Polynomial Solutions Modulo $p^s$ of Differential KZ and Dynamical Equations

We construct polynomial solutions modulo $p^s$ of the differential KZ and dynamical equations where $p$ is an odd prime number.


Introduction
The KZ equations were introduced by Knizhnik and Zamolodchikov [7] to describe the differential equations for conformal blocks on the Riemann sphere.Different versions of the KZ equations appear in mathematical physics, algebraic geometry and the theory of special functions, see, for example, [3,9].One of the important properties of the KZ equations is their realization as suitable Gauss-Manin connections.This construction gives a presentation of solutions of the KZ equations by multidimensional hypergeometric integrals, see [1,2,13].
The fact that certain integrals of closed differential forms over cycles satisfy a linear differential equation follows by Stokes' theorem from a suitable cohomological relation, in which the result of the application of the corresponding differential operator to the integrand of an integral equals the differential of a differential form of one degree lower.Such cohomological relations for the KZ equations associated with arbitrary Kac-Moody algebras were developed in [14].
The KZ equations possess a bispectrality property -they have a compatible system of dynamical equations with respect to associated dynamical parameters, see [4,5,9,11,18].
Let p be an odd prime.In [15,23], the differential KZ equations were considered modulo p s , and polynomial solutions modulo p s were constructed as analogs of the hypergeometric integrals.The construction was based on the fact that all cohomological relations described in [14] are defined over Z and can be reduced modulo p s .Studying solutions modulo of p s sheds light on solutions of the KZ equations both over the field of complex numbers and over p-adic fields, for example, see [16].
In this paper, we consider the joint system of the differential KZ and differential dynamical equations, the system introduced in [5], and construct polynomial solutions modulo p s of the joint system as analogs of the corresponding hypergeometric integrals with an exponential term.For this purpose, one needs to represent the exponential function e λt with an integer parameter λ by a polynomial in t modulo p s .This can be done after replacing λ with pλ.
An interesting problem is to study the p-adic limit of the constructed polynomial solutions modulo p s as s → ∞, see examples of this limit for the differential KZ equations in [22,23,24].
The joint system of the KZ and dynamical equations has many versions: differential KZ equations and differential dynamical equations, differential KZ equations and difference dynamical equations, difference KZ equations and differential dynamical equations, difference KZ equations and difference dynamical equations, see, for example, [4,9,11,18].The polynomial solutions modulo p s are constructed in this paper only for the original joint system of the differential KZ and differential dynamical equations, although there are examples of polynomial solutions modulo p s in other cases, see [10,12,22] and also Appendix A.
In the remainder of the introduction we consider an example.

Solutions over C
Consider the complex master function and the tuple of integrals where δ is a 1-cycle.
Theorem 1.1.The tuple I(z, λ) satisfies the joint system of KZ and dynamical equations The system of equations (1.2) and (1.3) is called the KZ equations of this example, the system of equations (1.4) is called the dynamical equation.The solutions I(z, λ) are called the hypergeometric solutions.
Proof .The proof uses the following identities: For j ̸ = i, we have This proves the first equation.Then gives the second equation.We also have The complex vector space of (multi-valued) solutions of the joint system of KZ and dynamical equations (1.2)-(1.4) is (2g + 1)-dimensional.Every solution of the joint system has the integral presentation (1.1) for a suitable cycle δ, see [8,Theorem 6.1] and an example in [8,Introduction].

Exponential function
We have 1) .
We set λ (m) = 0 for m < 0 by convention. Let , where Z p is the ring of p-adic integers and z = (z 1 , . . ., z 2g+1 ) are parameters.Consider the decomposition where each c k (λ, z) is a linear function in finitely many symbols λ (m) , m = 0, 1, . . ., whose coefficients lie in Z p [z]. Lemma 1.2.Let s, ℓ be positive integers.Then the coefficient of t ℓp s −1 in the series d dt e λt f (t, z) is divisible by p s , that is, all coefficients of the corresponding linear function in symbols λ (m) , m ≥ 0, are divisible by p s .
Proof .It is enough to prove the lemma for f (t) = t a .Then and

Remarks on p r
Let v p (a) denote the p-adic evaluation of a.
Let r 1 , r 2 be relatively prime positive integers.Denote r = r 1 /r 2 .Assume that r > 1/(p − 1).Then for a positive integer k, we have /k! does not grow as k → ∞, and so we get an infinite series rather than a polynomial.

Reformulation of the equations
Change the variable λ → pλ.Then the KZ and dynamical equations take the form ) ) For any positive integer s, we construct below some vectors of polynomials with coefficients in Z p which satisfy the KZ equations (1.5), (1.6) modulo p s if λ ∈ Z p and satisfy the dynamical equations (1.7) modulo p s if λ ∈ Z × p .

Solutions modulo p s
For a positive integer s, define Consider the Taylor expansions where each c o m (z) is a vector of polynomials in z with integer coefficients, and For any positive integer ℓ, denote All coordinates of this vector are polynomials in z, λ with coefficients in Z p .
We call such solutions the p s -hypergeometric solutions of the joint system of the KZ and dynamical equations.
Proof .The proof uses the following identities: Take the coefficient of t ℓp s −1 in both sides of the equation.As the result, we obtain modulo p s , We have .
Take the coefficient of t ℓp s −1 in both sides of the equation.As the result, we obtain modulo p s , does not contribute to this result by Lemma 1.2.We also have modulo p s , Take the coefficient of t ℓp s −1 in both sides of the equation.As the result, we obtain modulo p s , Notice that the coefficient of Then a p s -hypergeometric solution I ℓ (z, µ) is zero unless ℓ = 1, . . ., g.
Proof .The degree of the polynomial Ψ o s (t, z) with respect to t equals The degree of the polynomial E s (pλt) with respect to t is not greater than d(s) ≤ s p−1 p−2 + 1. Hence the degree of Ψ s (t, z, µ) is not greater than If inequality (1.8) holds, then the polynomial Ψ s (t, z, λ) does not have monomials of degree ℓp s −1 for ℓ > g. ■ For any p s -hypergeometric solution I ℓ (z, λ), consider its λ-independent term I ℓ (z, 0) = I ℓ 1 (z, 0), . . ., I ℓ 2g+1 (z, 0) .This is a vector of polynomials in z with integer coefficients.It is a solution modulo p s of the KZ equations (1.5) and (1.6) with λ = 0. We have since this sum is the coefficient of The solution I ℓ (z, λ) is a λ-deformation of the vector I ℓ (z, 0).
The proof is the same as the proof of Theorem 1.5.Theorem 1.5 is a special case of Theorem 1.8 for r = 1.
Notice that for degree reasons, Theorem 1.8 gives for every s only finitely many solutions I ℓ (z, λ).
Notice that this theorem gives infinitely many solutions I ℓ (z, λ).

Exposition of material
In Section 2, we describe the hypergeometric solutions of the joint system of the differential KZ and dynamical equations associated with sl 2 and explain their reduction to polynomial solutions modulo p s .In Section 2.6, we briefly comment on how the results of Section 2 are extended to the joint system of the differential KZ and dynamical equations associated with arbitrary simple Lie algebras.In Appendix A, we consider an example and explain how to construct the polynomial solutions modulo p s of qKZ difference equations.
2 The sl 2 differential KZ and dynamical equations

Equations
Let e, f , h be the standard basis of the complex Lie algebra sl 2 with relations [e, f ] = h, the Casimir element.
Given n, for any x ∈ sl 2 , let x (i) ∈ U (sl 2 ) ⊗n be the element equal to x in the i-th factor and to 1 in other factors.Similarly, for 1 ≤ i < j ≤ n, let Ω (i,j) ∈ U (sl 2 ) ⊗n be the element equal to Ω in the i-th and j-th factors and to 1 in other factors.
Let ⊗ n i=1 V i be a tensor product of sl 2 -modules and κ ∈ C × .The system of differential equations on a ⊗ n i=1 V i -valued function I(z 1 , . . ., z n , λ), ) is called the system of KZ and dynamical equations.The system depends on the parameter κ.

sl 2 -modules
For a nonnegative integer i, denote by L i the (i + 1)-dimensional module with a basis v i , f v i , . . ., f i v i and action ≥0 , with j s ≤ m s for s = 1, . . ., n, the vectors where 1 s = (0, . . ., 0, 1, 0, . . ., 0) with 1 staying at the s-th place.For w ∈ Z, introduce the weight subspace

Solutions over
For any function or differential form For J = (j 1 , . . ., j n ) ∈ I k , define the weight function For example, The function where δ(z, λ) in {(z, λ)} × C k t is a horizontal family of k-dimensional cycles of the twisted homology defined by the multivalued function Φ(t, z, λ), see, for example, [20,21].The cycles δ(z, λ) are multi-dimensional analogs of Pochhammer double loops.In Section 2.4, we sketch the proof of Theorem 2.1 following [5,14].The intermediate statement in this proof will be used later when constructing solutions modulo p s of the KZ and dynamical equations.The proof is based on the following cohomological relations.

Identities for differential forms
It is convenient to reformulate the definition of the hypergeometric integrals (2.3).Given k, n ∈ Z >0 and a multi-index J = (j 1 , . . ., j n ) with |J| ≤ k, denote Here a J,ℓ is the ℓ-th factor of the product in the right-hand side.Denote If |J| = k, then for any fixed z ∈ C n , we have the identity Example 2.3.For k = n = 2, we have The hypergeometric integrals (2.3) can be defined in terms of the differential forms α J : Theorem 2.4 ( [5,15]).
(i) We have the following algebraic identity for differential forms in t, z depending on parameter λ: where d t,z denotes the differential with respect to variables t, z.
(ii) For any fixed z, λ, we have the following algebraic identity for differential forms in t depending on parameters z, λ: where d t denotes the differential with respect to variables t.
The assumptions in part (ii) mean that all differentials dz 1 , . . ., dz n appearing in α must be put to zero to obtain identity (2.5).

Solutions modulo p
let p > 2 be a prime number such that p does not divide the numerator of κ.Change the variable λ → pλ in the KZ and dynamical equations (2.1), (2.2).Then the equations take the form Choose positive integers M l for l = 1, . . ., n, M i,j for 1 ≤ i < j ≤ n, and M 0 , such that Define the master polynomial This is a polynomial in t, z, λ with coefficients in Z p .Consider the Taylor expansion For any vector ℓ = (ℓ 1 , . . ., ℓ k ) with positive integer coordinates, denote All coordinates of this vector are polynomials in z, λ with coefficients in Z p .
We call such solutions the p s -hypergeometric solutions of the joint system of the KZ and dynamical equations.
Proof .The polynomial Ψ s (t, z, λ) satisfies identities (2.4) and (2.5) modulo p s .Taking the coefficient of t ℓ 1 p s −1 1 . . .t ℓ k p s −1 k in these identities kills the differentials with respect to t by Lemma 1.2.This proves the theorem.■ Remark 2.6.We observed in Section 1.7 that Theorem 1.5 can be generalized to Theorem 1.8 by replacing p with p r .Theorem 2.5 is generalized in the same way.We leave this exercise to readers.

Equations for other Lie algebras
The KZ and dynamical equations are defined for any simple Lie algebra g or more generally for any Kac-Moody algebra, see, for example, [5,14].Similarly to what is done in Section 2.5, we can construct polynomial solutions modulo p s of those KZ and dynamical equations.The construction of the polynomial solutions modulo p s in the sl 2 case is based on the algebraic identities for differential forms (2.4), (2.5).For an arbitrary Kac-Moody algebra, these algebraic identities were developed in [5,14].

A Solutions modulo p s of the qKZ equations
In Sections 1 and 2, we constructed solutions modulo p s of the differential KZ and dynamical equations by first p s -approximating the integrand of the hypergeometric solutions and then taking the coefficients of the monomials t ℓp s −1 in the Taylor expansion of the approximated integrand.In this appendix, we show that the same idea can be applied to the qKZ difference equations, but instead of considering the Taylor expansion of the approximated integrand and then taking the coefficients of the monomials t ℓp s −1 we first take the expansion of the p s -approximated integrand into a sum of Pochhammer polynomials [t] m and then take the coefficients of the Pochhammer polynomials with indices m = ℓp s − 1. See this approach in [10] for the qKZ equations with no exponential term.On the hypergeometric solutions of the qKZ difference equations see, for example, [17,19].
In this paper, we consider only a baby example of the qKZ equations which illustrates these constructions.More general qKZ equations are given by multidimensional Jackson integrals whose integrand is a product of exponential factors and ratios of gamma functions like in (A.1).To construct polynomial solutions modulo p s of the qKZ equations, we replace the exponential factors in the integrand by the product of the corresponding functions E r,s (t), replace the ratios of gamma functions by the corresponding Pochhammer polynomial as in (A.5), expand the result into Pochhammer polynomials and take the suitable coefficients of that expansion like in Theorem A.2.

Remark 1 . 10 .
Another possibility to extend the construction of polynomial solutions is to replace the ring Z p [p r ] by another p-adic ring, e.g., Z p [ζ], where ζ is a p m -th root of 1.
Let f (t) be a meromorphic function and a ∈ C. The sum