\dots+\partial_{n}^{2}.$

######  Theorem 2.1 ([SV1, FMTV]).

The vector

$\displaystyle I^{(\gamma)}(z,\lambda,\kappa)=\sum_{\sigma\in
S_{n}}I^{(\gamma)}_{\sigma}(z,\lambda,\kappa)v_{\sigma}$

is a solutions of the KZ equations,

(2.2) $\displaystyle\partial_{i}I=\Big{(}\kappa\sum_{j\neq
i}\frac{P^{(i,j)}}{z_{i}-z_{j}}+\lambda^{(i)}\Big{)}I,\qquad i=1,\dots,n.$

The $\lambda^{(i)}$ is the operator acting as a diagonal matrix in the i-th
factor and identically on all other factors in the tensor product.

######  Theorem 2.2 ([Ma, Ch]).

Let $I=\sum_{\sigma\in S_{n}}I_{\sigma}v_{\sigma}$ be a solution of the KZ
equations. Define

$\displaystyle\operatorname{m}(I)=\sum_{\sigma\in
S_{n}}I_{\sigma}\,,\qquad\operatorname{ch}(I)=\sum_{\sigma\in
S_{n}}(-1)^{\sigma}I_{\sigma}\,.$

Then $\varphi=\operatorname{m}(I)$ and $\psi=\operatorname{ch}(I)$ are
eigenfunctions of the Calogero-Moser operators,

(2.3) $\displaystyle\Big{(}-\Delta+\kappa(\kappa-1)\sum_{i\neq
j}\frac{1}{(z_{i}-z_{j})^{2}}\Big{)}\varphi=E\varphi\,,$ (2.4)
$\displaystyle\Big{(}-\Delta+\kappa(\kappa+1)\sum_{i\neq
j}\frac{1}{(z_{i}-z_{j})^{2}}\Big{)}\psi=E\psi\,,$

where $E=-(\lambda_{1}^{2}+\dots+\lambda_{n}^{2})$.

###  2.5. An integral formula for $\operatorname{ch}(I^{(\gamma)})$

######  Theorem 2.3.

We have

(2.5)
$\displaystyle\operatorname{ch}(I^{(\gamma)})(z,\lambda,\kappa)=\int_{\gamma}\Phi(t,z,\lambda,\kappa+1)dt.$

This formula is analogous to the vertex integral formulas in [FSTV, SmV1,
SmV2].

###### Proof.

Formula (2.5) is equivalent to the formula

(2.6) $\displaystyle\sum_{\sigma\in
S_{n}}(-1)^{\sigma}V_{\sigma}(t,z)=\frac{\prod_{1\leqslant a<b\leqslant
n}(z_{a}-z_{b})\prod_{i=1}^{n-2}\prod_{1\leqslant j<j^{\prime}\leqslant
n-i}(t_{i,j}-t_{i,j^{\prime}})^{2}}{\prod_{j=1}^{n-1}\prod_{a=1}^{n}(t_{1,j}-z_{a})\prod_{i=1}^{n-2}\prod_{j=1}^{n-i}\prod_{j^{\prime}=1}^{n-i-1}(t_{i,j}-t_{i+1,j^{\prime}})}.$

For example, for $n=3$ we have

$\displaystyle\frac{1}{(t_{1,2}-z_{2})(t_{1,1}-z_{1})(t_{2,1}-t_{1,1})}+\frac{1}{(t_{1,1}-z_{2})(t_{1,2}-z_{1})(t_{2,1}-t_{1,2})}$
$\displaystyle-$
$\displaystyle\frac{1}{(t_{1,2}-z_{1})(t_{1,1}-z_{2})(t_{2,1}-t_{1,1})}+\frac{1}{(t_{1,1}-z_{1})(t_{1,2}-z_{2})(t_{2,1}-t_{1,2})}$
$\displaystyle-$
$\displaystyle\frac{1}{(t_{1,2}-z_{2})(t_{1,1}-z_{3})(t_{2,1}-t_{1,1})}+\frac{1}{(t_{1,1}-z_{2})(t_{1,2}-z_{3})(t_{2,1}-t_{1,2})}$
$\displaystyle-$
$\displaystyle\frac{1}{(t_{1,2}-z_{3})(t_{1,1}-z_{1})(t_{2,1}-t_{1,1})}+\frac{1}{(t_{1,1}-z_{3})(t_{1,2}-z_{1})(t_{2,1}-t_{1,2})}$
$\displaystyle+$
$\displaystyle\frac{1}{(t_{1,2}-z_{3})(t_{1,1}-z_{2})(t_{2,1}-t_{1,1})}+\frac{1}{(t_{1,1}-z_{3})(t_{1,2}-z_{2})(t_{2,1}-t_{1,2})}$
$\displaystyle-$
$\displaystyle\frac{1}{(t_{1,2}-z_{1})(t_{1,1}-z_{3})(t_{2,1}-t_{1,1})}+\frac{1}{(t_{1,1}-z_{1})(t_{1,2}-z_{3})(t_{2,1}-t_{1,2})}$
$\displaystyle=$ $\displaystyle\frac{(t_{1,1}-t_{1,2})^{2}\prod_{1\leqslant
a<b\leqslant
3}(z_{a}-z_{b})}{(t_{2,1}-t_{1,1})(t_{2,1}-t_{1,2})\prod_{j=1}^{2}\prod_{a=1}^{3}(t_{1,j}-z_{a})}\,.$

Denote the left-hand in (2.6) by $L$. It is as a ratio of two polynomials
$P/Q$, where each of them is a homogeneous polynomial in variables $t,z$. We
may write

$\displaystyle
Q=\prod_{j=1}^{n-1}\prod_{a=1}^{n}(t_{1,j}-z_{a})\prod_{i=1}^{n-2}\prod_{j=1}^{n-i}\prod_{j^{\prime}=1}^{n-i-1}(t_{i,j}-t_{i+1,j^{\prime}})$

and $\deg Q-\deg P=\frac{n(n-1)}{2}$. It is easy to see that $L$ has the
following properties:

  1. (1)

for every $i=1,\dots,n-2$, the left-hand side $L$ is symmetric with respect to
permutations of the variables $t_{i,1},\dots,t_{i,n-i}$;

  2. (2)

if $t_{i,j}=t_{i,j^{\prime}}$ for some $i,j,j^{\prime}$ with $j\neq
j^{\prime}$, then $L$ equals zero;

  3. (3)

$L$ is anti-symmetric with respect to permutations of the variables
$z_{1},\dots,z_{n}$.

These remarks imply that

$\displaystyle P=C\prod_{1\leqslant a<b\leqslant
n}(z_{a}-z_{b})\prod_{i=1}^{n-2}\prod_{1\leqslant j<j^{\prime}\leqslant
n-i}(t_{i,j}-t_{i,j^{\prime}})^{2}$

where $C$ is a constant. We observe that $C=1$ by calculating any iterated
residue of both sides in (2.6). The theorem is proved. ∎

###  2.6. Reformulation of the equations

Change the variables $\lambda\mapsto p\lambda$ where $p$ is an odd prime
number111One can make a more general change of variables $\lambda\to
p^{r}\lambda$ where $r>1/(p-1)$. Then all the results below remain true for
this more general rescaling, cf. [EV2].. Then the KZ equations (2.2) take the
form

(2.7) $\displaystyle\partial_{i}I=\Big{(}\kappa\sum_{j\neq
i}\frac{P^{(i,j)}}{z_{i}-z_{j}}+p\lambda^{(i)}\Big{)}I,\qquad i=1,\dots,n,$

and the Calogero-Moser equations (2.3) and (2.4) take the form

(2.8) $\displaystyle\Big{(}-\Delta+\kappa(\kappa-1)\sum_{i\neq
j}\frac{1}{(z_{i}-z_{j})^{2}}\Big{)}\varphi=p^{2}E\varphi\,,$ (2.9)
$\displaystyle\Big{(}-\Delta+\kappa(\kappa+1)\sum_{i\neq
j}\frac{1}{(z_{i}-z_{j})^{2}}\Big{)}\psi=p^{2}E\psi\,.$

###  2.7. Truncated exponential function

Let $p$ be an odd prime number, $s$ a positive integer. Denote

(2.10) $\displaystyle d(s)=\Big{[}s\frac{p-1}{p-2}\Big{]}+1,\qquad
E_{s}(x)=\sum_{k=0}^{d(s)}\,\frac{x^{k}}{k!}\,.$

######  Lemma 2.4 ([EV2]).

If $\mu\in{\mathbb{Z}}_{p}$, then $e^{p\mu x}\in{\mathbb{Z}}_{p}[[x]],$
$E_{s}(p\mu x)\in{\mathbb{Z}}_{p}[x]$ and

$\displaystyle e^{p\mu x}$ $\displaystyle\equiv$ $\displaystyle E_{s}(p\mu
x)\pmod{p^{s}},$ (2.11) $\displaystyle\frac{\partial}{\partial x}E_{s}(p\mu
x)$ $\displaystyle\equiv$ $\displaystyle p\mu E_{s}(p\mu
x),\qquad\frac{\partial}{\partial\mu}E_{s}(p\mu x)\equiv pxE_{s}(p\mu
x)\pmod{p^{s}},$ (2.12) $\displaystyle E_{s}(p\mu(u+v))$ $\displaystyle\equiv$
$\displaystyle E_{s}(p\mu u)E_{s}(p\mu v)\pmod{p^{s}}.$

###  2.8. The $p^{s}$-approximations

Let

(2.13) $\displaystyle\kappa=\frac{r}{q}\,,\qquad\frac{1}{2}<\frac{r}{q}<1,$

where $r,q$ are relatively prime positive integers. Hence $q\geqslant 3$. Let
$p$ be a prime number of the form

(2.14) $\displaystyle p=kq+1$

for some positive integer $k$. Let $s$ be a positive integer.

Define

$\displaystyle\mathcal{E}_{s}(t,z,\lambda)=E_{s}\Big{(}p\lambda_{1}\sum_{a=1}^{n}z_{a}+\sum_{i=1}^{n-1}p(\lambda_{i+1}-\lambda_{i})\sum_{j=1}^{n-i}t_{i,j}\Big{)}.$

Define the master polynomial $\Phi_{s}(t,z,\lambda,\kappa)$ by the formula

$\displaystyle\phantom{aaa}\Phi_{s}(t,z,\lambda,\kappa)$ $\displaystyle=$
$\displaystyle\mathcal{E}_{s}(t,z,\lambda)\prod_{1\leqslant a<b\leqslant
n}(z_{a}-z_{b})^{\frac{(q-r)p^{s}+r}{q}}\prod_{j=1}^{n-1}\prod_{a=1}^{n}(t_{1,j}-z_{a})^{r\frac{p^{s}-1}{q}}$
$\displaystyle\times$ $\displaystyle\prod_{i=1}^{n-2}\prod_{1\leqslant
j<j^{\prime}\leqslant
n-i}(t_{i,j}-t_{i,j^{\prime}})^{2\frac{(q-r)p^{s}+r}{q}}\prod_{i=1}^{n-2}\prod_{j=1}^{n-i}\prod_{j^{\prime}=1}^{n-i-1}(t_{i,j}-t_{i+1,j^{\prime}})^{r\frac{p^{s}-1}{q}}.$

Set

$\displaystyle\Phi_{s,\sigma}(t,z,\lambda,\kappa)=\Phi_{s}(t,z,\lambda,\kappa)V_{\sigma}(t,z).$

Note that $\Phi_{s,\sigma}(t,z,\lambda,\kappa)$ is a polynomial in
$t,z,\lambda$.

Let $d=(d_{i,j})$ be a vector of positive integers. Denote by

$\displaystyle I_{s,\sigma,d}(z,\lambda,\kappa)$

the coefficient of the monomial $\prod_{i,j}t_{i,j}^{d_{i,j}p^{s}-1}$ in the
polynomial $\Phi_{s,\sigma}(t,z,\lambda,\kappa)$.

######  Theorem 2.5 ([EV2, SV2]).

For any $d$, the vector

$\displaystyle I_{s,d}(z,\lambda,\kappa)=\sum_{\sigma\in
S_{n}}I_{s,\sigma,d}(z,\lambda,\kappa)v_{\sigma}$

is a solution modulo $p^{s}$ of the KZ equations (2.7) with
$\kappa=\frac{r}{q}$,

(2.16) $\displaystyle\partial_{i}I\equiv\Big{(}\kappa\sum_{j\neq
i}\frac{P^{(i,j)}}{z_{i}-z_{j}}+p\lambda^{(i)}\Big{)}I\pmod{p^{s}},\qquad
i=1,\dots,n.$

######  Corollary 2.6.

The polynomials $\varphi_{s,d}=\operatorname{m}(I_{s,d})$ and
$\psi_{s,d}=\operatorname{ch}(I_{s,d})$ are eigenfunctions modulo $p^{s}$ of
the corresponding Calogero-Moser operators (2.8) and (2.9) with
$\kappa=\frac{r}{q}$ ,

(2.17) $\displaystyle\Big{(}-\Delta+\kappa(\kappa-1)\sum_{i\neq
j}\frac{1}{(z_{i}-z_{j})^{2}}\Big{)}\varphi_{s,d}(z,\lambda,\kappa)\equiv
p^{2}E\,\varphi_{s,d}(z,\lambda,\kappa)\pmod{p^{s}},$ (2.18)
$\displaystyle\Big{(}-\Delta+\kappa(\kappa+1)\sum_{i\neq
j}\frac{1}{(z_{i}-z_{j})^{2}}\Big{)}\psi_{s,d}(z,\lambda,\kappa)\equiv
p^{2}E\,\psi_{s,d}(z,\lambda,\kappa)\pmod{p^{s}},$

where $E=-(\lambda_{1}^{2}+\dots+\lambda_{n}^{2})$ .

Notice that $\Phi_{s,\sigma}(t,z,\lambda,\kappa)$ is a polynomial in $t$ of
degree independent on $\sigma$. Given $s$, there are at most finitely many
vectors $d=(d_{i,j})$ which produce nonzero vectors
$I_{s,d}(z,\lambda,\kappa)$. See below some remarks on the $n=2$ case.

###  2.9. The $p^{s}$-approximations for $\kappa=\frac{r}{q}+m$,
$m\in{\mathbb{Z}}$

In Section 2.8 we assumed that $p=kq+1$ and $\kappa=\frac{r}{q}\,,$ where
$\frac{1}{2}<\frac{r}{q}<1$, see (2.13) and (2.14), and constructed solutions
modulo $p^{s}$ of the KZ equations and eigenfunctions modulo $p^{s}$ of the
Calogero-Moser operators. We may modify this construction under the
assumptions that

(2.19)
$\displaystyle\kappa=\frac{r}{q}+m\,,\qquad\frac{1}{2}<\frac{r}{q}<1,\qquad
p=kq+1,$

where $r,q$ are relatively prime positive integers and $m$ is an integer.

Let $s$ be a positive integer. Under assumption (2.19) we define the master
polynomial $\Phi_{s}(t,z,\lambda,\kappa)$ by the formula

$\displaystyle\Phi_{s}(t,z,\lambda,\kappa)$ $\displaystyle=$
$\displaystyle\mathcal{E}_{s}(t,z,\lambda)\prod_{1\leqslant a<b\leqslant
n}(z_{a}-z_{b})^{\frac{(q-r)p^{s}+r}{q}+m}\prod_{j=1}^{n-1}\prod_{a=1}^{n}(t_{1,j}-z_{a})^{r\frac{p^{s}-1}{q}-m}$
$\displaystyle\times$ $\displaystyle\prod_{i=1}^{n-2}\prod_{1\leqslant
j<j^{\prime}\leqslant
n-i}(t_{i,j}-t_{i,j^{\prime}})^{2\frac{(q-r)p^{s}+r}{q}+2m}\prod_{i=1}^{n-2}\prod_{j=1}^{n-i}\prod_{j^{\prime}=1}^{n-i-1}(t_{i,j}-t_{i+1,j^{\prime}})^{r\frac{p^{s}-1}{q}-m}.$

Set

$\displaystyle\Phi_{s,\sigma}(t,z,\lambda,\kappa)=\Phi_{s}(t,z,\lambda,\kappa)V_{\sigma}(t,z).$

We assume that $s$ is such that

(2.21) $\displaystyle
p^{s}+(1-p^{s})\frac{r}{q}+m>0\qquad\operatorname{and}\qquad(p^{s}-1)\frac{r}{q}-m>0.$

Then the functions $\Phi_{s,\sigma}(t,z,\lambda,\kappa)$ are polynomials.

Let $d=(d_{i,j})$ be a vector of positive integers. Denote by
$I_{s,\sigma,d}(z,\lambda,\kappa)$ the coefficient of the monomial
$\prod_{i,j}t_{i,j}^{d_{i,j}p^{s}-1}$ in the polynomial
$\Phi_{s,\sigma}(t,z,\lambda,\kappa)$.

######  Theorem 2.7.

Under assumptions (2.19) and (2.21), for any positive integer $d$ the vector
$I_{s,d}(z,\lambda,\kappa)=\sum_{\sigma\in
S_{n}}I_{s,\sigma,d}(z,\lambda,\kappa)v_{\sigma}$ is a solution modulo $p^{s}$
of the KZ equations (2.7) with $\kappa=\frac{r}{q}+m$, and the polynomials
$\varphi_{s,d}(z,\lambda,\kappa)=\sum_{\sigma\in
S_{n}}I_{s,\sigma,d}(z,\lambda,\kappa)$ and
$\psi_{s,d}(z,\lambda,\kappa)=\sum_{\sigma\in
S_{n}}(-1)^{\sigma}I_{s,\sigma,d}(z,\lambda,\kappa)$ are eigenfunctions modulo
$p^{s}$ of the Calogero-Moser operators (2.17) and (2.18), respectively, with
$\kappa=\frac{r}{q}+m$.

The proof of this theorem is the same as the proof of Theorem 2.5.

We also have the following corollary of Theorem 2.3.

######  Corollary 2.8.

The polynomial $\psi_{s,d}(z,\lambda,\kappa)$ equals the coefficient of the
monomial
$\prod_{i,j}t_{i,j}^{d_{i,j}p^{s}-1}$ in the polynomial
$\Phi_{s}(t,z,\lambda,\kappa+1)$.

##  3\. The case $n=2$

###  3.1. Hypergeometric solutions

For $n=2$, we have $\lambda=(\lambda_{1},\lambda_{2})$, $z=(z_{1},z_{2})$. The
hypergeometric solutions of the KZ equations (2.16) involve a single
integration variable $t=t_{1,1}$. The master function is

(3.1)
$\displaystyle\Phi(t,z,p\lambda,\kappa)=e^{p\lambda_{1}(z_{1}+z_{2})+p(\lambda_{2}-\lambda_{1})t}(z_{1}-z_{2})^{\kappa}(t-z_{1})^{-\kappa}(t-z_{2})^{-\kappa}.$

The hypergeometric solutions have two coordinates,

$\displaystyle
I^{(\gamma)}(z,p\lambda,\kappa)=\left(I^{(\gamma)}_{1}(z,p\lambda,\kappa),\,I^{(\gamma)}_{2}(z,p\lambda,\kappa)\right),$

where

(3.2) $\displaystyle
I^{(\gamma)}_{i}(z,p\lambda,\kappa)=\int_{\gamma}\Phi(t,z,p\lambda,\kappa)\frac{dt}{t-z_{i}}\,,\qquad
i=1,2.$

Then

$\displaystyle\varphi^{(\gamma)}=I^{(\gamma)}_{1}(z,p\lambda,\kappa)+I^{(\gamma)}_{2}(z,p\lambda,\kappa),\quad\psi^{(\gamma)}=I^{(\gamma)}_{1}(z,p\lambda,\kappa)-I^{(\gamma)}_{2}(z,p\lambda,\kappa)$

are eigenfunctions of the Calogero-Moser operators,

(3.3)
$\displaystyle\Big{(}-\Delta+\frac{2\kappa(\kappa-1)}{(z_{1}-z_{2})^{2}}\Big{)}\varphi=p^{2}E\varphi\,,$
(3.4)
$\displaystyle\Big{(}-\Delta+\frac{2\kappa(\kappa+1)}{(z_{1}-z_{2})^{2}}\Big{)}\psi=p^{2}E\psi\,,$

where $\Delta=\partial_{1}^{2}+\partial_{2}^{2}$,
$E=-\lambda_{1}^{2}-\lambda_{2}^{2}$.

###  3.2. The $p^{s}$-approximations

Let $\kappa=\frac{r}{q}+m$ satisfies assumptions (2.19). Define

$\displaystyle\mathcal{E}_{s}(t,z,\lambda)$ $\displaystyle=$ $\displaystyle
E_{s}\big{(}p\lambda_{1}(z_{1}+z_{2})+p(\lambda_{2}-\lambda_{1})t\big{)},$
$\displaystyle\Phi_{s}(t,z,\lambda,\kappa)$ $\displaystyle=$
$\displaystyle\mathcal{E}_{s}(t,z,\lambda)(z_{1}-z_{2})^{p^{s}+(1-p^{s})\frac{r}{q}+m}\big{(}(t-z_{1})(t-z_{2})\big{)}^{(p^{s}-1)\frac{r}{q}-m},$
$\displaystyle\Phi_{s,i}(t,z,\lambda,\kappa)$ $\displaystyle=$
$\displaystyle\Phi_{s}(t,z,\lambda,\kappa)\frac{1}{t-z_{i}}\,,\quad i=1,2.$

Let $d$ be a positive integer. Denote by $I_{s,i,d}(z,\lambda)$ the
coefficient of the monomial $t^{dp^{s}-1}$ in the polynomial
$\Phi_{s,i}(t,z,\lambda,\kappa)$. Then for any positive integer $d$ the
polynomials

(3.5) $\displaystyle\varphi_{s,d}(z,\lambda,\kappa)$ $\displaystyle=$
$\displaystyle I_{s,1,d}(z,\lambda,\kappa)+I_{s,2,d}(z,\lambda,\kappa),$
$\displaystyle\psi_{s,d}(z,\lambda,\kappa)$ $\displaystyle=$ $\displaystyle
I_{s,1,d}(z,\lambda,\kappa)-I_{s,2,d}(z,\lambda,\kappa)$

are eigenfunctions modulo $p^{s}$ of the Calogero-Moser operators (2.8) and
(2.9) with $\kappa=\frac{r}{q}+m$,

(3.6)
$\displaystyle\Big{(}-\Delta+\frac{2\kappa(\kappa-1)}{(z_{1}-z_{2})^{2}}\Big{)}\varphi_{s,d}\,\equiv\,p^{2}E\varphi_{s,d}\pmod{p^{s}},$
(3.7)
$\displaystyle\Big{(}-\Delta+\frac{2\kappa(\kappa+1)}{(z_{1}-z_{2})^{2}}\Big{)}\psi_{s,d}\,\equiv\,p^{2}E\psi_{s,d}\,\pmod{p^{s}}.$

######  Lemma 3.1.

If

(3.8) $\displaystyle d>1\quad\operatorname{and}\quad
p^{s}>\frac{s-m}{1-\frac{r}{q}}\,,$

then $\varphi_{s,d}(z,\lambda,\kappa)$ and $\psi_{s,d}(z,\lambda,\kappa)$ are
zero polynomials.

###### Proof.

The degree of $\Phi_{s,i}(t,z,\lambda)$ with repsect to $t$ is not greater
than

$\displaystyle\Big{[}s\frac{p-1}{p-2}\Big{]}+1+2\frac{r}{q}(p^{s}-1)-2m-1\,,$

where $[\cdot]$ denotes the integer part. We have

$\displaystyle\Big{[}s\frac{p-1}{p-2}\Big{]}+1+2\frac{r}{q}(p^{s}-1)-2m-1-(2p^{s}-1)<\Big{(}2\frac{r}{q}-2\Big{)}p^{s}+2s-2m,$

and $(2\frac{r}{q}-2)p^{s}+2s-2m<0$ if the second inequality in (3.8) holds. ∎

######  Theorem 3.2.

  1. $\operatorname{(i)}$

Let $\kappa=\frac{r}{q}+m$ satisfy assumptions (2.19). Let a positive integer
$s$ satisfy assumptions (2.21). Then the $\varphi_{s,1}(z,\lambda,\kappa)$ is
divisible by $\lambda_{1}-\lambda_{2}$ modulo $p^{s}$ and is zero modulo $p$.

  2. $\operatorname{(ii)}$

Let $\kappa=\frac{r}{q}$ and $p$ satisfy assumptions (2.13) and (2.14). Then
the $\varphi_{s,1}(z,\lambda,\kappa)$ is nonzero modulo $p^{2}$ and the
polynomial $\psi_{s,1}(z,\lambda,\kappa)$ is nonzero modulo $p$.

###### Proof.

Let $\kappa=\frac{r}{q}+m$ . Let us prove that
$\varphi_{s,1}(z,\lambda,\kappa)$ is zero modulo $p$. It is enough to prove
that

(3.9)
$\displaystyle\varphi_{s,1}(z,0,\kappa)=I_{s,1,1}(z,0,\kappa)+I_{s,2,1}(z,0,\kappa)\equiv
0\pmod{p^{s}},$

since all powers of $\lambda$ come from
$E_{s}\big{(}p\lambda_{1}(z_{1}+z_{2})+p(\lambda_{2}-\lambda_{1})t\big{)}$
with positive powers of $p$. The congruence (3.9) follows from the fact that
the expression

$\displaystyle\Big{(}(p^{s}-1)\frac{r}{q}-m\Big{)}\Big{(}I_{s,1,1}(z,0,\kappa)+I_{s,2,1}(z,0,\kappa)\Big{)}$

equals the coefficient of $t^{p^{s}-1}$ in the polynomial
$\frac{\partial}{\partial t}\Phi_{s}(t,z,0,\kappa)$, and hence is congruent to
zero modulo $p^{s}$.

The fact that $\varphi_{s,1}(z,\lambda,\kappa)$ is divisible by
$\lambda_{1}-\lambda_{2}$ modulo $p^{s}$ also follows from the congruence
(3.9). Part (i) of the theorem is proved.

Let $\kappa=\frac{r}{q}$. Let us substitute $z=(z_{1},z_{2})=(0,-1)$,
$\lambda=(\lambda_{1},\lambda_{2})=(0,0)$ to $\psi_{s,1}$ and evaluate
$C=\psi_{s,1}(0,1,0,0,\kappa)$ modulo $p$. By Corollary 2.8 the number $C$ is
the coefficient of $t^{p^{s}-1}$ in
$t^{(p^{s}-1)\frac{r}{q}-1}(t+1)^{(p^{s}-1)\frac{r}{q}-1}$. Hence,
$C=\binom{A}{B}$, where $A=(p^{s}-1)\frac{r}{q}-1$,
$B=(p^{s}-1)\frac{q-r}{q}+1$.

Recall the assumptions: $\kappa=\frac{r}{q}\,,$ $\frac{1}{2}<\frac{r}{q}<1$
and $p=kq+1$. Then

$\displaystyle A=rk(1+p+\dots+p^{s-1})-1,\qquad
B=(q-r)k(1+p+\dots+p^{s-1})+1,$

where

$\displaystyle(q-r)k+1\leqslant rk-1<p-1.$

Applying Lucas’s theorem [L] we obtain

$\displaystyle\binom{A}{B}$ $\displaystyle\equiv$
$\displaystyle\binom{rk-1}{(q-r)k+1}\binom{rk-1}{(q-r)k}^{s-1}\not\equiv
0\pmod{p}.$

Thus $\psi_{s,1}(z,\lambda,\kappa)$ is not zero modulo $p$.

Assume that $\kappa=\frac{r}{q}$ and prove that the polynomial
$\varphi_{s,1}=I_{s,1,1}+I_{s,1,1}$ is nonzero modulo $p^{2}$. For that we
substitute $z=(z_{1},z_{2})=(0,-1)$,
$\lambda=(\lambda_{1},\lambda_{2})=(0,\lambda_{2})$ to $I_{s,1,1}$ and
$I_{s,2,1}$ and obtain two polynomials in $\lambda_{2}$. We denote by $D_{i}$
the coefficient of $\lambda_{2}$ in $I_{s,i,1}(0,-1,0,\lambda_{2})$ for
$i=1,2$. Then $D_{1}=p\binom{A+1}{B-1}$ and $D_{2}=p\binom{A}{B-2}$. Similarly
to the previous reasoning, Lucas’s theorem implies that $D_{1}+D_{2}\not\equiv
0\pmod{p^{2}}$. Theorem 3.2 is proved. ∎

###  3.3. Polynomials $\varphi_{s,d}(z,\lambda,\kappa)$ and
$\psi_{s,d}(z,\lambda,\kappa-1)$

The polynomials $\varphi_{s,d}(z,\lambda,\kappa)$ and
$\psi_{s,d}(z,\lambda,\kappa-1)$ are eigenfunctions modulo $p^{s}$ of the same
Calogero-Moser operator with the same eigenvalue,

$\displaystyle\Big{(}-\Delta+\frac{2\kappa(\kappa-1)}{(z_{1}-z_{2})^{2}}\Big{)}\mu\,\equiv\,p^{2}E\mu\,\pmod{p^{s}}.$

The next theorem shows that these eigenfunctions are proportional modulo
$p^{s}$.

######  Theorem 3.3.

If assumptions (2.19) and (2.21) hold for $\kappa=\frac{r}{q}+m$ and
$\kappa-1=\frac{r}{q}+m-1$, then

(3.10)
$\displaystyle(\lambda_{2}-\lambda_{1})\,p\,\psi_{s,d}(z,\lambda,\kappa-1)\,\equiv\,\kappa\,\varphi_{s,d}(z,\lambda,\kappa)\pmod{p^{s}}.$

###### Proof.

By Corollary 2.8, the polynomial $\psi_{s,d}(z,\lambda,\kappa-1)$ equals the
coefficient of $t^{dp^{s}-1}$ in $\Phi_{s}(t,z,\lambda,\kappa)$. We also have

$\displaystyle\frac{\partial}{\partial
t}\Phi_{s}(t,z,\lambda,\kappa)\equiv\Phi_{s}(t,z,\lambda,\kappa)\Big{(}(\lambda_{2}-\lambda_{1})p-\frac{\kappa}{t-z_{1}}-\frac{\kappa}{t-z_{2}}\Big{)}\pmod{p^{s}}.$

This gives (3.10), since the coefficient of $t^{dp^{s}-1}$ in
$\frac{\partial}{\partial t}\Phi_{s}(t,z,\lambda,\kappa)$ is congruent to 0
modulo $p^{s}$. ∎

###  3.4. Shift operator

Denote

$\displaystyle
L_{\kappa}=-\Delta+\frac{2\kappa(\kappa-1)}{(z_{1}-z_{2})^{2}}\,,\qquad
D_{\kappa}=\frac{\partial}{\partial z_{1}}-\frac{\partial}{\partial
z_{2}}-\frac{2\kappa}{z_{1}-z_{2}}\,.$

Then

$\displaystyle L_{\kappa+1}D_{\kappa}=D_{\kappa}L_{\kappa}.$

Hence, if $\varphi$ is an eigenfunction of $L_{\kappa}$ with some eigenvalue,
then $D_{\kappa}\varphi$ is an eigenfunction of $L_{\kappa+1}$ with the same
eigenvalue. The operator $D_{\kappa}$ is called the shift operator, see for
example [FVe].

For any positive integer $d$, the polynomial $\psi_{s,d}(z,\lambda,\kappa-1)$
is an eigenfunction modulo $p^{s}$ of $L_{\kappa}$ while the polynomial
$\psi_{s,d}(z,\lambda,\kappa-1)$ is an eigenfunction modulo $p^{s}$ of
$L_{\kappa+1}$.

######  Theorem 3.4.

We have

(3.11) $\displaystyle
D_{\kappa}\psi_{s,d}(z,\lambda,\kappa-1)\,\equiv\,\kappa\,\psi_{s,d}(z,\lambda,\kappa)\pmod{p^{s}}.$

###### Proof.

Direct calculation shows that

(3.12) $\displaystyle
D_{\kappa}\Phi_{s}(t,z,\lambda,\kappa)\equiv\kappa\Phi_{s}(t,z,\lambda,\kappa+1)\pmod{p^{s}}.$

Since $\psi_{s,d}(z,\lambda,\kappa-1)$ is the coefficient of $t^{dp^{s}-1}$ in
$\Phi_{s}(t,z,\lambda,\kappa)$, and $\psi_{s,d}(z,\lambda,\kappa)$ is the
coefficient of $t^{dp^{s}-1}$ in $\Phi_{s}(t,z,\lambda,\kappa+1)$, we obtain
(3.11). ∎

##  4\. The $p$-adic limit as $s\to\infty$ for $n=2$

###  4.1. Change of variables

Recall the eigenfunction equation (3.4). Its complex eigenfunctions are given
by the formula

(4.1)
$\displaystyle\psi^{(\gamma)}(z,\lambda,\kappa)=\int_{\gamma}e^{p\lambda_{1}(z_{1}+z_{2})+p(\lambda_{2}-\lambda_{1})t}(z_{1}-z_{2})^{\kappa+1}(t-z_{1})^{-\kappa-1}(t-z_{2})^{-\kappa-1}dt,$

see Theorem 2.3. Change the integration variable $t\to(z_{1}-z_{2})t+z_{2}$,
and choose $\gamma$, so that this function becomes

(4.2) $\displaystyle
e^{p(z_{1}\lambda_{1}+z_{2}\lambda_{2})}(z_{1}-z_{2})^{-\kappa}\int_{0}^{1}e^{p(\lambda_{2}-\lambda_{1})(z_{1}-z_{2})t}t^{-\kappa-1}(1-t)^{-\kappa-1}dt.$

The integral in (4.2) is related to Kummer’s (confluent hypergeometric)
function $M(a,b;x)$,

(4.3) $\displaystyle
M(a,b;x)={\frac{\Gamma(b)}{\Gamma(a)\Gamma(b-a)}}\sum_{{n=0}}^{\infty}\frac{\Gamma(a+n)x^{n}}{\Gamma(b+n)n!}=\sum_{{n=0}}^{\infty}\frac{(a)_{n}x^{n}}{(b)_{n}n!}={}_{1}F_{1}(a;b;x),$

where $(a)_{n}=a(a+1)\dots(a+n-1)$. Kummer’s function is a solution of
Kummer’s equation

$\displaystyle x{\frac{d^{2}w}{dx^{2}}}+(b-x){\frac{dw}{dx}}-aw=0$

and has an integral representation

$\displaystyle
M(a,b;x)={\frac{\Gamma(b)}{\Gamma(a)\Gamma(b-a)}}\int_{0}^{1}e^{{xt}}t^{{a-1}}(1-t)^{{b-a-1}}\,dt,$

if $\operatorname{Re}\,b>\operatorname{Re}a>0$. Thus,

(4.4)
$\displaystyle\int_{0}^{1}e^{p(\lambda_{2}-\lambda_{1})(z_{1}-z_{2})t}t^{-\kappa-1}(t+1)^{-\kappa-1}dt=\frac{\Gamma(-\kappa)^{2}}{\Gamma(-2\kappa)}M(-\kappa,-2\kappa;p(\lambda_{2}-\lambda_{1})(z_{1}-z_{2})).$

###  4.2. The $p$-adic limit

We $p^{s}$-approximate the integral in the left-hand side of (4.4) by
polynomials as in the previous sections and show that the corresponding monic
polynomials $p$-adically tend to the power series expansion in
$p(\lambda_{2}-\lambda_{1})(z_{1}-z_{2})$ of the function
$M(-\kappa,-2\kappa;p(\lambda_{2}-\lambda_{1})(z_{1}-z_{2}))$. The transition
from the $p^{s}$-approximating polynomials to the monic polynomials
corresponds to the prefactor $\frac{\Gamma(-\kappa)^{2}}{\Gamma(-2\kappa)}$ in
(4.4).

Notice that our $p^{s}$-approximation procedure does not require the use of
the integration cycle $\gamma$. Therefore, the $p^{s}$-approximation procedure
independently of any cycle identifies the solution to Kummer’s equation that
is analytic at $x=0$.

Denote $x=(\lambda_{2}-\lambda_{1})(z_{1}-z_{2})$ and

$\displaystyle M(x)$ $\displaystyle:=$ $\displaystyle
M(-\kappa,-2\kappa,px)=\frac{\Gamma(-2\kappa)}{\Gamma(-\kappa)^{2}}\int_{0}^{1}e^{pxt}t^{-\kappa-1}(t+1)^{-\kappa-1}dt$
$\displaystyle=$
$\displaystyle\sum_{{n=0}}^{\infty}\frac{(-\kappa)_{n}}{(-2\kappa)_{n}n!}\,p^{n}x^{n}=1+\frac{-\kappa}{-2\kappa}\,px+\frac{(-\kappa)(-\kappa+1)}{(-2\kappa)(-2\kappa+1)2}\,p^{2}x^{2}\,+\,\dots.$

Denote

(4.6) $\displaystyle c_{n}=\frac{(-\kappa)_{n}}{(-2\kappa)_{n}n!}.$

Let $p=kq+1$ and $\kappa=\frac{r}{q}$ satisfy (2.13) and (2.14). Denote by
$M_{s}(x)$ the coefficient of $t^{p^{s}-1}$ in

$\displaystyle\Psi_{s}(t)\,=\,E_{s}(pxt)\,t^{(p^{s}-1)\frac{r}{q}-1}(1-t)^{(p^{s}-1)\frac{r}{q}-1}\,.$

Then

(4.7) $\displaystyle
M_{s}(x)=\sum_{n=0}^{d(s)}(-1)^{(p^{s}-1)\frac{q-r}{q}-n+1}\binom{(p^{s}-1)\frac{r}{q}-1}{(p^{s}-1)\frac{q-r}{q}-n+1}p^{n}x^{n}.$

Define the normalized $p^{s}$-approximating polynomial:

(4.8) $\displaystyle
M^{0}_{s}(x):=\frac{M_{s}(x)}{M_{s}(0)}=\sum_{n=0}^{d(s)}c_{s,n}\,p^{n}x^{n},\qquad
c_{s,n}=\frac{\big{(}\frac{-r}{q}+p^{s}\frac{r-q}{q}\big{)}_{n}}{\big{(}\frac{-2r}{q}+p^{s}\frac{2r-q}{q}\big{)}_{n}\,n!}\,.$

######  Theorem 4.1.

For any $n$ the coefficient $c_{s,n}$ $p$-adically tends to $c_{n}$ and
$s\to\infty$.

###### Proof.

The statement follows since both $p^{s}\frac{2r-q}{q}$ and
$p^{s}\frac{r-q}{q}$ $p$-adically tend to zero as
$s\to\infty$. ∎

We have a stronger statement.

######  Theorem 4.2.

Let $p=kq+1$ and $\kappa=\frac{r}{q}$ satisfy (2.13) and (2.14). Then the
power series

(4.9) $\displaystyle M(x)\ =\
\sum_{{n=0}}^{\infty}\frac{\big{(}-\frac{r}{q}\big{)}_{n}}{\big{(}-\frac{2r}{q}\big{)}_{n}\,n!}\,p^{n}x^{n}$

$p$-adically uniformly converges for all $x\in{\mathbb{Z}}_{p}$ . The sequence
of polynomials $\big{(}M_{s}^{0}(x)\big{)}_{s\geqslant 0}$ uniformly
$p$-adically converges on ${\mathbb{Z}}_{p}$ to the analytic function $M(x)$.

###### Proof.

Recall that
$c_{n}=\frac{\big{(}-\frac{r}{q}\big{)}_{n}}{\big{(}-\frac{2r}{q}\big{)}_{n}\,n!}$.
Let $\operatorname{ord}_{p}a$ denote the $p$-adic order of $a$. It follows
from [Dw, Section 1], also from [AS, Section 5], that

(4.10)
$\displaystyle\operatorname{ord}_{p}r_{n}\,\geqslant\,-\,\frac{n}{p-1}\,-\,\log_{p}n\,-\,1.$

Hence

(4.11)
$\displaystyle\operatorname{ord}_{p}(c_{n}p^{n})\,\geqslant\,n\,\frac{p-2}{p-1}\,-\,\log_{p}n\,-\,1.$

Hence, given $d>0$, there exists $N_{0}>0$ such that

(4.12)
$\displaystyle\operatorname{ord}_{p}\left(\sum_{{n=N}}^{\infty}c_{n}p^{n}x^{n}\right)>{d}\,,$

for all $N\geqslant N_{0}$, if $\operatorname{ord}_{p}x\geqslant 0$. Thus, the
series $M(x)$ $p$-adically uniformly converges for all $x\in{\mathbb{Z}}_{p}$
.

The inequality (4.11) also implies that there exists a real number $A$ such
that $\operatorname{ord}_{p}(c_{n}p^{n})>A$ for all $n$.

Let us compare the coefficients $c_{n}$ and $c_{s,n}$. Under our assumptions
we have the following $p$-adic presentations,

(4.13)
$\displaystyle\phantom{aaaaaaa}\frac{-r}{q}=kr(1+p+p^{2}+\dots),\qquad\frac{-2r}{q}=k(2r-q)-1+k(2r-q)(p+p^{2}+\dots),$

where $0\leqslant k(2r-q)-1<p$.

The polynomial $M^{0}_{s}(x)$ is of degree $d(s)<3s$ in $x$. It is clear from
(4.13) that if $n\leqslant d(s)$, then

$\displaystyle\frac{c_{s,n}}{c_{n}}=1+\tilde{c}_{s,n}\quad\operatorname{where}\quad\operatorname{ord}_{p}\tilde{c}_{s,n}\geqslant
s.$

Hence

$\displaystyle\operatorname{ord}_{p}(c_{s,n}-c_{n})=\operatorname{ord}_{p}\Big{(}c_{n}\Big{(}\frac{c_{s,n}}{c_{n}}-1\Big{)}\Big{)}=\operatorname{ord}_{p}(c_{n}\tilde{c}_{s,n})\geqslant
A+s,$

and

(4.14)
$\displaystyle\operatorname{ord}_{p}\Big{(}M^{0}_{s}(x)-\sum_{n=0}^{d(s)}c_{n}p^{n}x^{n}\Big{)}\geqslant
A+s$

if $\operatorname{ord}_{p}x\geqslant 0$. The inequalities (4.12) and (4.14)
imply that the sequence of polynomials $\big{(}M_{s}^{0}(x)\big{)}_{s\geqslant
0}$ uniformly $p$-adically converges on ${\mathbb{Z}}_{p}$ to the analytic
function $M(x)$. ∎

##  5\. Some properties of $p^{s}$-truncated Calogero-Moser model

###  5.1. Arithmetic with the Planck constant

In previous sections we combined a $p^{s}$-approximation of solutions to KZ
equations with an artificial rescaling $\lambda\rightarrow p\lambda$. Let us
consider a procedure which makes this rescaling natural. To this aim introduce
the Planck constant $\hbar$ in the KZ equations and Calogero-Moser operator:

(5.1) $\displaystyle\phantom{aaaa}\hbar\partial_{i}I=\Big{(}\kappa\sum_{j\neq
i}\frac{P^{(i,j)}}{z_{i}-z_{j}}+\lambda^{(i)}\Big{)}I,\qquad i=1,\dots,n,$
$\displaystyle\Big{(}-\hbar^{2}\Delta+\kappa(\kappa+\hbar)\sum_{j\neq
i}\frac{1}{(z_{i}-z_{j})^{2}}\Big{)}\Psi=E\Psi.$

The Matsuo-Cherednik map holds at any $\hbar$. Let us divide the KZ equations
(5.1) by $\hbar$ and introduce $\tilde{\kappa}=\kappa/\hbar$. Then

(5.2) $\displaystyle\partial_{i}I=\Big{(}\tilde{\kappa}\sum_{j\neq
i}\frac{P^{(i,j)}}{z_{i}-z_{j}}+\hbar^{-1}\lambda^{(i)}\Big{)}I,\qquad
i=1,\dots,n.$

These equations coincide with equations (2.7),

$\displaystyle\partial_{i}I=\Big{(}\kappa\sum_{j\neq
i}\frac{P^{(i,j)}}{z_{i}-z_{j}}+p\lambda^{(i)}\Big{)}I,\qquad i=1,\dots,n,$

upon the identification $\kappa=\tilde{\kappa}$ and $p=\hbar^{-1}$.

For example, for $n=2$, the complex solutions to the KZ equations (5.2) are
given by periods of the master function

(5.3)
$\displaystyle\Phi(t,z,\hbar^{-1}\lambda,\widetilde{\kappa})=e^{\hbar^{-1}\lambda_{1}(z_{1}+z_{2})+\hbar^{-1}(\lambda_{2}-\lambda_{1})t}(z_{1}-z_{2})^{\tilde{\kappa}}(t-z_{1})^{-\tilde{\kappa}}(t-z_{2})^{-\tilde{\kappa}},$

which is identical to the master function in (3.1) if we identify
$\kappa=\tilde{\kappa}$ and $p=\hbar^{-1}$.

At the Calogero-Moser side, we also obtain the coupling constant
$\tilde{\kappa}$ and rescaled energy $E\rightarrow\hbar^{-2}E$. In this case,
the conventional quantum mechanics associated with the master function $\Phi$
in (5.3) is present. Thus, we can question the physical meaning of our
$\hbar^{-s}$-approximation procedure. Our approach differs from the standard
small $\hbar$ semiclassical series expansion or potential $\hbar^{-1}$ series
expansion at large $\hbar$. Instead, we investigate the arithmetic properties
of the series expansion of the wave function
$\Psi(z,\lambda,\tilde{\kappa},\hbar)$ modulo $\hbar^{-s}$ when $\hbar^{-1}$
is a prime number.

Interestingly, it turns out that the identification $\hbar^{-1}=p$ naturally
emerges in $p$-adic quantum mechanics [FO, HB], where
$z,\lambda\in\mathbb{F}_{p}$ and ${\mathbb{F}}_{p}$ is the field with $p$
elements. In this context, the $p^{s}$-truncation procedure for the Calogero-
Moser system corresponds to selecting particular polynomials in $z,\lambda$ in
the modulo $\hbar^{-s}$-series expansion of the wave function
$\Psi(z,\lambda,\tilde{\kappa},\hbar)$. As we take the limit $s\to\infty$, we
obtain an $\hbar^{-1}$-adic version of the quantum mechanical wave function,
while the finite $s$ case corresponds to a specific approximation of the
$p$-adic limit.

###  5.2. Semiclassical limit modulo $p^{s}$

Recall the semiclassical limit of the Matsuo-Cherednik correspondence which is
known as classical-quantum duality in the integrability framework. In the
semiclassical limit the integral representations for the solutions of the KZ
equations get localized at the saddle points of the master function
$\Phi(t,z,\lambda,\kappa)$ with respect to the $t$-variables. The saddle point
equations for the master function coincide with the system of the Bethe Anzatz
equations in the Gaudin model for the Bethe roots $t^{0}=(t_{i,j}^{0})$, see
[RV]. The vector weights $V(t,z)$ in the integral representations for
solutions of the KZ equations become the Bethe vectors - eigenfunctions of
Gaudin Hamiltonians $H_{i}$ ,

$\displaystyle
H_{i}(z,\lambda,\kappa)V(t^{0},z)=m_{i}(t^{0},z,\lambda,\kappa)V(t^{0},z),\quad
m_{i}=\frac{\partial\ln\Phi(t^{0},z,\lambda,\kappa)}{\partial z_{i}}\,,\quad
i=1,\dots,n.$

At the Calogero-Moser side as $\hbar\to 0$, we get the classical model with
the phase space variables $z=(z_{1},\dots,z_{n})$, $m=(m_{1},\dots,m_{n})$.
Given $\lambda=(\lambda_{1},\dots,\lambda_{n})$, the symmetric powers
$\lambda_{1}^{k}+\dots+\lambda_{n}^{k}$, $k=1,2,\dots$, give the values of all
standard commuting Hamiltonians of the classical Calogero-Moser system. There
are two natural types of Lagrangian submanifolds in the classical phase space:
a submanifold $L_{1}$ of the first type is defined by fixing all the
$z$-coordinates, while a submanifold $L_{2}$ of the second type is defined by
fixing all the $\lambda$-parameters. If we evaluate the momenta $m_{i}$ of
Calogero-Moser particles at points of $L_{1}\cap L_{2}$ , they coincide with
eigenvalues $m_{i}=\frac{\partial\ln\Phi(z,t^{0},\lambda,\kappa)}{\partial
z_{i}}$ of the Gaudin Hamiltonians evaluated at the solutions $t^{0}$ of the
Bethe Ansatz equations, see [MTV2, GK, GZZ].

The truncation of the semiclassical limit of solutions of KZ equations in
terms of the truncation of the Bethe Ansatz equations modulo $p$ has been
considered in [V2] where the algebra $\mathfrak{sl}_{2}$ and its
representations are considered over the field ${\mathbb{F}}_{p}$. It was shown
that solutions of the Bethe Ansatz equations over ${\mathbb{F}}_{p}$ give
eigenvectors of the Gaudin Hamiltonians over ${\mathbb{F}}_{p}$.

We check the consistency of the semiclassical limit with the truncation
procedure for the simplest case $n=2$. That is, we evaluate the momenta
$(m_{1},m_{2})$ of particles at the intersection of $L_{1}\cap L_{2}$, where
$L_{1}$ is defined by fixing $(z_{1},z_{2})$ and $L_{2}$ is defined by fixing
$(\lambda_{1},\lambda_{2})$, and compare them with the eigenvalues of the
Gaudin Hamiltonians evaluated at the solutions of the Bethe equations.

For $n=2$, the phase space has coordinates $(z_{1},z_{2},m_{1},m_{2})$. The
intersection of the two Lagrangian submanifords reads as

(5.4) $\displaystyle m_{1}+m_{2}=p(\lambda_{1}+\lambda_{2}),\qquad
m_{1}^{2}+m_{2}^{2}-\frac{2\kappa^{2}}{(z_{1}-z_{2})^{2}}=p^{2}(\lambda_{1}^{2}+\lambda_{2}^{2})\,,$

where the rescaling $\lambda\rightarrow p\lambda$ is taken into account.

The Bethe ansatz equation associated with the master function

(5.5)
$\displaystyle\Phi(t,z,p\lambda,\kappa)=e^{p\lambda_{1}(z_{1}+z_{2})+p(\lambda_{2}-\lambda_{1})t}(z_{1}-z_{2})^{\kappa}(t-z_{1})^{-\kappa}(t-z_{2})^{-\kappa}$

is the equation

(5.6) $\displaystyle
p(\lambda_{2}-\lambda_{1})-\frac{\kappa}{t-z_{1}}-\frac{\kappa}{t-z_{2}}=0.$

If $t$ is a solution of the Bethe ansatz equation, then

$\displaystyle
m_{1}=p\lambda_{1}+\frac{\kappa}{z_{1}-z_{2}}+\frac{\kappa}{t-z_{1}}\,,\qquad
m_{2}=p\lambda_{1}+\frac{\kappa}{z_{2}-z_{1}}+\frac{\kappa}{t-z_{2}}\,,$

are eigenvalues of the corresponding Gaudin Hamiltonians
$H_{1}(z,\lambda,\kappa)$ and $H_{2}(z,\lambda,\kappa)$, respectively, see
[RV]. It was shown in [MTV2] that if $t$ is a solution of the Bethe ansatz
equation (5.6), then $(z_{1},z_{2},m_{1},m_{2})$ satisfy equations (5.4).
Indeed,

$\displaystyle
m_{1}+m_{2}=2p\lambda_{1}+\Big{(}\frac{\kappa}{t-z_{1}}+\frac{\kappa}{t-z_{1}}-p(\lambda_{2}-\lambda_{1})\Big{)}+p(\lambda_{2}-\lambda_{1})=p(\lambda_{1}+\lambda_{2}),$
$\displaystyle
m_{1}^{2}+m_{2}^{2}-\frac{2\kappa^{2}}{(z_{1}-z_{2})^{2}}=\Big{(}p\lambda_{1}+\frac{\kappa}{z_{1}-z_{2}}+\frac{\kappa}{t-z_{1}}\Big{)}^{2}+\Big{(}p\lambda_{1}+\frac{\kappa}{z_{2}-z_{1}}+\frac{\kappa}{t-z_{2}}\Big{)}^{2}$
$\displaystyle-\
\frac{2\kappa^{2}}{(z_{1}-z_{2})^{2}}=2p^{2}\lambda_{1}^{2}+2p\lambda_{1}\Big{(}\Big{(}\frac{\kappa}{t-z_{1}}+\frac{\kappa}{t-z_{1}}-p(\lambda_{2}-\lambda_{1})\Big{)}+p(\lambda_{2}-\lambda_{1})\Big{)}$
$\displaystyle+\
\kappa^{2}\Big{(}\frac{2}{(z_{1}-z_{2})(t-z_{1})}+\frac{1}{(t-z_{1})^{2}}+\frac{2}{(z_{2}-z_{1})(t-z_{2})}+\frac{1}{(t-z_{2})^{2}}\Big{)}$
$\displaystyle=\
2p^{2}\lambda_{1}\lambda_{2}+\kappa^{2}\Big{(}\frac{2}{(t-z_{1})(t-z_{2})}+\frac{1}{(t-z_{1})^{2}}+\frac{1}{(t-z_{2})^{2}}\Big{)}$
$\displaystyle=\
2p^{2}\lambda_{1}\lambda_{2}+\Big{(}\frac{\kappa}{t-z_{2}}+\frac{\kappa}{t-z_{1}}\Big{)}^{2}=2p^{2}\lambda_{1}\lambda_{2}+p^{2}(\lambda_{2}-\lambda_{1})^{2}=p^{2}(\lambda_{1}^{2}+\lambda_{2}^{2}).$

In this calculation, the Bethe Ansatz equation was utilized and the expression
$\frac{\kappa}{t-z_{1}}+\frac{\kappa}{t-z_{1}}$ was repeatedly replaced with
$p(\lambda_{2}-\lambda_{1})$. Suppose we were able to identify a value for $t$
that solves the Bethe Ansatz equations up to a high power of $p$. In that
case, the calculated values for $m_{1}$ and $m_{2}$ corresponding to that
particular $t$ would fulfill equations (5.4) up to a high power of $p$. Here
is how we solve the Bethe Ansatz equation (5.6) modulo a high power of $p$.
Equation (5.6) is a quadratic equation with solutions

$\displaystyle t_{\pm}(z,\lambda,\kappa)$ $\displaystyle=$
$\displaystyle\frac{z_{1}+z_{2}}{2}+\frac{\kappa}{p(\lambda_{2}-\lambda_{1})}\pm\frac{\kappa}{p(\lambda_{2}-\lambda_{1})}\sqrt{1+\Big{(}p\frac{(\lambda_{2}-\lambda_{1})(z_{1}-z_{2})}{2\kappa}\Big{)}^{2}}$
$\displaystyle=$
$\displaystyle\frac{z_{1}+z_{2}}{2}+\frac{\kappa}{p(\lambda_{2}-\lambda_{1})}\pm\frac{\kappa}{p(\lambda_{2}-\lambda_{1})}\sum_{k=0}^{\infty}\binom{\frac{1}{2}}{k}\Big{(}p\frac{(\lambda_{2}-\lambda_{1})(z_{1}-z_{2})}{2\kappa}\Big{)}^{2k}\,.$

The root $t_{-}(z,\lambda,\kappa)$ is of order 0 and the root
$t_{+}(z,\lambda,\kappa)$ is of order $-1$. A truncation of the series in
(LABEL:root) modulo a high power of $p$ gives us a solution of the Bethe
Ansatz equations modulo a high power of $p$. Then the corresponding
$m_{1}(t_{\pm}(z,\lambda,\kappa),z,\lambda,\kappa)$ and
$m_{2}(t_{\pm}(z,\lambda,\kappa),z,\lambda,\kappa)$ solve equations (5.4)
modulo a high power of $p$.

##  6\. Discussion

In this note, by utilizing the Matsuo-Cherednik duality, we have obtained a
$p^{s}$-truncated version of the Calogero-Moser eigenfunctions and explored
their properties. The analysis conducted for the simplest rational Calogero-
Moser system can be extended to encompass more general systems such as the
Ruijsenaars-Schneider or Toda systems. To address the Ruijsenaars-Schneider
model, it is necessary to initiate the truncation procedure starting from the
truncated version of the integral representation for hypergeometric solutions
to the qKZ equations. Additionally, it would be interesting to investigate the
commutativity between the bispectral duality and the $p^{s}$-truncation
method. We have investigated the $s\rightarrow\infty$ limit of the $p^{s}$
truncation to the $p$-adic case for the $n=2$ case only. More general analysis
for $n>2$ is required.

It is known for a while [OP, E] that the Calogero-Moser eigenfunctions are
closely related to the representations of Lie groups, for instance to
representations of the $SU(N)$ group for the $N$-particle Calogero-Moser
system. In particular, useful representations of wave functions for
trigonometric and rational Calogero system have the form

$\displaystyle\Psi(\lambda,z,\kappa)=Tr_{\lambda}V_{\kappa}e^{itz},$

where $V_{\kappa}$ is the intertwiner between an $SL(N)$ Verma module
$M_{\lambda}$ and a finite dimensional representation while $t$ belongs to
Cartan subalgebra, see [EK, FV]. It would be interesting to get the similar
representation for the truncated Calogero-Moser eigenfunctions in terms of
twisted intertwiner for the group in characteristic $p$. Probably the results
from [BFG] or procedure of Verma module truncation developed in [V1] could be
useful.

Another point concerns the relation of the Calogero-Moser operator with the
rational spherical Cherednik algebra $H_{c}$ parametrized by $c$ and defined
as follows [Ch2]. Consider the operators $x_{i}$ which multiply polynomials
$p(x_{1}\dots x_{n})$ by $x_{i}$ and the Dunkl operators acting as

$\displaystyle D_{i}(p)=\frac{\partial p}{\partial x_{i}}+c\sum_{j\neq
i}\frac{s_{ij}p-p}{x_{i}-x_{j}},$

$[D_{i},D_{j}]=0$. The operators $(x_{i},y_{i},s)\in H_{c}$ obey the following
relations

$\displaystyle[x,x^{\prime}]=0,\qquad
sxs^{-1}=s(x),\qquad[y,y^{\prime}]=0,\qquad sys^{-1}=s(y),$
$\displaystyle[y,x]=<y,x>-c\sum_{s\in S}<y,\alpha_{s}><\alpha_{s}^{v},x>s,$

where $(x,x^{\prime})\in V^{*}=\operatorname{Span}(x_{1},\dots x_{n})$,
$(y,y^{\prime})\in V=\operatorname{Span}(D_{1},\dots D_{n})$, $s\in S_{n}$.
When $c=\frac{r}{q}\,,$ the rational Cherednik algebra has a finite-
dimensional polynomial representation $L_{r/q}$ [BEG]. To make the link to the
Calogero-Moser operator we have to consider the spherical subalgebra of
$H_{c}$ defined as $eH_{c}e$ where

$\displaystyle e=\frac{1}{|S_{n}|}\sum_{s\in S_{n}}s.$

The finite-dimensional polynomial representation of spherical subalgebra is
$eL_{r/q}$ and corresponding polynomials are eigenfunctions of the Calogero-
Moser operator $H_{CM}=\sum_{i=1}^{N}y_{i}^{2}$.

It would be interesting to clarify if $p^{s}$ truncated polynomial Calogero-
Moser eigenfunctions can be related to the truncation of the finite-
dimensional polynomial representation of spherical subalgebra of the rational
Cherednik algebra. Notice also that at the rational Calogero-Moser coupling
$c=\kappa=\frac{r}{q}$ the graded dimensions of the space of polynomial
Calogero-Moser eigenfunctions with the eigenvalue $E=0$ were identified with
the HOMFLY invariants of the $T_{r,q}$ torus knots [GORS, Ch3, EGL]. It would
be interesting to determine if the $p^{s}$ truncation procedure for the torus
knot invariants is available.

The quantum Calogero-Moser system and its generalizations appear in many
physical systems hence our findings are of some interest for them. Let us
restrict ourselves by short comments concerning the relation with the
perturbed topological gauge theories in two and three dimensions and
$\Omega$-deformed supersymmetric non-Abelian gauge theories in four and five
dimensions. The quantum trigonometric $N$-particle Calogero-Moser system can
be derived from quantum $SU(N)$ 2D Yang–Mills theory on the cylinder with the
Wilson line in the finite-dimensional representation [GN2]. Similarly the
quantum $N$-body trigonometric Ruijsenaars-Schneider system can be derived
from the $SU(N)$ Chern-Simons theory on $T^{2}\times R^{1}$ with two Wilson
lines inserted properly [GN1], (see, [G] for the review). This realization via
the perturbed topological theories provides the representation of the
Calogero-Moser eigenfunctions in terms of twisted intertwiners. The truncation
procedure discussed in this note presumably can be formulated at the level of
Lagrangians for such perturbed topological gauge theories.

The $N$-particle quantum Calogero-Moser system describes the wave function of
the corresponding surface defect in the $\mathcal{N}=2$ SUSY $SU(N)$
$\Omega$-deformed gauge theory with the massive adjoint matter in four
dimensions where $\kappa$ parameter is identified with the mass of the adjoint
matter [NS1, NS2, NS3]. Similarly the quantum Ruijsenaars-Schneider system
corresponds to the lifting of four-dimensional system to five-dimensional
gauge theory with one compact coordinate. The $\Omega$ \- deformation of SYM
theory involves two equivariant parameters $\epsilon_{1},\epsilon_{2}$ with
respect to two $U(1)$ rotations in ${\mathbb{R}}^{4}$. In the Nekrasov-
Shatashvili limit one considers $\epsilon_{1}=0$ and it turns out that
$\epsilon_{2}$ becomes the effective Planck constant $\hbar$ in the quantum
Calogero-Moser or Toda model for the surface defect, see, for instance, the
rigorous derivation in [NT, JLN].

The truncation procedure we have discussed could have at least two interesting
applications for $\mathcal{N}=2$ SUSY gauge theories. First, from the quantum
Calogero-Moser or Toda wave functions we can derive prepotentials which
describe low-energy effective actions of SUSY gauge theories depending on the
equivariant parameters of $\Omega$-deformation [N]. It would be interesting to
get the $p^{s}$ truncated version of the prepotentials somewhat similarly to
the derivation of the $p^{s}$ truncated version of superpotential or vertex
function discussed in [SmV1, SmV2].

Secondly, as we have mentioned above if the Planck constant has the special
form $\hbar=\frac{1}{p}$ where $p$ is prime number the interesting
$\hbar^{-s}$ truncation is possible for the wave function. In the
$\Omega$-deformed $\mathcal{N}=2$ SUSY gauge theories in the Nekrasov-
Shatashvili limit $\hbar=\epsilon_{2}$ hence if $\epsilon_{2}=\frac{1}{p}$
with $p$ prime we can develop the $\epsilon_{2}^{-s}$ truncation procedure. In
the $s\rightarrow\infty$ limit we can presumably get the proper $p$-adic limit
of the $\Omega$-deformed SUSY theories with the surface defects.

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