GHOSTS AND CONGRUENCES FOR $\boldsymbol {p}^{\boldsymbol {s}}$-APPROXIMATIONS OF HYPERGEOMETRIC PERIODS

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 Knizhnik–Zamolodchikov (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.

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) $$ \begin{align} x(1-x) I" +(1-2x)I'-\tfrac14I=0, \end{align} $$

whose analytic solution at the origin

(1-2) $$ \begin{align} _2F_1\bigg(\frac12,\frac 12; 1; x\bigg) = \frac 1\pi\,\int_1^\infty t^{-1/2}(t-1)^{-1/2}(t-x)^{-1/2}\,dt =\sum_{k = 0}^\infty\binom{-1/2}{k}^2x^k \end{align} $$

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

(1-3) $$ \begin{align} F_s(x)=\sum_{k = 0}^{p^s-1}\binom{-1/2}{k}^2x^k \quad\text{for } s=1,2,\ldots, \end{align} $$

which are nothing but truncations of the infinite sum in Equation (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, named 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].

Dwork indicates in [Dw] that in the p-adic case, he adopts a point of view similar to that of Igusa in [Ig] on the modulo p solutions of Equation (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 in Equation (1-1). At the same time, Igusa’s modulo p solution

(1-4) $$ \begin{align} g(x)=\sum_{k = 0}^{(p-1)/2}\binom{(p-1)/2}{k}^2x^k \end{align} $$

of Equation (1-1), though indeed coinciding with Dwork’s $F_1(x)$ modulo p, hints at a somewhat different way for approximating the function in Equation (1-2) p-adically through different truncations of the infinite sum in Equation (1-2). Notice the difference of the limits of summations in Equations (1-3) and (1-4).

This recipe seemed to escape its own development until recently, when Schechtman and Varchenko constructed in [SV2] polynomial solutions modulo p to general Knizhnik–Zamolodchikov (KZ) equations, recovering Igusa’s polynomial as a particular case.

The principal goal of this paper is to show that the p-approximation approach in [SV2] goes in parallel with that of Dwork in [Dw] and leads to Dwork-type congruences for solutions of the KZ equations, at least for the cases in which the technicality of proofs does 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 Equation (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 Equation (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

(1-5) $$ \begin{align} P_s(x) = (-1)^{(p^s-1)/2} \sum_{k=0}^{(p^s-1)/2}\binom{(p^s-1)/2}{k}^2 x^k. \end{align} $$

We prove the Dwork-type congruence,

(1-6) $$ \begin{align} P_{s+1}(x) P_{s-1}(x^p) \equiv P_{s}(x) P_{s}(x^p) \,(\mathrm{mod}\,{p^s}), \end{align} $$

in Theorem 4.2.

In Section 6, we consider the simplest example of the KZ connection. 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 new p-adic feature since the corresponding complex KZ connection has no proper invariant subbundle due to the irreducibility of its monodromy representation.

Notice that usually, the invariant subbundles of the KZ connection over ${\mathbb C}$ 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 a 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

(1-7) $$ \begin{align} F_{s+1}(x) F_{s-1}(x^p) \equiv F_{s}(z) F_{s}(x^p) \,(\mathrm{mod}\,{p^s}) \quad\text{for}\; s=1,2,\ldots, \end{align} $$

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, another set of congruences, à la Lucas, was established for the sequence of coefficients $A(n)$ :

(1-8) $$ \begin{align} \frac{A(n+mp^s)}{A([n/p]+mp^{s-1})}\equiv\frac{A(n)}{A([n/p])}\,(\mathrm{mod}\,{p^s}) \quad\text{for}\; m,n\in{\mathbb Z}_{\geqslant0} \text{ and } s=1,2,\ldots \end{align} $$

(see [Dw, Corollary 1(ii), page 36]). These two different-looking families of congruences in Equations (1-7) and (1-8) are both known as Dwork congruences, and to distinguish between the two, we name them type I and type II, respectively.

Our Igusa-inspired $p^s$ -approximations of solutions of a hypergeometric equation, like the $p^s$ -approximation in Equation (1-5), are of dual nature. Although congruences in Equation (1-6) look like congruences in Equation (1-7) 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 the congruences

(1-9) $$ \begin{align} &\frac{A(n+mp^s;x)}{A([n/p]+mp^{s-1};x^p)}\equiv\frac{A(n;x)}{A([n/p];x^p)}\,(\mathrm{mod}\,{p^s}) \quad\text{for } m,n\in{\mathbb Z}_{\geqslant0} \text{ and } s=1,2,\ldots. \end{align} $$

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 $p^s$ -approximation polynomials in parameter x. (We make this explicit in the remark after Theorem 4.4.)

In summary, our principal tools for establishing the existence of p-adic convergence are the Dwork-type congruences in Equation (1-9), of which the required type-I congruences are particular instances.

Initially, general theorems toward Dwork’s congruences were given by Mellit [Me] and independently by Samol and van Straten [SvS]. They were generalized in [MV, Vl]. As these results are insufficiently general for us, we extend Dwork-type congruences further using the elegant method from Mellit’s unpublished preprint [Me]. Our Dwork-type congruences are displayed in Section 2, and their power is illustrated by the congruences in Equation (1-6) from Theorem 4.2 and by several other quite different applications in later sections. In particular, see Section 6 for applications to the KZ equations.

In Section 7, we conjecture some stronger congruences for the polynomials $P_s(x)$ .

We finish the introduction with a remark on the differential KZ equations. The differential KZ equations were discovered by physicists Knizhnik and Zamolodchikov [KZ] as differential equations satisfied by conformal blocks in conformal field theory. It was realized later that versions of KZ equations appear in different situations, for example, as quantum differential equations for Nakajima quiver varieties [MO]. The KZ equations are closely related to quantum integrable systems and the Bethe ansatz method.

The KZ equations were identified with suitable Gauss–Manin connections in [SV1], and integral representations for solutions of the KZ equations were constructed in the form of multidimensional hypergeometric integrals depending on parameters. Integral representations have connected the KZ equations with the theory of special functions and the hypergeometric functions.

In [SV2], a new side of the KZ equations was observed, namely, polynomial solutions modulo p of the KZ equations were constructed. Those are certain vectors of polynomials with integer coefficients which solve the KZ differential equations modulo p. Those polynomial solutions are, in some sense, p-approximations of the multidimensional hypergeometric solutions of the KZ equations. This brings us a general problem of studying arithmetic properties of solutions of the KZ equations with possible applications to enumerative geometry of Nakajima varieties. The present paper is a step in this direction.

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 $t=(t_1,\ldots ,t_r)$ 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 $a=(a_0,a_1,\ldots ,a_{l-1})$ , 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 $a'=(a_1,\ldots ,a_{l-1})$ 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 $m=(m_0,\ldots ,m_{l-1})$ of integers from $\{1,\ldots ,p-1\}$ , denote by $\operatorname {CT}_t(\Lambda ^m)$ the constant term of the Laurent polynomial $\Lambda (t)^{m_0+m_1p+m_2p^2+\cdots +m_{l-1}p^{l-1}}$ .

Theorem 2.1 [Me].

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

$$ \begin{align*} \operatorname{CT}_t(\Lambda^{a*b}) \operatorname{CT}_t(\Lambda^{a'*c}) \equiv \operatorname{CT}_t(\Lambda^{a'*b}) \operatorname{CT}_t(\Lambda^{a*c}) \,(\mathrm{mod}\,{p^{l(a)}}). \end{align*} $$

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

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 $(N_0, N_1,\ldots ,N_{l-1})$ of convex polytopes is called $\operatorname {admissible}$ if for any $0\leqslant i\leqslant j \leqslant l-1$ ,

$$ \begin{align*} (N_i + pN_{i+1}+ \cdots + p^{j-i}N_j)\cap p^{j-i+1}{\mathbb Z}^r = \{0\}. \end{align*} $$

2.3 Definition of ghosts

Let $\Lambda (t,z)$ be a Laurent polynomial in some variables $t=(t_1,\ldots ,t_r)$ , $z=(z_1,\ldots ,z_{r'})$ 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,

   $$ \begin{align*} \Lambda(t,z)^{p^m} = R_{0}(\Lambda)(t^{p^{m}}, z^{p^{m}}) +R_{1}(\Lambda)(t^{p^{m-1}}, z^{p^{m-1}})+\cdots + R_{m}(\Lambda)(t, z). \end{align*} $$

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

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

$$ \begin{align*} R_m(\Lambda)(t,z) = \Lambda(t,z)^{p^m} - \Lambda(t^p,z^p)^{p^{m-1}}, \quad R_0(\Lambda)(t,z) = \Lambda(t,z). \end{align*} $$

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,

$$ \begin{align*} N(R_m(\Lambda))\subset p^mN(\Lambda). \end{align*} $$

2.4 Composed ghosts

Let $\lambda = (\Lambda _0(t,z), \ldots , \Lambda _{l-1}(t,z))$ be a tuple of Laurent polynomials with coefficients in ${\mathbb Z}_p$ . We decompose the product

$$ \begin{align*} \tilde \lambda(t,z) := \Lambda_0(t,z) (\Lambda_1(t,z))^p\cdots (\Lambda_{l-1}(t,z))^{p^{l-1}} \end{align*} $$

into the sum of ghost terms of $\Lambda _0, \ldots , \Lambda _{l-1}$ . As the result, we obtain that $\tilde \lambda $ is the sum of the products

$$ \begin{align*} R_{m,\lambda}(t,z) &:= R_{m_0}(\Lambda_0)(t,z)\cdot R_{m_1}(\Lambda_1)(t^{p^{1-m_1}},z^{p^{1-m_1}})\cdot R_{m_2}(\Lambda_2)(t^{p^{2-m_2}},z^{p^{2-m_2}})\cdots \\& \quad R_{m_{l-1}}(\Lambda_{l-1})(t^{p^{l-1-m_{l-1}}},z^{p^{l-1-m_{l-1}}}), \end{align*} $$

where $m=(m_0,\ldots ,m_{l-1})$ runs over the set of all l-tuples of integers satisfying ${0\leqslant m_i \leqslant i}$ . Clearly,

$$ \begin{align*} R_{m,\lambda}(t,z) \equiv 0\, (\mathrm{mod}\,{p^{|m|}}) \end{align*} $$

and

$$ \begin{align*} N(R_{m,\lambda}(t,z)) \subset N(\Lambda_0(t,z)) + pN(\Lambda_1(t,z)) + \cdots + p^{l-1} N(\Lambda_{l-1}(t,z)). \end{align*} $$

2.5 Indecomposable tuples

Denote by $S_k$ the set of all k-tuples $m=(m_0,\ldots ,m_{k-1})$ 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 $m'*m"$ for $m',m"\in S$ . 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 $i\in \{1,\ldots ,k-1\}$ , there exists $j\geqslant i$ such that $m_j>j-i$ , that is, $j\geqslant i>j-m_j$ . The number of such i for a given j is $m_j$ . The total number of such 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 $m^1,\ldots , m^r$ such that $m=m^1*\cdots *m^r$ .

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

$$ \begin{align*} m=m^1*\cdots *m^r = n^1*\cdots *n^s \end{align*} $$

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 $m^1*\cdots *m^{r-1} = n^1*\cdots *n^{s-1}$ , 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 $\lambda =(\Lambda _0(t,z),\Lambda _1(t,z),\ldots ,\Lambda _{l-1}(t,z))$ of Laurent polynomials with coefficients in ${\mathbb Z}_p$ , define

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

We have

$$ \begin{align*} I_\lambda(x,z) \equiv 0\, (\mathrm{mod}\,{p^{l-1}}) \end{align*} $$

by Lemma 2.3 and

$$ \begin{align*} N(I_{\lambda}(t,z)) \subset N(\Lambda_0(t,z)) + pN(\Lambda_1(t,z)) + \cdots + p^{l-1} N(\Lambda_{l-1}(t,z)). \end{align*} $$

Lemma 2.5. We have

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

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

Proof. We have

$$ \begin{align*} \tilde \lambda(t,z) = \sum_{m\in S_{l}} R_{m,\lambda}(t,z). \end{align*} $$

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

(2-1) $$ \begin{align} R_{m,\lambda}(t,z) = R_{m^1,\lambda^1}(t,z) R_{m^2,\lambda^2}(t^{p^{l(\lambda^1)}},z^{p^{l(\lambda^1)}}) \cdots R_{m^s,\lambda^s}(t^{p^{l(\lambda^1)+\cdots+l(\lambda^{s-1})}},z^{p^{l(\lambda^1)+\cdots+l(\lambda^{s-1})}}). \end{align} $$

This product contributes to the expansion of the product

(2-2) $$ \begin{align} I_{\lambda^1}(t,z) I_{\lambda^2}(t^{p^{l(\lambda^1)}},z^{p^{l(\lambda^1)}}) \cdots I_{\lambda^s}(t^{p^{l(\lambda^1)+\cdots +l(\lambda^{s-1})}},z^{p^{l(\lambda^1)+\cdots+l(\lambda^{s-1})}}) \end{align} $$

into the sum, and conversely each summand in the expansion of Equation (2-2) comes from Equation (2-1) for a unique indecomposable factorization $m=m^1*\cdots *m^s$ .

2.7 Admissible tuples of Laurent polynomials

Definition 2.6. A tuple $\lambda =(\Lambda _0(t,z),\Lambda _1(t,z),\ldots ,\Lambda _{l-1}(t,z))$ of Laurent polynomials is called $\operatorname {admissible}$ if the tuple $(N(\Lambda _0(t,z)), N(\Lambda _1(t,z)),\ldots ,N(\Lambda _{l-1}(t,z)))$ 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 $\lambda =(\Lambda _0(t,z),\Lambda _1(t,z),\ldots ,\Lambda _{l-1}(t,z))$ be an admissible tuple of Laurent polynomials with coefficients in ${\mathbb Z}_p$ and $\lambda =\lambda ^1*\cdots *\lambda ^s$ . Then,

$$ \begin{align*} \operatorname{CT}_t\!\bigg(\prod_{i=1}^s I_{\lambda^i}(t^{p^{l(\lambda^1)+\cdots +l(\lambda^{i-1})}},z^{p^{l(\lambda^1)+\cdots+l(\lambda^{i-1})}})\bigg)(z) =\,\prod_{i=1}^s \operatorname{CT}_t (I_{\lambda^i}(t,z))(z^{p^{l(\lambda^1)+\cdots+l(\lambda^{i-1})}}). \end{align*} $$

Proof. We have

$$ \begin{align*} N(I_{\lambda^1}(t,z)) \subset N(\Lambda_0(t,z)) + pN(\Lambda_1(t,z)) + \cdots + p^{l(\lambda^1)-1} N(\Lambda_{l(\lambda^1)-1}(t,z)). \end{align*} $$

Hence,

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

and

$$ \begin{align*} &\operatorname{CT}_t\!\bigg(\prod_{i=1}^s I_{\lambda^i}(t^{p^{l(\lambda^1)+\cdots +l(\lambda^{i-1})}},z^{p^{l(\lambda^1)+\cdots+l(\lambda^{i-1})}})\bigg)(z) \\ &\quad = \operatorname{CT}_t(I_{\lambda^1}(t,z))(z) \operatorname{CT}_t\!\bigg( \prod_{i=2}^s I_{\lambda^i}(t^{p^{l(\lambda^2)+\cdots+l(\lambda^{i-1})}},z^{p^{l(\lambda^2)+\cdots+l(\lambda^{i-1})}})\bigg) (z^{p^{l(\lambda^1)}}). \end{align*} $$

Thus, by induction on s, we prove the statement.

Corollary 2.8. We have

$$ \begin{align*} \operatorname{CT}_t(\tilde \lambda)(z) =\sum_{\lambda=\lambda^1*\cdots *\lambda^s} \operatorname{CT}_t(I_{\lambda^1})(z)\cdot \operatorname{CT}_t(I_{\lambda^2})(z^{p^{l(\lambda^1)}}) \cdots \operatorname{CT}_t(I_{\lambda^s})(z^{p^{l(\lambda^1)+\cdots+l(\lambda^{s-1})}}), \end{align*} $$

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 $b,c,a'$ can be empty. Assume that the tuples $a*b, a*c, a'*b, a'*c$ of Laurent polynomials are admissible. Then,

(2-3) $$ \begin{align} \operatorname{CT}_t(\widetilde{a*b})(z)\, \operatorname{CT}_t(\widetilde{a'*c})(z^p) \,\equiv\, \operatorname{CT}_t(\widetilde{a'*b})(z^p) \,\operatorname{CT}_t(\widetilde{a*c})(z) \,(\mathrm{mod}\,{p^{l(a)}}). \end{align} $$

Proof. The left-hand side and right-hand side of Equation (2-3) are

$$ \begin{align*} \sum_{{a*b=x^1*\cdots *x^q\atop a'*c=y^1*\cdots *y^s}} \prod_{i=1}^q \operatorname{CT}_t(I_{x^i})(z^{p^{l(x^1)+\cdots+l(x^{i-1})}}) \prod_{i=1}^s \operatorname{CT}_t(I_{y^j})(z^{p^{1+l(y^1)+\cdots+l(y^{j-1})}}) \end{align*} $$

and

(2-4) $$ \begin{align} \sum_{{a'*b=x^1*\cdots *x^q\atop a*c=y^1*\cdots *y^s}} \prod_{i=1}^q \operatorname{CT}_t(I_{x^i})(z^{p^{1+l(x^1)+\cdots+l(x^{i-1})}}) \prod_{i=1}^s \operatorname{CT}_t(I_{y^j})(z^{p^{l(y^1)+\cdots+l(y^{j-1})}}), \end{align} $$

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

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

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

(2-5) $$ \begin{align} l(a)+l(b)+l(c) \geqslant q+s+1. \end{align} $$

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:

(2-6) $$ \begin{align} \prod_{i=1}^q \operatorname{CT}_t(I_{x^i})(z^{p^{l(x^1)+\cdots+l(x^{i-1})}}) \prod_{i=1}^s \operatorname{CT}_t(I_{y^j})(z^{p^{1+l(y^1)+\cdots+l(y^{j-1})}}), \end{align} $$

the summand corresponding to the presentation $a\kern1.2pt{*}\kern1.2ptb\kern1.2pt{=}\kern1.2pt x^1\kern1.2pt{*}\kern1.2pt\cdots *x^q$ , $a'\kern1.2pt{*}\kern1.2ptc\kern1.2pt{=}\kern1.2pt y^1 \kern1.2pt{*}\cdots \kern1.2pt{*}\kern1.2pt y^s$ .

Lemma 2.10. There exist indices $i\geqslant 1$ and $j\geqslant 0$ such that

(2-7) $$ \begin{align} l(x^1)+\cdots+l(x^i) = l(y^1)+\cdots+l(y^j) + 1\leqslant l(a). \end{align} $$

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 $\{2, \ldots ,l(a)\}$ cannot be represented simultaneously as $l(x^1)+\cdots +l(x^i)$ and $l(y^1)+\cdots +l(y^j) + 1$ . Therefore, the sum of the total number of $i\geqslant 1$ , such that $l(x^1)+\cdots +l(x^i)\leqslant l(a)$ , and the total number of $j\geqslant 1$ , such that $l(y^1)+\cdots +l(y^j) + 1\leqslant l(a)$ 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 Equation (2-5). 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 in Equation (2-6). Choose the minimal indices $i\geqslant 1$ and $j\geqslant 0$ such that Equation (2-7) holds. Then it is easy to see that

(2-8) $$ \begin{align} \hspace{-12pt}a'*b=y^1*\cdots *y^j *x^{i+1}*\cdots *x^q, \quad a*c =x^1*\cdots *x^i * y^{j+1}*\cdots *x^s, \end{align} $$

and the summand in Equation (2-4) corresponding to the presentations in Equation (2-8) equals the product in Equation (2-6). 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) $$ \begin{align} I^{(C)}(x) =\int_C t^\alpha(t-1)^\beta (t-x)^\gamma\,dt, \end{align} $$

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) $$ \begin{align} x(1-x) I" +((\alpha+\beta+2\gamma)x - (\alpha+\gamma))I' -\gamma(\alpha+\beta+\gamma+1) I=0. \end{align} $$

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

$$ \begin{align*} \mathcal D = x(1-x) \frac{d^2}{d x^2} +((\alpha+\beta+2\gamma)x - (\alpha+\gamma))\frac{d}{d x} -\gamma(\alpha+\beta+\gamma+1). \end{align*} $$

Then,

(3-3) $$ \begin{align} d_t\bigg(\gamma \frac{t(t-1)}{t-x}\Phi(t,x)\bigg) = \mathcal D \, \Phi(t,x) \,dt. \end{align} $$

The differential equation in Equation (3-2) turns into the standard hypergeometric differential equation

$$ \begin{align*} x(1-x)I"+(c-(a+b+1)x)I'-abI=0 \end{align*} $$

if $\alpha =a-c$ , $\beta =c-b-1$ , $\gamma =-a$ . If $c\notin {\mathbb Z}_{\leqslant 0}$ , then for a suitable choice of C and multiplication of the integral by a constant, the integral in Equation (3-1) can be expanded as a power series

$$ \begin{align*} _2F_1(a,b;c;x) ={}_2F_1\biggl(\begin{matrix} a, \, b \\ c \end{matrix}\biggm| x\biggr) =\sum_{k=0}^\infty\frac{(a)_k(b)_k}{k!(c)_k}x^k. \end{align*} $$

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 Equation (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

$$ \begin{align*} \alpha_s\equiv \alpha, \quad \beta_s\equiv \beta, \quad \gamma_s\equiv\gamma \,(\mathrm{mod}\,{p^s}). \end{align*} $$

Define the master polynomial

$$ \begin{align*} \Phi_s(t,x) = t^{\alpha_s}(t-1)^{\,\beta_s} (t-x)^{\gamma_s} \end{align*} $$

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,

$$ \begin{align*} I_s(x) = (-1)^{\alpha_s+\beta_s+\gamma_s-p^s+1}\sum_{k_1+k_2=\alpha_s+\beta_s+\gamma_s-p^s+1} \binom{\beta_s}{k_1}\binom{\gamma_s}{k_2}\,x^{k_2}. \end{align*} $$

The polynomial $I_s(x)$ has integer coefficients.

Theorem 3.1. The polynomial $I_s(x)$ is a solution of the hypergeometric differential equation in Equation (3-2) modulo $p^s$ ,

$$ \begin{align*} \mathcal D I_s(x) \,\in\, p^s {\mathbb Z}_p[x]. \end{align*} $$

Proof. The theorem follows from Equation (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 [SV2].

4 Function $_2F_1(\tfrac 12, \tfrac 12; 1,x)$

4.1 Polynomials $P_s(x)$

The function

(4-1) $$ \begin{align} _2F_1\bigg(\frac12,\frac 12; 1; x\bigg) = \frac 1\pi\,\int_1^\infty t^{-1/2}(t-1)^{-1/2}(t-x)^{-1/2}\,dt =\sum_{k = 0}^\infty\binom{-1/2}{k}^2x^k \end{align} $$

satisfies the hypergeometric differential equation

(4-2) $$ \begin{align} x(1-x) I" +(1-2x)I'-\tfrac14I=0. \end{align} $$

Define the master polynomial

$$ \begin{align*} \Phi_{p^s}(t,x) = t^{(p^s-1)/2}(t-1)^{(p^s-1)/2}(t-x)^{(p^s-1)/2}. \end{align*} $$

The number $M=({p^s-1})/2=({p-1})/2+({p-1})/2 p+\cdots +({p-1})/2 p^{s-1}$ is the unique positive integer such that $1\leqslant M\leqslant p^s$ and $M\equiv -1/2 \ \,(\mathrm {mod}\,{p^s})$ . 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,

(4-3) $$ \begin{align} P_s(x) = (-1)^{(p^s-1)/2} \sum_{k=0}^{(p^s-1)/2}\binom{(p^s-1)/2}{k}^2 x^k. \end{align} $$

Define $P_0(x)=1$ .

Recall the hypergeometric function ${}_2F_1(a,b;c;x)$ . Then,

(4-4) $$ \begin{align} P_s(x) = (-1)^{(p^s-1)/2} {}_2F_1\bigg(\frac{1-p^s}2, \frac{1-p^s}2; 1; x\bigg). \end{align} $$

The polynomial $P_s(x)$ is a solution of the hypergeometric equation in Equation (4-2) modulo $p^s$ . This follows from Theorem 3.1 or from Equation (4-4).

4.2 Baby congruences

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

(4-5) $$ \begin{align} \varphi_{s+1}(x)\varphi_{s-1}(x^p)\equiv\varphi_s(x)\varphi_s(x^p)\,(\mathrm{mod}\,{p^s}). \end{align} $$

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

Lemma 4.1. The master polynomials $\Phi _{s}(t,x)$ satisfy the baby congruence

(4-6) $$ \begin{align} \Phi_{s+1}(t,x)\Phi_{s-1}(t^p,x^p) \equiv \Phi_{s}(t,x)\Phi_{s}(t^p,x^p) \,(\mathrm{mod}\,{p^s}). \end{align} $$

Proof. The lemma follows from Equation (4-5).

4.3 Congruences for $P_s(x)$

Theorem 4.2. The approximation polynomials $P_s(x)$ satisfy the congruence

(4-7) $$ \begin{align} P_{s+1}(x) P_{s-1}(x^p) \equiv P_{s}(x) P_{s}(x^p) \,(\mathrm{mod}\,{p^s}). \end{align} $$

This theorem follows from a more general Theorem 4.4.

Using Equation (4-4), we may rewrite Equation (4-7) as the congruence

(4-8) $$ \begin{align} &{}_2F_1\bigg(\dfrac{1-p^{s+1}}2, \dfrac{1-p^{s+1}}2; 1; x\bigg) {}_2F_1\bigg(\dfrac{1-p^{s-1}}2, \dfrac{1-p^{s-1}}2; 1; x^p\bigg) \notag \\ & \quad \equiv {}_2F_1\bigg(\dfrac{1-p^s}2, \dfrac{1-p^s}2; 1; x\bigg){}_2F_1\bigg(\dfrac{1-p^s}2, \dfrac{1-p^s}2; 1; x^p\bigg) \,(\mathrm{mod}\,{p^s}). \end{align} $$

Let $\alpha $ be a rational number which is a p-adic unit, $\alpha = \alpha _0 + \alpha _1 p+ \alpha _2p^2+\cdots $ . Denote by $[\alpha ]_s$ the sum of the first s summands. Then the congruence in Equation (4-8) takes the form:

$$ \begin{align*} &{}_2F_1\big([-\tfrac12]_{s+1}, [-\tfrac12]_{s+1};1;x\big)\, {}_2F_1\big([-\tfrac12]_{s-1}, [-\tfrac12]_{s-1};1;x^p\big) \notag \\ &\quad \equiv {}_2F_1\big([-\tfrac 12]_{s}, [-\tfrac12]_{s};1;x\big)\, {}_2F_1\big([-\tfrac12]_{s}, [-\tfrac12]_{s};1;x^p\big) \,(\mathrm{mod}\,{p^s}). \end{align*} $$

4.4 Coefficients of master polynomials

Consider

$$ \begin{align*} \hat\Phi_s(t,x) &:= t^{-(p^s-1)}\Phi_s(t,x) = t^{-(p^s-1)/2}((t-1)(t-x))^{(p^s-1)/2} \\ &= ((t-1)(1-x/t))^{(p^s-1)/2} =\sum_{j=-(p^s-1)/2}^{(p^s-1)/2} C_{s,j}(x)t^j, \end{align*} $$

where

$$ \begin{align*} C_{s,j}(x) = (-1)^{(({p^s-1})/2)-j}\sum_{m} \binom{\frac{p^s-1}2}{m+j} \binom{\frac{p^s-1}2}{m}\, x^m. \end{align*} $$

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

$$ \begin{align*} C_{s,j}(x) = (-1)^{(({p^s-1})/2)-j} \binom{\frac{p^s-1}2}{j} {}_2F_1\bigg( \frac{1-p^s}2, \frac{1-p^s}2+j; j+1; x\bigg) \quad\text{for}\; j\geqslant0, \end{align*} $$

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

(4-9) $$ \begin{align} C_{s,-j}(x) = x^j C_{s,j}(x). \end{align} $$

We expand the congruence in Equation (4-6) into a congruence of polynomials $C_{s,j}(x)$ . The constant term in t gives us

(4-10) $$ \begin{align} \sum_{k} C_{s+1,kp}(x)C_{s-1,-k}(x^p) \equiv \sum_{k} C_{s,kp}(x)C_{s,-k}(x^p) \,(\mathrm{mod}\,{p^s}). \end{align} $$

The following Theorem 4.4 establishes the congruences of individual pairs of terms in Equation (4-10).

Theorem 4.4. For any k appearing in Equation (4-10),

(4-11) $$ \begin{align} C_{s+1,kp}(x)C_{s-1,-k}(x^p) \equiv C_{s,kp}(x)C_{s,-k}(x^p) \,(\mathrm{mod}\,{p^s}). \end{align} $$

In particular, for $k=0$ , we have the congruence in Equation (4-7).

Proof. Every index k appearing in Equation (4-10) can be written uniquely as

$$ \begin{align*} \phantom{aaa} k= k_0 + k_1p+\cdots + k_{s-2}p^{s-2}, \quad -(p-1)/2\leqslant k_i \leqslant (p-1)/2. \end{align*} $$

Using Equation (4-9), we reformulate Equation (4-11) as

(4-12) $$ \begin{align} C_{s+1,kp}(x)C_{s-1,k}(x^p) \equiv C_{s,kp}(x)C_{s,k}(x^p)\ (\mod {p^{3}}) \end{align} $$

and prove it. We have

$$ \begin{align*} C_{s+1,kp}(x) &= \operatorname{CT}_t\bigg[\hat\Phi_1(t,x) \bigg(\prod_{i=0}^{s-2} ( t^{-k_i}\hat\Phi_1(t,x))^{p^{i+1}}\bigg) \hat\Phi_1(t,x)^{p^s}\bigg], \\ C_{s-1,k}(x) &= \operatorname{CT}_t\bigg[\prod_{i=0}^{s-2} ( t^{-k_i}\hat\Phi_1(t,x))^{p^{i}}\bigg], \\ C_{s,kp}(x) &=\operatorname{CT}_t\bigg[\hat\Phi_1(t,x) \prod_{i=0}^{s-2} ( t^{-k_i}\hat\Phi_1(t,x))^{p^{i+1}}\bigg], \\ C_{s,k}(x) &= \operatorname{CT}_t\bigg[\bigg(\prod_{i=0}^{s-2} ( t^{-k_i}\hat\Phi_1(t,x))^{p^{i}}\bigg) \hat\Phi_1(t,x)^{p^{s-1}}\bigg]. \end{align*} $$

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

$$ \begin{align*} \hat\Phi_1(t,x),\, t^{-k_0}\hat\Phi_1(t,x), \,\ldots, \,t^{-k_{s-2}}\hat\Phi_1(t,x),\, \hat\Phi_1(t,x) \end{align*} $$

is admissible in the sense of Definition 2.6. Now the application of Theorem 2.9 gives the congruence in Equation (4-12) and hence the congruence in Equation (4-11).

Remark 4.5. Denote $A(n,x) :={}_2F_1(-n,-n;1;x) = \sum _k \binom {n}{k}^2 x^k$ . Let

$$ \begin{align*} n = n_0+n_1p+ \cdots + n_{s-1}p^{s-1}, \quad [n/p] = n_1 +\cdots + n_{s-1}p^{s-2}, \end{align*} $$

where $0\leqslant n_i<p$ . Then for any $m\in {\mathbb Z}_{\geqslant 0}$ ,

$$ \begin{align*} A(n +mp^s, x) A([n/p],x^p) \equiv A(n,x) A([n/p]+mp^{s-1},x^p) \,(\mathrm{mod}\,{p^s}). \end{align*} $$

The proof follows from Theorem 2.9 and the identity

$$ \begin{align*} A(n,x) = \operatorname{CT}_t[((t+1)(1+x/t))^n]. \end{align*} $$

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

$$ \begin{align*} D_{\alpha,r} = \{ x\in {\mathbb Z}_p \mid |x-\omega(\alpha)|_p < r \}. \end{align*} $$

Denote

$$ \begin{align*} \bar P_s(x) := (-1)^{(p^s-1)/2} P_s(x) = {}_2F_1\big(\big[\!-\tfrac 12\big]_{s},\big[\!-\tfrac12\big]_{s};1;x\big); \end{align*} $$

see Equation (4-4). Denote

$$ \begin{align*} \mathfrak D = \{x\in {\mathbb Z}_p \mid |\bar P_1(x)|_p=1\}. \end{align*} $$

Theorem 4.6. For $s\geqslant 1$ , the rational function ${\bar P_{s+1}(x)}/{\bar P_s(x^p)}$ is regular on $\mathfrak D$ . The sequence $({\bar P_{s+1}(x)}/{\bar P_s(x^p)})_{s\geqslant 1}$ uniformly converges on $\mathfrak D$ . The limiting analytic function $f(x)$ equals the ratio ${F(x)}/{F(x^p)}$ on the disc $D_{0,1}$ , where $F(x):={}_2F_1(1/2,1/2;1;x)$ is defined by the convergent power series in Equation (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

$$ \begin{align*} \{x\in {\mathbb Z}_p \mid |\bar P_1(x^p)|_p=1\} = \bigcup_{\alpha\in{\mathbb F}_p,\, |\bar P_1(\omega(\alpha))|_p =1} D_{\alpha,1} = \mathfrak D \end{align*} $$

for the same reason.

We have $\bar P_s(x) \equiv \bar P_1(x) \bar P_1(x^p)\cdots \bar P_1(x^{p^{s-1}}) \ \,(\mathrm {mod}\,{p})$ . Indeed, the polynomial $ P_s(x)$ is the coefficient of $x^{p^s-1}$ in the master polynomials $\Phi _{p^s}(t,x)$ and

$$ \begin{align*} \Phi_{p^s}(t,x) \equiv \Phi_{p}(t,x)\Phi_{p}(t^{p},x^{p})\cdots \Phi_{p}(t^{p^{s-1}},x^{p^{s-1}}) \,(\mathrm{mod}\,{p}). \end{align*} $$

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 ${\bar P_{s+1}(x)}/{\bar P_s(x^p)}$ are regular on $\mathfrak D$ .

The congruence in Equation (4-7) implies that

$$ \begin{align*} \bigg|\frac{\bar P_{s+1}(x)}{\bar P_{s}(x^p)} - \frac{\bar P_{s}(x)}{\bar P_{s-1}(x^p)}\bigg|_p\leqslant p^{-s} \quad\operatorname{for}\ x\in\mathfrak D. \end{align*} $$

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 $f(x) ={F(x)}/{F(x^p)}$ on $D_{0,1}$ . The theorem is proved.

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

$$ \begin{align*} F_s(x) = \sum_{k=0}^{p^s-1}\binom{-1/2}{k}^2 x^k, \end{align*} $$

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

(4-13) $$ \begin{align} g(x) =\sum_{k=0}^{(p-1)/2} \binom{-1/2}{k}^2x^k \end{align} $$

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

It is easy to see that the two sequences of rational functions $({\bar P_{s+1}(x)}/{\bar P_s(x^p)})_{s\geqslant 1}$ and $({F_{s+1}(x)}/{F_s(x^p)})_{s\geqslant 1}$ have the same limiting functions on the same domain. Indeed, $\bar P_1(x)\equiv g(x) \ \,(\mathrm {mod}\,{p})$ 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 ${\bar P_{s+1}(x)}/{\bar P_s(x^p)}$ of approximation polynomials multiplied by $(-1)^{(p-1)/2}$ .

5 Function $_2F_1(2/3, \tfrac 13; 1;x)$

5.1 Two hypergeometric integrals

The integral

$$ \begin{align*} I^{(C)}(x) =\int_C t^{-1/3}(t-1)^{-1/3} (t-x)^{-2/3}\,dt, \end{align*} $$

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

(5-1) $$ \begin{align} x(1-x) I" +(1-2x)I'-\tfrac29I=0. \end{align} $$

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

$$ \begin{align*} {}_2F_1\bigg(\frac23,\frac 13; 1,x\bigg) = \sum_{k = 0}^\infty\binom{-1/3}{k}\binom{-2/3}{k}\,x^k. \end{align*} $$

The integral

$$ \begin{align*} J^{(D)}(x) =\int_D t^{-2/3}(t-1)^{-2/3} (t-x)^{-1/3}\,dt, \end{align*} $$

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 $_2F_1(2/3,\tfrac 13; 1,x)$ .

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 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

$$ \begin{align*} \Phi_s(t,x) = t^{(p^s-1)/3}(t-1)^{(p^s-1)/3}(t-x)^{2(p^s-1)/3}. \end{align*} $$

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

$$ \begin{align*} Q_s(x) = (-1)^{(p^s-1)/3}\sum_{k} \binom{2(p^s-1)/3}{k} \binom{(p^s-1)/3}{k}\, x^k. \end{align*} $$

Define $Q_0(x)=1$ . We have

(5-2) $$ \begin{align} Q_s(x) = {}_2F_1\bigg(\frac{2-2p^s}3, \frac{1-p^s}3; 1; x\bigg), \end{align} $$

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

The polynomial $Q_s(x)$ is a solution of the hypergeometric equation in Equation (5-1) modulo $p^s$ . This follows from Theorem 3.1 or from Equation (5-2).

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

$$ \begin{align*} \Psi_s(t,x) = t^{2(p^s-1)/3}(t-1)^{2(p^s-1)/3}(t-x)^{(p^s-1)/3}. \end{align*} $$

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

$$ \begin{align*} R_s(x) = \sum_{k} \binom{2(p^s-1)/3}{k} \binom{(p^s-1)/3}{k}\, x^k. \end{align*} $$

Define $R_0(x)=1$ . We have

(5-3) $$ \begin{align} Q_s(x) = {}_2F_1\bigg(2\frac{1-p^s}3, \frac{1-p^s}3; 1; x\bigg). \end{align} $$

The master polynomials satisfy the baby congruences,

(5-4) $$ \begin{align} \begin{aligned} &\kern0.2pt\Phi_{s+1}(t,x) \Phi_{s-1}(t^p,x^p) \equiv \Phi_{s}(t,x) \Phi_{s}(t^p,x^p) \,(\mathrm{mod}\,{p^s}),\\ &\Psi_{s+1}(t,x) \Psi_{s-1}(t^p,x^p) \equiv \Psi_{s}(t,x) \Psi_{s}(t^p,x^p) \,(\mathrm{mod}\,{p^s}) , \end{aligned}\qquad \end{align} $$

by Equation (4-5).

Theorem 5.1. For $p=3\ell +1$ , the approximation polynomials $R_s(x)$ and $Q_s(x)$ satisfy the congruences

(5-5) $$ \begin{align} &Q_{s+1}(x) Q_{s-1}(x^p) \equiv Q_{s}(x) Q_{s}(x^p) \,(\mathrm{mod}\,{p^s}), \end{align} $$

(5-6) $$ \begin{align} &R_{s+1}(x) R_{s-1}(x^p) \equiv R_{s}(x) R_{s}(x^p) \,(\mathrm{mod}\,{p^s}). \end{align} $$

Proof. Denote $\hat \Phi _1(t,x) = (t-1)^{\ell }(1-x/t)^{2\ell }$ . Then, $Q_{s+1}(x) = \operatorname {CT}_t[\hat \Phi _1(t,x)^{1+p+\cdots +p^s}]$ . It is easy to see that the $(s+1)$ -tuple $(\hat \Phi _1(t,x),\hat \Phi _1(t,x), \,\ldots , \hat \Phi _1(t,x))$ of Laurent polynomials is admissible in the sense of Definition 2.6. Now the application of Theorem 2.9 gives the congruence in Equation (5-5). The congruence in Equation (5-6) is proved in the same way applied to the formula $R_{s+1}(x) = \operatorname {CT}_t[\hat \Psi _1(t,x)^{1+p+\cdots +p^s}]$ , where $\hat \Psi _1(t,x) = (t-1)^{2\ell }(1-x/t)^{\ell }$ . The congruence in Equation (5-6) also follows from Equation (5-5) since $R_s(x)=Q_s(x)$ .

Equations (5-2) and (5-3) imply that for $p=3\ell +1$ , $s\geqslant 1$ ,

$$ \begin{align*} &{}_2F_1\bigg(\dfrac{2-2p^{s+1}}3, \dfrac{1-p^{s+1}}3; 1; x\bigg) {}_2F_1\bigg(\dfrac{2-2p^{s-1}}3, \dfrac{1-p^{s-1}}3; 1; x^p\bigg) \notag\\ &\quad \equiv {}_2F_1\bigg(\dfrac{2-2p^s}3, \dfrac{1-p^s}3; 1; x\bigg) {}_2F_1\bigg(\dfrac{2-2p^s}3, \dfrac{1-p^s}3; 1; x^p\bigg) \,(\mathrm{mod}\,{p^s}). \end{align*} $$

Using the expansions

$$ \begin{align*} -1/3 &= \ell + \ell p+\ell p^2+\cdots, &\quad -2/3 &= 2\ell + 2\ell p+2\ell p^2+\cdots,\\ (p^s-1)/3 &= \ell + \ell p+\ell p^2+\cdots + \ell p^{s-1}, &\ \ \, (p^s-2)/3 &= 2\ell + 2\ell p+2\ell p^2+\cdots + 2\ell p^{s-1}, \end{align*} $$

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

(5-7) $$ \begin{align} &{}_2F_1\big([-\tfrac23]_{s+1}, [-\tfrac13]_{s+1};1;x\big)\, {}_2F_1\big([-\tfrac23]_{s-1}, [-\tfrac13]_{s-1};1;x^p\big) \notag \\ &\quad \equiv {}_2F_1\big([-\tfrac 23]_{s}, [-\tfrac13]_{s};1;x\big)\, {}_2F_1\big([-\tfrac23]_{s}, [-\tfrac13]_{s};1;x^p\big) \,(\mathrm{mod}\,{p^s}). \end{align} $$

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

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

$$ \begin{align*} \Phi_s(t,x) &= t^{(2p^s-1)/3}(t-1)^{(2p^s-1)/3}(t-x)^{(p^s-2)/3}, \quad \!\!\operatorname{odd}\,s,\\ \Phi_s(t,x) &= t^{(p^s-1)/3}(t-1)^{(p^s-1)/3}(t-x)^{2(p^s-1)/3}, \quad \operatorname{even}\,s. \end{align*} $$

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

$$ \begin{align*} Q_s(x) &= (-1)^{(2p^s-1)/3}\sum_{k} \binom{(2p^s-1)/3}{k} \binom{(p^s-2)/3}{k}\, x^k, \quad \!\!\operatorname{odd}\,s,\\ Q_s(x) &= (-1)^{(p^s-1)/3}\sum_{k} \binom{2(p^s-1)/3}{k} \binom{(p^s-1)/3}{k}\, x^k, \quad \operatorname{even}\,s. \end{align*} $$

Define $Q_0(x)=1$ . We have

(5-8) $$ \begin{align} \begin{aligned} \kern1pt Q_s(x) &= -\, {}_2F_1\bigg(\frac{2-p^s}3, \frac{1-2p^s}3; 1; x\bigg), \quad\! \!\!\operatorname{odd}\,s,\\ Q_s(x) &= {}_2F_1\bigg(\frac{2-2p^s}3, \frac{1-p^s}3; 1; x\bigg),\ \quad \operatorname{even}\,s. \end{aligned}\qquad \end{align} $$

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 in Equation (5-1) modulo $p^s$ . This follows from Theorem 3.1 or from Equation (5-8).

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

$$ \begin{align*} \Psi_s(t,x) &= t^{(p^s-2)/3}(t-1)^{(p^s-2)/3}(t-x)^{(2p^s-1)/3}, \quad\, \operatorname{odd}\,s,\\ \Psi_s(t,x) &= t^{2(p^s-1)/3}(t-1)^{2(p^s-1)/3}(t-x)^{(p^s-1)/3}, \quad \!\operatorname{even}\,s. \end{align*} $$

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

$$ \begin{align*} R_s(x) &= (-1)^{(p^s-2)/3}\sum_{k} \binom{(2p^s-1)/3}{k} \binom{(p^s-2)/3}{k}\, x^k, \quad \operatorname{odd}\,s, \\ R_s(x) &= (-1)^{2(p^s-1)/3}\sum_{k} \binom{2(p^s-1)/3}{k} \binom{(p^s-1)/3}{k}\, x^k, \quad\!\! \operatorname{even}\,s. \notag \end{align*} $$

Define $R_0(x)=1$ . We have

(5-9) $$ \begin{align} \begin{aligned} &\kern2pt R_s(x) = -\, {}_2F_1\bigg(\frac{2-p^s}3, \frac{1-2p^s}3; 1; x\bigg), \quad \operatorname{odd}\,s,\\ &R_s(x) = {}_2F_1\bigg(\frac{2-2p^s}3, \frac{1-p^s}3; 1; x\bigg),\quad \ \quad \!\!\operatorname{even}\,s. \end{aligned}\qquad \end{align} $$

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 in Equation (5-1) modulo $p^s$ . This follows from Theorem 3.1 or from Equation (5-9).

Lemma 5.2. The master polynomials satisfy the baby congruences,

(5-10) $$ \begin{align} &\Phi_{s+1}(t,x) \Psi_{s-1}(t^p,x^p) \equiv \Phi_{s}(t,x) \Psi_{s}(t^p,x^p) \,(\mathrm{mod}\,{p^s}), \end{align} $$

(5-11) $$ \begin{align} &\Psi_{s+1}(t,x) \Phi_{s-1}(t^p,x^p) \equiv \Psi_{s}(t,x) \Phi_{s}(t^p,x^p) \,(\mathrm{mod}\,{p^s}). \end{align} $$

Proof. We prove Equation (5-10) for an odd s. The case of an even s and the congruence in Equation (5-11) are proved similarly. The left-hand side of Equation (5-10) for an odd $s=2k+1$ equals

$$ \begin{align*} t^{(p^{s+1}-1)/3}(t-1)^{(p^{s+1}-1)/3}(t-x)^{2(p^{s+1}-1)/3} t^{2p(p^{s-1}-1)/3}(t^p-1)^{2(p^{s-1}-1)/3}(t^p-x^p)^{(p^{s-1}-1)/3}, \end{align*} $$

while the right-hand side equals

$$ \begin{align*} t^{(2p^s-1)/3}(t-1)^{(2p^s-1)/3}(t-x)^{(p^s-2)/3} t^{p(p^s-2)/3}(t^p-1)^{(p^s-2)/3}(t^p-x^p)^{(2p^s-1)/3}. \end{align*} $$

Now the congruence in Equation (5-10) for an odd s follows from Equation (4-5).

Theorem 5.3. For $p=3\ell +2>2$ , the approximation polynomials $R_s(x)$ and $Q_s(x)$ satisfy the congruences

(5-12) $$ \begin{align} & Q_{s+1}(x) R_{s-1}(x^p) \equiv Q_{s}(x) R_{s}(x^p) \,(\mathrm{mod}\,{p^s}), \end{align} $$

(5-13) $$ \begin{align} & R_{s+1}(x) Q_{s-1}(x^p) \equiv R_{s}(x) Q_{s}(x^p) \,(\mathrm{mod}\,{p^s}). \end{align} $$

Proof. We prove Equation (5-12) for an odd s. The case of an even s and the congruence in Equation (5-13) are proved similarly. Denote

$$ \begin{align*} f(t,x) &= (t-1)^{(2p-1)/3}(1-x/t)^{(p-2)/3}=(t-1)^{2\ell+1}(1-x/t)^{\ell}, \\ g(t,x) &= (t-1)^{(p-2)/3}(1-x/t)^{(2p-1)/3}= (t-1)^{\ell}(1-x/t)^{2\ell+1}. \end{align*} $$

It is easy to see that for an odd s,

$$ \begin{align*} Q_{s+1}(x) &=\operatorname{CT}_t[(t-1)^{(p^{s+1}-1)/3}(1-x/t)^{2(p^{s+1}-1)/3}] \\ &=\operatorname{CT}_t[f(t,x) g(t,x)^{p} \cdots f(t,x)^{p^{s-1}} g(t,x)^{p^s} ], \\ R_{s-1}(x) &=\operatorname{CT}_t[(t-1)^{2(p^{s-1}-1)/3}(1-x/t)^{(p^{s-1}-1)/3}]\\ &=\operatorname{CT}_t[g(t,x) f(t,x)^{p} \cdots g(t,x)^{p^{s-3}}f(t,x)^{p^{s-2}} ], \end{align*} $$

$$ \begin{align*} Q_{s}(x) &= \operatorname{CT}_t[(t-1)^{(2p^{s}-1)/3}(1-x/t)^{(p^{s}-2)/3}]\\ &=\operatorname{CT}_t[f(t,x) g(t,x)^{p} \cdots g(t,x)^{p^{s-2}} f(t,x)^{p^{s-1}} ],\\ R_{s}(x) &=\operatorname{CT}_t[(t-1)^{(p^{s}-2)/3}(1-x/t)^{(2p^{s}-1)/3}]\\ &=\operatorname{CT}_t[g(t,x) f(t,x)^{p} \cdots f(t,x)^{p^{s-2}}g(t,x)^{p^{s-1}} ]. \end{align*} $$

Observe that the $(s+1)$ -tuple of Laurent polynomials $(f(t,x),g(t,x),\ldots , f(t,x), g(t,x))$ is admissible in the sense of Definition 2.6. Now the application of Theorem 2.9 gives the congruence in Equation (5-12) for an odd s.

Remark 5.4. In general, we may take any admissible tuple of Laurent polynomials and obtain the corresponding Dwork congruences. For example, the tuples $(f,g,f,f,f,g,g,f,\ldots )$ and $(f,f,\ldots )$ are admissible.

Using Equations (5-8) and (5-9), we may reformulate the congruences in Equations (5-12) and (5-13) as

(5-14) $$ \begin{align} &{}_2F_1\bigg(\dfrac{2-p^{s+1}}3, \dfrac{1-2p^{s+1}}3; 1; x\bigg) {}_2F_1\bigg(\dfrac{2-p^{s-1}}3, \dfrac{1-2p^{s-1}}3; 1; x^p\bigg) \notag \\ &\quad \equiv {}_2F_1\bigg(\dfrac{2-2p^s}3, \dfrac{1-p^s}3; 1; x\bigg) {}_2F_1\bigg(\dfrac{2-2p^s}3, \dfrac{1-p^s}3; 1; x^p\bigg) \,(\mathrm{ mod}\,{p^s}), \ \ \operatorname{odd} \,s, \end{align} $$

(5-15) $$ \begin{align} &{}_2F_1\bigg(\dfrac{2-2p^{s+1}}3, \dfrac{1-p^{s+1}}3; 1; x\bigg) {}_2F_1\bigg(\dfrac{2-2p^{s-1}}3, \dfrac{1-p^{s-1}}3; 1; x^p\bigg) \notag\\ &\quad \equiv {}_2F_1\bigg(\dfrac{2-p^s}3, \dfrac{1-2p^s}3; 1; x\bigg) {}_2F_1\bigg(\dfrac{2-p^s}3, \dfrac{1-2p^s}3; 1; x^p\bigg) \,(\mathrm{ mod}\,{p^s}), \ \ \operatorname{even} \,s. \end{align} $$

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

Consider the p-adic presentations

(5-16) $$ \begin{align} -1/3 &= 2\ell +1 + \ell p+(2\ell+1)p^2+\ell p^3+\cdots , \end{align} $$

(5-17) $$ \begin{align} -2/3 &= \ell + (2\ell+1)p+\ell p^2+ (2\ell+1)p^3+\cdots. \end{align} $$

Recall that $[-1/3]_s$ (respectively, $[-2/3]_s$ ) is the sum of the first s summands in Equation (5-16) (respectively, (5-17)). Then the congruences in Equations (5-14) and (5-15) imply that for $p=3\ell +2$ , $s\geqslant 1$ ,

(5-18) $$ \begin{align} &{}_2F_1\big([-\tfrac23]_{s+1}, [-\tfrac13]_{s+1};1;x\big)\, {}_2F_1\big([-\tfrac23]_{s-1}, [-\tfrac13]_{s-1};1;x^p\big) \notag\\ &\quad \equiv {}_2F_1\big([-\tfrac 23]_{s}, [-\tfrac13]_{s};1;x\big)\, {}_2F_1\big([-\tfrac23]_{s}, [-\tfrac13]_{s};1;x^p\big) \,(\mathrm{mod}\,{p^s}). \end{align} $$

5.4 Limits of $\overline {Q}_s(x)$

Define

$$ \begin{align*} \bar Q_s(x) = {}_2F_1\big(\big[\!-\tfrac 23\big]_{s}, \big[\!-\tfrac13\big]_{s};1;x\big). \end{align*} $$

Then for any prime $p>3$ ,

$$ \begin{align*} \bar Q_{s+1}(x)\bar Q_{s-1}(x^p) \equiv \bar Q_{s}(x)\bar Q_{s}(x^p) \,(\mathrm{mod}\,{p^s}) \end{align*} $$

by Equations (5-7) and (5-18).

Theorem 5.5. For any prime $p>3$ and integer $s\geqslant 1$ , the rational function ${\bar Q_{s+1}(x)}/{\bar Q_s(x^p)}$ is regular on the domain

$$ \begin{align*} \mathfrak D = \{ x\in{\mathbb Z}_p\ |\ |\bar Q_1(x)|_p=1\}. \end{align*} $$

The sequence $({\bar Q_{s+1}(x)}/{\bar Q_s(x^p)})_{s\geqslant 1}$ uniformly converges on $\mathfrak D$ . The limiting analytic function $f(x)$ equals the ratio ${F(x)}/{F(x^p)}$ on the disc $D_{0,1}$ , where $F(x):={}_2F_1(2/3,1/3;1;x)$ is defined by the corresponding convergent power series.

Proof. The proof is the same as the proof of Theorem 4.6.

5.5 Remark

Although the congruences in Equations (5-7) and (5-18) look the same for $p=3\ell +1$ and $p=3\ell +2$ , the proofs of them are different as already presented. The proof of Equation (5-7) 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 Equation (5-18) for $p=3\ell +2$ uses the interaction of the two master polynomials $\Phi _s(t,x)$ and $\Psi _s(t,x)$ . See the baby congruences in Equations (5-4), (5-10), (5-11).

5.6 Congruences related to $-\tfrac 15$ , $-\tfrac 25$ , $-\tfrac 35$ , $-\tfrac 45$

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 $s\geqslant 1$ ,

$$ \begin{align*} &{}_2F_1\big([-\tfrac45]_{s+1}, [-\tfrac15]_{s+1};1;x\big)\, {}_2F_1\big([-\tfrac35]_{s-1}, [-\tfrac25]_{s-1};1;x^p\big) \notag \\ &\quad \equiv {}_2F_1\big([-\tfrac 45]_{s}, [-\tfrac15]_{s};1;x\big)\, {}_2F_1\big([-\tfrac35]_{s}, [-\tfrac25]_{s};1;x^p\big) \,(\mathrm{mod}\,{p^s}), \end{align*} $$

$$ \begin{align*} &{}_2F_1\big([-\tfrac35]_{s+1}, [-\tfrac25]_{s+1};1;x\big)\, {}_2F_1\big([-\tfrac45]_{s-1}, [-\tfrac15]_{s-1};1;x^p\big) \notag\\ &\quad \equiv {}_2F_1\big([-\tfrac 35]_{s}, [-\tfrac25]_{s};1;x\big)\, {}_2F_1\big([-\tfrac45]_{s}, [-\tfrac15]_{s};1;x^p\big) \,(\mathrm{mod}\,{p^s}).\end{align*} $$

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

$$ \begin{align*} &{}_2F_1\big([-\tfrac45]_{s+1}, [-\tfrac15]_{s+1};1;x\big)\, {}_2F_1\big([-\tfrac45]_{s-1}, [-\tfrac15]_{s-1};1;x^p\big) \notag\\ &\quad \equiv {}_2F_1\big([-\tfrac 45]_{s}, [-\tfrac15]_{s};1;x\big)\, {}_2F_1\big([-\tfrac45]_{s}, [-\tfrac15]_{s};1;x^p\big) \,(\mathrm{mod}\,{p^s}), \end{align*} $$

$$ \begin{align*} & {}_2F_1\big([-\tfrac35]_{s+1}, [-\tfrac25]_{s+1};1;x\big)\, {}_2F_1\big([-\tfrac35]_{s-1}, [-\tfrac25]_{s-1};1;x^p\big) \notag \\ &\quad \equiv {}_2F_1\big([-\tfrac 35]_{s}, [-\tfrac25]_{s};1;x\big)\, {}_2F_1\big([-\tfrac35]_{s}, [-\tfrac25]_{s};1;x^p\big) \,(\mathrm{mod}\,{p^s}). \end{align*} $$

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 are 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

$$ \begin{align*} \Omega = \sum_i \,h_i\otimes h_i \ \ \in \ {\mathfrak g} \otimes {\mathfrak g}, \end{align*} $$

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 are the system of differential equations on a V-valued function $I(z_1,\ldots ,z_n)$ ,

$$ \begin{align*} \frac{\partial I}{\partial z_i}\ =\ \frac 1\kappa\,\sum_{j\ne i}\, \frac{\Omega_{i,j}}{z_i-z_j} I, \quad i=1,\ldots,n, \end{align*} $$

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) $$ \begin{align} \begin{aligned} \frac{\partial I}{\partial z_1} &= {\frac 12} \bigg( \frac{\Omega_{12}}{z_1 - z_2} +\frac{\Omega_{13}}{z_1 - z_3} \bigg) I , \quad \frac{\partial I}{\partial z_2} = {\frac 12} \bigg( \frac{\Omega_{21}}{z_2 - z_1}+\frac{\Omega_{23}}{z_2 - z_3} \bigg) I ,\quad\ \ \\ \frac{\partial I}{\partial z_3} &= {\frac 12} \bigg( \frac{\Omega_{31}}{z_3 - z_1}+\frac{\Omega_{32}}{z_3 - z_2}\bigg) I, \quad 0= I_1+I_2+I_3, \end{aligned} \end{align} $$

where $\Omega _{ij}=\Omega _{ji}$ and

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

Denote

$$ \begin{align*} &H_1(z) = {\dfrac 12} \bigg( \dfrac{\Omega_{12}}{z_1 - z_2}+\dfrac{\Omega_{13}}{z_1 - z_3} \bigg), \quad H_2(z)={\dfrac 12}\bigg( \dfrac{\Omega_{21}}{z_2 - z_1} +\dfrac{\Omega_{23}}{z_2 - z_3} \bigg), \\ & H_3(z)= {\dfrac 12}\bigg( \dfrac{\Omega_{31}}{z_3 - z_1} +\dfrac{\Omega_{32}}{z_3 - z_2}\bigg), \quad \nabla_i^{\operatorname{KZ}} = \dfrac{\partial}{\partial z_i} - H_i(z), \quad i=1,2,3. \end{align*} $$

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

$$ \begin{align*} \nabla_i^{\operatorname{KZ}}I=0, \quad i=1,2,3,\quad I_1+I_2+I_3 =0. \end{align*} $$

System in Equation (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

$$ \begin{align*} \Phi(t,z) = (t-z_1)^{-1/2}(t-z_2)^{-1/2}(t-z_3)^{-1/2} \end{align*} $$

and the column 3-vector

(6-2) $$ \begin{align} I^{(C)}(z) = (I_1(z),I_2(z),I_3(z)):= \int_{C} \bigg(\frac {\Phi(t,z)}{t-z_1}, \frac {\Phi(t,z)}{t-z_2}, \frac {\Phi(t,z)}{t-z_3}\bigg)\,dt , \end{align} $$

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 (See [V4]).

The function $I^{(C)}(z)$ is a solution of the system in Equation (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) $$ \begin{align} -\frac 12\, \bigg(\frac {\Phi(t,z)}{t-z_1} + \frac{\Phi(t,z)}{t-z_2} + \frac {\Phi(t,z)}{t-z_3}\bigg)\, =\, \frac{\partial\Phi}{\partial t}(t,z), \end{align} $$

(6-4) $$ \begin{align} \bigg(\frac{\partial }{\partial z_i}-\frac12 \sum_{j\ne i} \frac {\Omega_{i,j}}{z_i-z_j} \bigg) \bigg(\frac {\Phi(t,z)}{t-z_1}, \frac{\Phi(t,z)}{t-z_2},\frac {\Phi(t,z)}{t-z_3}\bigg)\, = \frac{\partial \Psi^i}{\partial t} (t,z), \end{align} $$

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

Theorem 6.2 (See [V1, Equation (1.3)]).

All solutions of the system in Equation (6-1) have this form. Namely, the complex vector space of solutions of the form in Equation (6-2) is $2$ -dimensional.

6.3 Solutions as vectors of first derivatives

Consider the elliptic integral

$$ \begin{align*} T(z) = T^{(C)}(z) = \int_C \Phi(t,z)\,dt. \end{align*} $$

Then,

$$ \begin{align*} I^{(C)}(z) = \, 2\, \bigg(\frac {\partial T^{(C)}}{\partial z_1}, \frac {\partial T^{(C)}}{\partial z_2}, \frac {\partial T^{(C)}}{\partial z_3}\bigg). \end{align*} $$

Denote $\nabla T = ( {\partial T}/{\partial z_1}, {\partial T}/{\partial z_2}, {\partial T}/{\partial z_3})$ . Then the column gradient vector of the function $T(z)$ satisfies the following system of (KZ) equations:

$$ \begin{align*} \nabla_i^{\operatorname{KZ}} \nabla T =0, \quad i=1,2,3,\quad \frac {\partial T}{\partial z_1} + \frac {\partial T}{\partial z_2}+ \frac {\partial T}{\partial z_3}=0. \end{align*} $$

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

$$ \begin{align*} \Phi_s(t,z) = ((t-z_1)(t-z_2)(t-z_3))^{(p^s-1)/2}. \end{align*} $$

Define the column 3-vector

$$ \begin{align*} I_s(z)=(I_{s,1}(z), I_{s,2}(z), I_{s,3}(z)) \end{align*} $$

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

$$ \begin{align*} \bigg(\frac {\Phi_s(t,z)}{t-z_1}, \frac {\Phi_s(t,z)}{t-z_2}, \frac {\Phi_s(t,z)}{t-z_3}\bigg). \end{align*} $$

Theorem 6.3 [V4].

The polynomial $I_s(z)$ is a solution of the system in Equation (6-1) modulo $p^s$ .

Proof. We have the following modifications of the identities in Equations (6-3), (6-4):

(6-5) $$ \begin{align}\begin{aligned} &\qquad\quad\frac {p^s-1}2\, \bigg(\frac {\Phi_s(t,z)}{t-z_1} + \frac{\Phi_s(t,z)}{t-z_2} + \frac {\Phi_s(t,z)}{t-z_3}\bigg)\, =\, \frac{\partial\Phi_s}{\partial t}(t,z), \\&\ \bigg(\frac{\partial }{\partial z_i} + \frac {p^s-1}2 \sum_{j\ne i} \frac {\Omega_{i,j}}{z_i-z_j} \bigg) \bigg(\frac {\Phi_s(t,z)}{t-z_1}, \frac{\Phi_s(t,z)}{t-z_2},\frac {\Phi_s(t,z)}{t-z_3}\bigg)\, = \frac{\partial \Psi_s^i}{\partial t} (t,z),\end{aligned}\end{align} $$

where $\Psi _s^i(t,z)$ is the column $3$ -vector $(0,\ldots ,0,-{\Phi _s(t,z)}/({t-z_i}),0,\ldots ,0)$ 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 $\Phi _s(t,z)$ ,

(6-6) $$ \begin{align} T_s(z) = (-1)^{(p^s-1)/2}\sum_{k_1+k_2+k_3 = (p^s-1)/2} \binom{(p^s-1)/2}{k_1}\binom{(p^s-1)/2}{k_2}\binom{(p^s-1)/2}{k_3} z_1^{k_1}z_2^{k_2}z_3^{k_3}. \end{align} $$

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

$$ \begin{align*} T_s(1,z_2,0) = P_s(z_2), \end{align*} $$

where $P_s(x)$ is defined in Equation (4-3). The gradient vector

$$ \begin{align*}\nabla T_s :=({\partial T_s}/{\partial z_1}, {\partial T_s}/{\partial z_2}, {\partial T_s}/{\partial z_3}) \end{align*} $$

of the $p^s$ -approximation polynomial $T_s(z)$ is a solution modulo $p^s$ of the system in Equation (6-1) since

$$ \begin{align*} \nabla T_s = \frac{1-p^s}2\,(I_{s,1}(z), I_{s,2}(z), I_{s,3}(z)). \end{align*} $$

Lemma 6.4. For $s\geqslant 1$ , the master polynomials satisfy the baby congruences,

$$ \begin{align*} \Phi_{s+1}(t,z) \Phi_{s-1}(t^p,z_1^p,z_2^p,z_3^p) \equiv \Phi_{s}(t,z) \Phi_{s}(t^p,z_1^p,z_2^p,z_3^p) \,(\mathrm{mod}\,{p^s}). \end{align*} $$

Theorem 6.5. For $s\geqslant 1$ ,

(6-7) $$ \begin{align} T_{s+1}(z_1,z_2,z_3) \, T_{s-1}(z_1^p,z_2^p,z_3^p) \,\equiv\, T_{s}(z_1,z_2,z_3)\, T_{s}(z_1^p,z_2^p,z_3^p) \,(\mathrm{mod}\,{p^s}). \end{align} $$

Proof. Let $h(t,z) = t^{1-p} ((t-z_1)(t-z_2)(t-z_3))^{(p-1)/2}$ . Then,

$$ \begin{align*} T_s(z) = \operatorname{CT}_t[h(t,z) h(t,z)^p\cdots h(t,z)^{p^{s-1}}]. \end{align*} $$

The tuple of Laurent polynomials $(h(t,z),h(t,z), \ldots )$ is admissible in the sense of Definition 2.6. Now the application of Theorem 2.9 gives the congruence in Equation (6-7).

6.6 Limits of $T_s(z)$

Denote $\bar T_s(z) := (-1)^{(p^s-1)/2} T_s(z)$ ; see Equation (6-6),

(6-8) $$ \begin{align} \mathfrak D = \{(z_1,z_2,z_3)\in {\mathbb Z}_p^3 \,\|\bar T_1(z_1,z_2,z_3)|_p=1\}. \end{align} $$

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 ${\mathbb Z}_p$ ,

$$ \begin{align*} \mathfrak D ={\bigcup}^o D_{\alpha,1}\times D_{\beta,1}\times D_{\gamma,1}, \end{align*} $$

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,

$$ \begin{align*} \mathfrak D = \{(z_1,z_2,z_3)\in {\mathbb Z}_p^3 \,\mid |\bar T_1(z_1^p,z_2^p,z_3^p)|_p=1\}. \end{align*} $$

Denote

$$ \begin{align*} \mathfrak E =\{ (1,z_2,0)\in {\mathbb Z}_p^3\ |\ |z_2|_p<1 \}. \end{align*} $$

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 ${\bar T_{s+1}(z)}/{\bar T_s(z^p)}$ is regular on $\mathfrak D$ . The sequence $({\bar T_{s+1}(z)}/{\bar T_s(z^p)})_{s\geqslant 1}$ uniformly converges on $\mathfrak D$ . The limiting analytic function $f(z)$ , restricted to $\mathfrak E$ , equals the ratio ${F(z_2)}/{F(z_2^p)}$ , where $F(z_2):={}_2F_1(1/2,1/2;1;z_2)$ is defined by the convergent power series in Equation (4-1). We also have $|f(z)|_p=1$ for every $z\in \mathfrak D$ .

Proof. Similarly to the proof of Theorem 4.6, we have $\bar T_s(z) \equiv \bar T_1(z) \bar T_1(z^p)\cdots \bar T_1(x^{p^{s-1}}) (\mathrm { mod}\,{p})$ . 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 ${\bar T_{s+1}(z)}/{\bar T_s(z^p)}$ are regular on $\mathfrak D$ .

The congruence in Equation (6-7) implies that

$$ \begin{align*} \bigg|\frac{\bar T_{s+1}(z)}{\bar T_{s}(z^p)} - \frac{\bar T_{s}(z)}{\bar T_{s-1}(z^p)}\bigg|_p\leqslant p^{-s} \quad\operatorname{for}\ z\in\mathfrak D. \end{align*} $$

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

We have $\bar T_s(1,z_2,0) = \bar P_s(z_2) = \sum _{k} \binom {(p^s-1)/2}{k}^2z_2^k$ . 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 $f(1,z_2,0) ={F(z_2)}/{F(z^p)}$ on $\mathfrak E$ . The theorem is proved.

Remark 6.8. The analytic function $f(z)$ of Theorem 6.10 exhibits behavior very different from the behavior of the corresponding ratio $T^{(C)}(z)/T^{(C)}(z^p)$ of complex elliptic integrals.

By Theorem 6.10, 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 ${F(z_1)}/{F(z_1^p)}$ , ${F(z_2)}/{F(z_2^p)}$ , ${F(z_3)}/{F(z_3^p)}$ , respectively.

In the complex case, for the ratio $T^{(C_1)}(z)/T^{(C_1)}(z^p)$ to be equal to ${F(z_1)}/{F(z_1^p)}$ 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 ${F(z_2)}/{F(z_2^p)}$ 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 ${F(z_3)}/{F(z_3^p)}$ 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 $z_3=0$ . However, 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

$$ \begin{align*} f_s(z)=T_{s}(z)/T_{s-1}(z^p), \quad \eta^{(i)}_s(z) = \frac{\partial T_s}{\partial z_i}(z)/T_s(z), \quad \eta^{(ij)}_s(z) = \frac{\partial^2 T_s}{\partial z_i\partial z_j}(z)/T_s(z). \end{align*} $$

Theorem 6.9. 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

(6-9) $$ \begin{align} &\eta^{(1)} + \eta^{(2)} + \eta^{(3)}=0, \qquad\qquad\quad\ \ \,\end{align} $$

(6-10) $$ \begin{align} &\kern1pt\eta^{(j1)} + \eta^{(j2)} + \eta^{(j3)}=0, \quad j=1,2,3, \end{align} $$

(6-11) $$ \begin{align} &\frac{\partial}{\partial z_j}\eta^{(i)}=\eta^{(ji)} - \eta^{(i)}\eta^{(j)}.\ \,\ \ \,\qquad\qquad\end{align} $$

Proof. Denote $\delta _i=z_i({\partial }/{\partial z_i})$ . By Theorem 6.7, the sequence $(f_s)$ uniformly converges to the analytic function f on $\mathfrak D$ . Therefore, the sequence of the derivatives $(({\partial }/{\partial z_i})f_s)$ uniformly converges on $\mathfrak D$ to $({\partial }/{\partial z_i})f$ . Hence, the sequence $((\delta _if_s)/f_s)$ uniformly converges on $\mathfrak D$ to the function $(\delta _if)/f$ . At the same time,

$$ \begin{align*} \frac{\delta_if_s}{f_s}(z) =\frac{\delta_i T_s}{T_s}(z)- p\, \frac{\delta_i T_{s-1}}{T_{s-1}}(z^p) \end{align*} $$

and, more generally,

$$ \begin{align*} \frac{\delta_if_{s-k}}{f_{s-k}}(z^{p^k}) =\frac{\delta_iT_{s-k}}{T_{s-k}}(z^{p^k}) -p\,\frac{\delta_iT_{s-k-1}}{T_{s-k-1}}(z^{p^{k+1}}) \quad\text{for } k=0,1,\ldots,s. \end{align*} $$

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

$$ \begin{align*} \sum_{k=0}^{r-1}\,p^k\, \frac{\delta_if_{s-k}}{f_{s-k}}(z^{p^k}) \,=\, \frac{\delta_iT_s}{T_s}(z)\,-\,p^r\, \frac{\delta_i T_{s-r}}{T_{s-r}}(z^{p^r}). \end{align*} $$

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

$$ \begin{align*} \sum_{k=0}^\infty \,p^k\,\frac{\delta_i f}{f}(z^{p^k}) =\lim_{s\to\infty} \frac{\delta_iT_s}{T_s}(z). \end{align*} $$

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

$$ \begin{align*} \eta^{(i)}(x) =\lim_{s\to\infty} \frac{\frac{\partial}{\partial z_i}T_s}{T_s}(z) =\frac1{z_i}\sum_{k=0}^\infty \,p^k\,\frac{\delta_i f}{f}(z^{p^k}). \end{align*} $$

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 $\eta ^{(ijk\ldots )}$ . Note that Equation (6-11) comes out from differentiating logarithmic derivatives.

Equations (6-9) and (6-10) follow from Equation (6-5). The theorem is proved.

Theorem 6.10. We have the following system of equations on $\mathfrak D$ :

(6-12) $$ \begin{align} \begin{aligned} &\hspace{-12pt} \begin{pmatrix} \eta^{(11)} \\ \eta^{(12)} \\ \eta^{(13)} \end{pmatrix} = {\dfrac 12} \bigg( \dfrac{\Omega_{12}}{z_1 - z_2}+\dfrac{\Omega_{13}}{z_1 - z_3} \bigg) \begin{pmatrix} \eta^{(1)} \\ \eta^{(2)} \\ \eta^{(3)} \end{pmatrix}, \quad \begin{pmatrix} \eta^{(21)} \\ \eta^{(22)} \\ \eta^{(23)} \end{pmatrix} = {\dfrac 12} \bigg( \dfrac{\Omega_{21}}{z_2 - z_1}+\dfrac{\Omega_{23}}{z_2 - z_3} \bigg) \begin{pmatrix} \eta^{(1)} \\ \eta^{(2)} \\ \eta^{(3)} \end{pmatrix},\\ &\begin{pmatrix} \eta^{(31)} \\ \eta^{(32)} \\ \eta^{(33)} \end{pmatrix}= {\dfrac 12} \bigg( \dfrac{\Omega_{31}}{z_3 - z_1}+\dfrac{\Omega_{33}}{z_3 - z_2} \bigg) \begin{pmatrix} \eta^{(1)} \\ \eta^{(2)} \\ \eta^{(3)} \end{pmatrix}, \quad \eta^{(1)}+\eta^{(2)}+ \eta^{(3)} =0. \end{aligned} \end{align} $$

Proof. The theorem follows from Theorems 6.3 and 6.10.

Theorem 6.11. The column vector

$$ \begin{align*} \vec \eta(z):=(\eta^{(1)}(z), \eta^{(2)}(z), \eta^{(3)}(z)) \end{align*} $$

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 Equation (6-12) written as

(6-13) $$ \begin{align} \frac\partial{\partial z_i} \vec \eta = (H_i -\eta^{(i)})\,\vec\eta,\quad i=1,2,3. \end{align} $$

Hence, $\vec \eta (z)$ equals zero identically on $\mathfrak D$ . On the other hand, $\eta ^{(2)}(1,0,0) = F'(0)/F(0) = 1/4$ 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,

$$ \begin{align*} [\nabla^{\operatorname{KZ}}_i, \nabla^{\operatorname{KZ}}_j]=0 \quad \text{for all }\,i,j. \end{align*} $$

The flat sections of the KZ connection are solutions of the system in Equation (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,

$$ \begin{align*} \mathcal L := \bigcup_{a\in \mathfrak D}\,\mathcal L_a \,\to\, \mathfrak D \end{align*} $$

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

Theorem 6.12. 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 the sections $\nabla _i s(z)$ , $i=1,2,3$ , also are sections of $\mathcal L \,\to \, \mathfrak D$ .

Proof. The theorem follows from Equation (6-13).

Remark 6.13. 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 in Equation (6-1). Such a function is a solution of the system of equations $ {\partial u}/{\partial z_i} = - \eta ^{(i)} u$ , $i=1,2,3$ . This system is compatible since $ {\partial \eta ^{(j)}}/{\partial z_i} = \eta ^{(ij)} - \eta ^{(i)}\eta ^{(j)} = {\partial \eta ^{(i)}}/{\partial z_j}$ .

Remark 6.14. 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 6.15. 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; see [V4].

Remark 6.16. 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; see [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 $\Phi _s(t+z_3,z) = ((t-(z_1-z_3))(t-(z_2-z_3))t)^{(p^s-1)/2}$ . It is easy to see that $U_1(z)=T_1(z)\ \,(\mathrm {mod}\,{p})$ . Similarly to Theorem 6.5, we conclude that

$$ \begin{align*} \phantom{aaa} U_{s+1}(z_1,z_2,z_3)\, U_{s-1}(z_1^p,z_2^p,z_3^p) \,\equiv\, U_{s}(z_1,z_2,z_3)\, U_{s}(z_1^p,z_2^p,z_3^p) \,(\mathrm{mod}\,{p^s}). \end{align*} $$

Hence, the sequence $({U_{s+1}(z)}/{U_s(z^p)})_{s\geqslant 1}$ uniformly converges to an analytic function on the domain $\mathfrak D$ defined in Equation (6-8). The vector-valued polynomial $\nabla U_s(z) =({\partial U_s}/{\partial z_1}, {\partial U_s}/{\partial z_2}, {\partial U_s}/{\partial z_3} )$ is a solution modulo $p^s$ of the KZ equations in Equation (6-1); see [V4, Theorem 9.1], and see the proof of Theorem 6.3. Consider the function

$$ \begin{align*} \vec{\mu}=(\mu^{(1)}, \mu^{(2)},\mu^{(3)}):=\lim_{s\to\infty} \frac{\nabla U_s}{U_s} \end{align*} $$

defined on the same domain $\mathfrak D$ . Similarly to the proofs of Theorems 6.9–6.12, we conclude that the function $\vec {\mu }(z)$ is nonzero on $\mathfrak D$ and its values span an analytic line subbundle

$$ \begin{align*} \mathcal M := \bigcup_{a\in \mathfrak D}\,\mathcal M_a \,\to\, \mathfrak D \end{align*} $$

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

Proof. The proof rests on the following two lemmas.

Lemma 6.18. 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 6.19. For $i=1,2,3$ , the functions ${\partial T_s}/{\partial z_i}(z)/ T_s(z)$ and ${\partial U_s}/{\partial z_i}(z)/ U_s(z)$ 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 $\omega (x) =F'(x)/F(x)$ , where $F(x)\ \mathrm{is\ defined\ as} {}_2F_1(1/2, 1/2; 1;x)$ . We have $\omega (x) = \lim _{s\to \infty }({P_s'(x)}/{P_s(x)})$ on $D_{0,1}$ . Introduce new variables

$$ \begin{align*} u_1=z_1-z_3, \quad u_2=\frac{z_2-z_3}{z_1-z_3}, \quad u_3=z_1+z_2+z_3, \end{align*} $$

and a vector-valued function

$$ \begin{align*} \vec\omega(u) = \frac1{u_1} (\!- 1/2 - \omega(u_2) u_2,\, \omega(u_2),\, 1/2 + \omega(u_2)(u_2-1)). \end{align*} $$

Define

$$ \begin{align*} \hat{\mathfrak D}_0 = \{(z_1,z_2,z_3)\in {\mathbb Q}_p^3\ |\ z_i\ne z_j\quad \text{for all } i\ne j\}. \end{align*} $$

For any $\sigma =(i,j,k)\in S_3$ , define

$$ \begin{align*} &\phantom{aaaaa} \hat {\mathfrak D}_1^\sigma =\bigg\{(z_1,z_2,z_3)\in \hat {\mathfrak D}_0\ \bigg|\ \ \frac{z_j-z_k}{z_i-z_k}\in {\mathbb Z}_p,\ \bigg| g\bigg(\frac{z_j-z_k}{z_i-z_k}\bigg)\bigg|_p = 1\bigg\}, \\ &\hat{\mathfrak D}_2^\sigma = \bigg\{(z_1,z_2,z_3)\in \hat {\mathfrak D}_0\ \bigg|\ \frac{z_i-z_k}{z_j-z_k} \in \hat {\mathfrak D}_1^\sigma\bigg\}, \quad \hat{\mathfrak D}^\sigma= \hat{\mathfrak D}_1^\sigma\cup\hat {\mathfrak D}_2^\sigma, \quad \hat{\mathfrak D} = \sum_{\sigma\in S_3}\hat{\mathfrak D}^\sigma, \end{align*} $$

where $g(\lambda )$ is the Igusa polynomial in Equation (4-13).

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 6.20. 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 identify 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

$$ \begin{align*} U_s(z) &= (z_1-z_3)^{(p^s-1)/2} P_s\bigg(\dfrac{z_2-z_3}{z_1-z_3}\bigg), \quad \dfrac{\partial U_s}{\partial z_2} = (z_1-z_3)^{(p^s-1)/2-1} P_s'\bigg(\dfrac{z_2-z_3}{z_1-z_3}\bigg) \\ \dfrac{\partial U_s}{\partial z_1} &= \dfrac{p^s-1}2\,\operatorname{const}\,(z_1-z_3)^{(p^s-1)/2-1} P_s\bigg(\dfrac{z_2-z_3}{z_1-z_3}\bigg)\\ &\quad- \operatorname{const}\,(z_1-z_3)^{(p^s-1)/2-1} \dfrac{z_2-z_3}{z_1-z_3} P_s'\bigg(\dfrac{z_2-z_3}{z_1-z_3}\bigg). \end{align*} $$

Hence,

$$ \begin{align*} \dfrac 1{U_s(z)}\bigg(\dfrac{\partial U_s}{\partial z_1}, \dfrac{\partial U_s}{\partial z_2}, \dfrac{\partial U_s}{\partial z_3}\bigg) = \dfrac 1{u_1}\bigg(\dfrac{p^s-1}2 - u_2 \dfrac{P_s'(u_2)}{P_s(u_2)},\,\dfrac{P_s'(u_2)}{P_s(u_2)}, \,-\dfrac{p^s-1}2 + (u_2-1) \dfrac{P_s'(u_2)}{P_s(u_2)}\bigg). \end{align*} $$

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 $\overline {P}_s(x)$

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

$$ \begin{align*} \bar P_{4}(x) \bar P_{2}(x^p) - \bar P_{3}(x) \bar P_{3}(x^p) \equiv 0\,(\mathrm{mod}\,{p^{3}}). \end{align*} $$

In particular, for the coefficient of $x^{N_0+N_1p+N_2p^2 +N_3p^3}$ in $\bar P_4(x) \bar P_2(x^p) -\bar P_3(x)\bar P_3(x^p)$ ,

$$ \begin{align*} &\sum_{\genfrac{}{}{0pt}1{\scriptstyle k_1+l_1=N_1}{\scriptstyle k_2+l_2=N_2}} \Bigg(\binom{(p^4-1)/2}{N_0+k_1p+k_2p^2+N_3p^3}^2 \binom{(p^2-1)/2}{l_1+l_2p}^2 \notag \\ &\qquad - \binom{(p^3-1)/2}{N_0+k_1p+k_2p^2}^2 \binom{(p^3-1)/2}{l_1+l_2p+N_3p^2}^2\,\Bigg) \equiv 0 \,(\mathrm{mod}\,{p^{3}}). \end{align*} $$

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

$$ \begin{align*} A(a,b;c,c';d,d') = \binom{(p^4-1)/2}{a+cp+dp^2+bp^3}^2 \binom{(p^2-1)/2}{c'+d'p}^2 -\binom{(p^3-1)/2}{a+cp+dp^2}^2 \binom{(p^3-1)/2}{c'+d'p+bp^2}^2 \end{align*} $$

and

(7-1) $$ \begin{align} &B(a,b;c,c';d,d') = \operatorname{Sym} A(a,b;c,c';d,d')\notag\\ &\quad :=A(a,b;c,c';d,d') + A(a,b;c',c;d,d')+A(a,b;c,c';d',d)+A(a,b;c',c;d',d). \end{align} $$

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

More generally, define

$$ \begin{align*} k=(k^{(1)}, \ldots,k^{(s)}),\quad k^{(i)}=(k^{(i)}_1, k^{(i)}_2), \end{align*} $$

$$ \begin{align*} A(a,b;k) = \binom{(p^{s+2}-1)/2}{a+\sum_{i=1}^sk^{(i)}_1p^i+bp^{s+1}}^2 \binom{(p^{s}-1)/2}{\sum_{i=1}^sk^{(i)}_2p^{i-1}}^2 -\binom{(p^{s+1}-1)/2}{a+\sum_{i=1}^sk^{(i)}_1p^i}^2 \binom{(p^{s+1}-1)/2}{\sum_{i=1}^sk^{(i)}_2p^{i-1}+bp^{s}}^2. \end{align*} $$

Set

$$ \begin{align*} B(a,b;k) = \operatorname{Sym} A(a,b;k), \end{align*} $$

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 Equation (7-1).

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 Papers [BV, Vl]

After reading this paper, Masha Vlasenko was able to invent a new proof of our congruences in Theorems 4.2 and 5.1 (private communication). Her proof was based on the results of [BV, Vl].