• E -mail: vadim.schechtman@math.univ-toulouse.fr E -mail: anv@email.unc.edu , supported in part by NSF grants DMS-1362924, DMS-1665239

The KZ equations were discovered by physicists Vadim Knizhnik and Alexander Zamolodchikov [KZ] to describe the differential equations for conformal blocks on sphere in the Wess-Zumino-Witten model of conformal field theory. As I.M. Gelfand said, the KZ equations are remarkable differential equations discovered by physicists, defined in terms of a Lie algebra and whose monodromy is described by the corresponding quantum group. It turned out that the KZ equations are realized as suitable Gauss-Manin connections and its solutions are represented by multidimensional hypergeometric integrals, see [CF,DJMM,Mat,SV1,SV2,SV3]. 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 form of one degree less. Such cohomological relations for the KZ equations associated with Kac-Moody algebras were developed in [SV3].

The goal of this paper is to construct polynomial solutions of the KZ differential equations over a finite field F p with p elements, where p is a prime number, as analogs of the hypergeometric solutions constructed in [SV3]. Our construction is based on the fact that all cohomological relations described in [SV3] are defined over Z and can be reduced modulo p. We learned how to construct polynomial solutions in this situation out of hypergeometric solutions from the remarkable paper by Yu.I. Manin [Ma], see a detailed exposition of Manin's idea in Section "Manin's Result: The Unity of Mathematics" in the book [Cl] by H.C. Clemens. In the remainder of the introduction we consider the example of one-dimensional hypergeometric and p-hypergeometric integrals as an illustration of our constructions and results. The multidimensional case is considered in Sections 2-4. where

The integrals are over a closed (Pochhammer) curve γ in C -{z 1 , . . . , z n } on which one fixes a uni-valued branch of the master function to make the integral well-defined. Starting from such a curve chosen for given {z 1 , . . . , z n }, the vector I (γ) (z) can be analytically continued as a multivalued holomorphic function of z to the complement in C n to the union of the diagonal hyperplanes z i = z j .

Theorem 1.1. The vector I (γ) (z) satisfies the algebraic equation

and the differential KZ equations:

where

all other diagonal entries are

and the remaining off-diagonal entries are all zero.

Remark. The vector I (γ) (z) depends on the choice of the curve γ. Different curves give different solutions of the same KZ equations and all solutions of equations (1.3) and (1.4) are obtained in this way, if κ, m 1 , . . . , m n are generic.

Remark. The differential equations (1.4) are the KZ differential equations with parameter κ associated with the Lie algebra sl 2 and the singular weight subspace of weight |m| -2 of the tensor product of sl 2 -modules with highest weights m 1 , . . . , m n , see Section 2.

Remark. The KZ equations define a flat connection over the complement in C n to the union of all diagonal hyperplanes,

for all j, k.

Theorem 1.1 is a classical statement probably known in 19th century. Much more general algebraic and differential equations satisfied by analogous multidimensional hypergeometric integrals were considered in [SV3]. Theorem 1.1 is discussed as an example in [V2, Section 1.1].

Below we give a proof of Theorem 1.1. A modification of this proof in Section 1.2 will produce for us polynomial solutions of the equations (1.3) and (1.4) modulo a prime p.

Proof of Theorem 1.1. Equations (1.3) and (1.4) are implied by the following cohomological identities. We have

where d t denotes the differential with respect to the variable t. This identity and Stokes' theorem imply equation (1.3). Denote

For any i = 1, . . . , n, let W i (t, z) be the vector of (0, . . . , 0, -1 t-z i , 0, . . . , 0) with nonzero element at the i-th place. Then

The proof of this identity is straightforward. Much more general identities of this type see in [SV3,Lemmas 7.5.5 and 7.5.7], cf. identities in Section 2.4.

Identity (1.8) and Stokes' theorem imply the KZ equation (1.4).

, where

In this case, the curve γ(z) may be thought of as a closed path on the elliptic curve

Each of these integrals is an elliptic integral. Such an integral is a branch of analytic continuation of a suitable Euler hypergeometric function up to change of variables.

That means that we project m a , κ, 2 to F p , calculate -ma κ , mam b 2κ in F p and then choose positive integers M a , M a,b satisfying these equations.

Fix an integer q. Consider the master polynomial

and the Taylor expansion with respect to the variable t of the vector of polynomials

where the Ī(i) (z, q) are n-vectors of polynomials in z with integer coefficients. Let

) n be the canonical projection of Ī(i) (z, q).

Theorem 1.2. For any integer q and positive integer l, the vector of polynomials I (lp-1) (z, q) satisfies equations (1.3) and (1.4).

The parameters q and lp -1 are analogs of cycles γ in Section 1.1.

Proof. To prove that I (lp-1) (z, q) satisfies (1.3) and (1.4) we consider the Taylor expansions at t = q of both sides of equations (1.6) and (1.8), divide them by dt, and then project the coefficients of (t -q) lp-1 to (F p [z]) n . The projections of the right-hand sides equal zero since d(t lp )/dt = lpt lp-1 ≡ 0 (mod p). (1.11) where ci = (c i 1 , . . . , ci n ). Let c i be the projection of ci to (F p [z]) n . Then the vector of polynomials

1 2 (I 1 , . . . , I i-1 , -I i + 2I j , I i+1 , . . . , I j-1 , 2I i -I j , I j+1 , . . . , I n )

≡ (-I 1 , . . . , -I i-1 , I i + I j , -I i+1 , . . . , -I j-1 , I i + I j , -I j+1 , . . . , -I n ) (mod 3). Equation (1.3) has the form I 1 (z) + • • • + I n (z) = 0. We may choose the master polynomial

Choose a nonnegative integer l. Then the vector I(z, q) := I (3l-1) (z, q) = (I 1 (z, q), . . . , I n (z, q)) of Theorem 1.2 has coordinates

and is a solution of (1.3) and (1.4) with values in (F 3 [z]) n for any q = 0, 1, 2. Expanding these solutions into polynomials homogeneous in z we obtain solutions in homogeneous polynomials, which stabilize with respect to n as follows. The vector I [r] (z) = (I n (z)), with coordinates

is a solution of (1.3) and (1.4) with values in (F 3 [z]) n if r ≡ n (mod 3) and r < n. Thus, the vector I [0] (z), with coordinates

is a solution with values in (F 3 [z]) n for n ≡ 0 (mod 3); the vector I (1) (z), with coordinates

is a solution for n ≡ 1 (mod 3) and so on. Note that the sum in (1.14) is the m-th elementary symmetric function in z 1 , . . . , z j , . . . , z n .

Solutions provided by Theorem 1.2 depend on parameters q, lp -1. In this example all solutions I [r] (z) can be obtained by putting q = 0 and varying lp -1 only.

1.3. Relation of polynomial solutions to integrals over F p . For a polynomial

Recall that the sum t∈Fp t i equals -1 if (p -1) i and equals zero otherwise.

(1.17) Theorem 1.3. Fix x 1 , . . . , x n , q ∈ F p . Consider the vector of polynomials

of Section 1.2. Assume that deg t F (t, x 1 , . . . , x n ) < 2p -2. Consider the polynomial solution I (p-1) (z 1 , . . . , z n , q) of equations (1.3) and (1.4) defined in front of Theorem 1.2. Then

This integral is a p-analog of the hypergeometric integral (1.2).

x 3 ) be the projective closure of the affine curve

as the sum over all points P ∈ Γ(x 1 , x 2 , x 3 ), where h(P ) is defined.

x 3 ) be the vector of polynomials appearing in the solution (1.12) of the KZ equations of Example 1.2 for n = 3. Then

Remark. Theorems 1.2 and 1.4 say that the integrals Γ(x 1 ,x 2 ,x 3 ) 1 t-x j are polynomials in x 1 , x 2 , x 3 ∈ F p and the triple of polynomials

Proof of Theorem 1.4. The proof is analogous to the reasoning in [Ma,Section 2] and [Cl]. The value of 1/(t -x j ) at the infinite point of Γ(x 1 , x 2 , x 3 ) equals zero. It is easy to see that

where the last equality is by formula (1.17).

Remark. In [Ma,Section 2] and in [Cl], an equation analogous to (1.21) is considered, where the left-hand side is the number of points on Γ(x 1 , x 2 , x 3 ) over F p and the right-hand side is the reduction modulo p of a solution of a second order Euler hypergeometric differential equation. Notice that the number of points on Γ(x 1 , x 2 , x 3 ) is the discrete integral over Γ(x 1 , x 2 , x 3 ) of the constant function h = 1. See details in Section "Manin's Result: The Unity of Mathematics" in [Cl].

Example 1.6. This example is a variant of Example 1.5. Let x 1 , x 2 , x 3 , x 4 ∈ F p . Let Γ(x 1 , x 2 , x 3 , x 4 ) be the projective closure of the affine curve

Let p > 3 be a prime. Let

be the vector of polynomials appearing in the solution (1.12) of the KZ equations of Example 1.2 for n = 4. Then 1,2,3,4. (1.23) Example 1.7. Let κ = 3,n = 3,. Choose the master polynomial

Consider the Taylor expansion

x 3 ) be the projective closure of the affine

over F p . The curve has 3 points at infinity. Theorem 1.5. Let p be a prime such that 3 (p -1). Let

x 3 ) be the vector of polynomials appearing in the solution (1.25) of the KZ equations. Then for j = 1, 2, 3 we have

Proof. The value of 1/(t -x j ) at infinite points of Γ equals zero. It is easy to see that

Notice that t∈Fp

= 0 since the polynomial under the sum is of degree p -2 which is less than p -1. The last equality in (1.28) is by formula (1.17).

Example 1.8.

Consider the Taylor expansion

where bi = ( bi 1 , bi 2 , bi 3 ). Let b i be the projection of bi to (F p [z]) 3 . Then the vector

Consider the Taylor expansion

). Let c i be the projection of ci to (F p [z]) 3 . Then the vector

is a solution of the corresponding KZ differential equations over F p [z] and 2I 1 (z) + 2I 2 (z) + I 3 (z) = 0.

For distinct x 1 , x 2 , x 3 ∈ F p let Γ(x 1 , x 2 , x 3 ) be the projective closure of the affine curve

over F p . The curve has genus 2 and one point at infinity. Theorem 1.6. Let p be a prime such that 3 divides p -1.

x 3 ) be the vector of polynomials appearing in the solution (1.30) of the KZ equations with n = 3,

x 3 ) be the vector of polynomials appearing in the solution (1.32) of the KZ equations with n = 3, κ = 3, m 1 = m 2 = 2, m 3 = 1. Then for j = 1, 2, 3 we have

Proof. The value of 1/(t -x j ) at infinite points of Γ equals zero. It is easy to see that

1.5. Resonances over C and F p . Under assumptions of Section 1.1 assume that

Then the vector I (γ) (z), defined in (1.1), in addition to the algebraic equation (1.3) and differential equations (1.4) satisfies the algebraic equation

Equation (1.36) follows from the cohomological relation:

Relation (1.36) manifests resonances in conformal field theory, where solutions of KZ equations represent conformal blocks and conformal blocks satisfy algebraic equations analogous to (1.36), see [FSV1,FSV2], Section 3.6.2 in [V2]. In conformal field theory the numbers m 1 , . . . , m n , κ are natural numbers. In that case the master function Φ(t, z) is an algebraic function and the hypergeometric integrals become integrals of algebraic forms over cycles lying on suitable algebraic varieties. The monodromy of the hypergeometric integrals I (γ) (z) in that case was studied in Sections 13 and 14 of [V1].

Relation (1.36) has an analog over F p .

Theorem 1.7. Under assumptions of Theorem 1.2 let I (lp-1) (z, q) ∈ F p [z] n be the polynomial solution of equations (1.3) and (1.4) described in Theorem 1.2. Assume that

Proof. The theorem follows from (1.37) similarly to the proof of Theorem 1.2. Namely, we consider the Taylor expansions at t = q of both sides of equation (1.37), divide them by dt, and then project the coefficients of (t -q) lp-1 to F p [z]. The projection coming from d t (tΦ) equals zero since d(t lp )/dt = lpt lp-1 ≡ 0 (mod p). The projection coming from 1 -n j=1 m j κ Φdt equals zero by (1.38). The projection coming from -n j=1 z j m j κ Φ dt t-z j

gives (1.39).

Example 1.9.

). Choose a positive integer r, such that r ≡ n (mod 3) and r < n. Then the vector I [r] (z) given by (1.14) satisfies equations (1.3), (1.4), and

n (z) ≡ 0 (mod 3). 1.6. Exposition of material. In Section 2 we describe the hypergeometric solutions of the KZ equations associated with sl 2 and explain their reduction to polynomial solutions over F p . In Section 3 we describe the resonance relations for sl 2 conformal blocks and construct their reduction over F p . In Section 4 we explain how the results of Section 2 and 3 are extended to the KZ equations associated with simple Lie algebras. This article was inspired by lectures on hypergeometric motives by Fernando Rodriguez-Villegas in May 2017 at MPI in Bonn. The authors thank him for stimulating discussions. We were also motivated by the classical paper by Yu.I. Manin [Ma], from which we learned how to construct solutions of differential equations over F p from cohomological relations between algebraic differential forms. The authors thank A. Buium, Yu.I. Manin, and W. Zudilin for useful discussions and the referee for comments and suggestions contributed to improving the presentation.

The article was conceived during the Summer 2017 Trimester program "K-Theory and Related Fields" of the Hausdorff Institute for Mathematics (HIM), Bonn. The authors are thankful to HIM for stimulating atmosphere and working conditions. The first author is grateful to Max Planck Institute for Mathematics for hospitality during a visit in June 2017.

In this section we describe solutions of the KZ equations associated with the Lie algebra sl 2 . The solutions to the KZ equations over C in the form of multidimensional hypergeometric integrals are known since the end of 1980s. The polynomial solutions of the KZ equations over F p in the form of F p -analogs of the multidimensional hypergeometric integrals are new.

2.1. sl 2 KZ equations. Let e, f, h be standard basis of the complex Lie algeba sl 2 with

is called the Casimir element. Given n, 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 the other factors. For i = 1, . . . , n and distinct z 1 , . . . , z n ∈ C introduce

the Gaudin Hamiltonians. For any κ ∈ C × and any i, k, we have

and for any x ∈ sl 2 and i we have

V i be a tensor product of sl 2 -modules. The system of differential equations

2.2. Irreducible sl 2 -modules. For a nonnegative integer i denote by L i the irreducible i + 1-dimensional module with basis v i , f v i , . . . , f i v i and action h.

0 , with j s m s for s = 1, . . . , n, the vectors

form a basis of L ⊗m . We have

For λ ∈ Z, introduce the weight subspace

Remark. The sl 2 -action on the sum of singular weight subspaces SingL ⊗m [|m| -2k] generates the entire sl 2 -module L ⊗m . If I(z 1 , . . . , z n ) is an L ⊗m -valued solution of the KZ equations, then for any x ∈ sl 2 the function x.I(z 1 , . . . , z n ) is also a solution, see (2.4). These observations show that in order to construct all L ⊗m -valued solutions of the KZ equations it is enough to construct all SingL ⊗m [|m| -2k]-valued solutions for all k and then generate the other solutions by the sl 2 -action.

For any function or differential form

For J = (j 1 , . . . , j n ) ∈ I k define the weight function

The function

is the L ⊗m [|m| -2k]-valued vector weight function.

Consider the L ⊗m [|m| -2k]-valued function (2.10) where γ(z) in {z} × C k t is a horizontal family of k-dimensional cycles of the twisted homology defined by the multivalued function Φ k,n,m (t, z, m), see [SV3,V1,V2]. The cycles γ(z) are multi-dimensional analogs of Pochhammer double loops. This theorem and its generalizations can be found, for example, in [CF,DJMM,SV1,SV2,SV3].

The solutions in Theorem 2.1 are called the multidimensional hypergeometric solutions of the KZ equations. The coordinate functions (2.11) are called the multidimensional hypergeometric functions associated with the master function Φ k,n,m .

The fact that solutions in Theorem 2.1 take values in SingL ⊗m [|m| -2k] may be reformulated as follows. For any J ∈ I k-1 , we have n s=1

J+1s (z) = 0, (2.12) where we set I (γ)

The pair consisting of the KZ equations (1.4) and hypergeometric solutions (1.2) is identified with the pair consisting of the KZ equations (2.5) and hypergeometric solutions (2.10) with values in SingL ⊗m [|m| -2]. In this case the system of equations in (2.12) is identified with equation (1.3).

2.4. Proof of Theorem 2.1. We sketch the proof following [SV3]. The reason to present a proof is to show later in Section 2.5 how a modification of this reasoning leads to a construction of polynomial solutions of the KZ equations over F p .

The proof of Theorem 2.1 is a generalization of the proof of Theorem 1.1 and is based on cohomological relations.

It is convenient to reformulate the definition of the hypergeometric integral (2.10). Given k, n ∈ Z >0 and a multi-index J = (j 1 , . . . , j n ) with |J| k, introduce a differential form

which is a logarithmic differential form on C n × C k with coordinates z, t. If |J| = k, then for any z ∈ C n we have on {z} × C k the identity

The hypergeometric integrals (2.10) can be defined in terms of the differential forms η J :

We shall use the following algebraic identities for logarithmic differential forms.

Proof. Identity (2.14) is the spacial case of Theorem 6.16.2 in [SV3] for the Lie algebra sl 2 . Identity (2.15) is a special case of Theorem 7.5.2" in [SV3] for the Lie algebra sl 2 .

Corollary 2.3. On C n × C k we have

where the differential is taken with respect to variables z, t. Now we deduce from identity (2.14) the following formula (2.20). Since |J| = k -1, we can write (2.17) where the dots denote the terms having differentials dz i and c J,l (t, z) are rational functions of the form P J,l (t, z)

where P J,l (t, z) is a polynomial in t, z with integer coefficients. Also for any s = 1, . . . , n we have (2.19) where the dots denote the terms having differentials dz i . Formula (2.14) implies that for any J with |J| = k -1 we have the identity

where d t denotes the differential with respect to the variables t. Now we deduce from identity (2.16) the following formula (2.23). Fix i ∈ {1, . . . , n}. For any

where the dots denote the terms which contain dz j with j = i, and the coefficients c J,i,l (t, z) are rational functions in t, z of the form

where P J,i,l (t, z) is a polynomial in t, z with integer coefficients.

Formula (2.16) implies that for any i ∈ {1, . . . , n} we have

where d t denotes the differential with respect to the variables t.

Integrating both sides of equations (2.20) and (2.23) over γ(z) and using Stokes' theorem we obtain equations (2.12) and (2.5) for the vector I (γ) (z) in (2.10). Theorem 2.1 is proved.

let p > 2 be a prime number such that p does not divide the numerator of κ. In this case equations (2.12) and (2.5) are well-defined over the field F p and we may discuss their polynomial solutions in F p [z 1 , . . . , z n ].

Choose positive integers M s for s = 1, . . . , n, M i,j for 1 i < j n, and M 0 , such that

Fix integers q = (q 1 , . . . , q k ). Let t = (t 1 , . . . , t k ), z = (z 1 , . . . , z n ) be variables. Define the master polynomial

Consider the Taylor expansion of the vector

) is a polynomial in t, z with integer coefficients due to the positivity of the integers M s , M i,j , M 0 and the definition of the weight functions W J (t, z). Hence the Taylor coefficients Ī(i 1 ,...,i k ) (z, q) are vectors of polynomials in z with integer coefficients. Let I (i 1 ,...,i k ) (z, q) ∈ (F p [z]) dim L ⊗m [|m|-2k] be their canonical projection modulo p.

Theorem 2.4. For any integers q = (q 1 , . . . , q k ) and positive integers l = (l 1 , . . . , l k ), the vector of polynomials I(z, q) := I (l 1 p-1,...,l k p-1) (z, q) satisfies equations (2.12) and (2.5).

The parameters q, l 1 p -1, . . . , l k p -1 are analogs of cycles γ in Section 2.3. Proof. To prove that I (l 1 p-1,...,l k p-1) (z, q) satisfies (2.12) and (2.5) consider the Taylor expansions at (t 1 , . . . , t k ) = (q 1 , . . . , q k ) of both sides of equations (2.20) and (2.23), divide them by dt 1 ∧ • • • ∧ dt k . Notice that the Taylor expansions are well defined due to formulas (2.18) and (2.22).

Project the Taylor coefficients of (t 1 -q 1 ) l 1 p-1 . . . |m|-2k] . Then the terms coming from the d t ( )-summands equal zero since d(t l i p i )/dt i = l i pt l i p-1 i ≡ 0 (mod p), and we obtain equations (2.12) and (2.5).

Example 2.2. Let p = 3, κ = 4, n = 5, k = 2, m 1 = • • • = m 5 = 1. Notice that in this case κ ≡ 1 (mod 3) and we may set κ = 1.

The set I k consists of ten elements J = (j 1 , . . . , j 5 ) with j i ∈ {0, 1} and

(mod 3). The other Ω (i,j) act similarly. The system of equations (2.12) on I(z) = J∈I k I J (z)f J v m consists of five equations. The first is I (1,1,0,0,0) (z) + I (1,0,1,0,0) (z) + I (1,0,0,1,0) (z) + I (1,0,0,0,1) (z) ≡ 0 (mod 3), where z = (z 1 , . . . , z 5 ), the other are similar. Let t = (t 1 , t 2 ). We may choose the master polynomial

Fix integers q = (0, 0) and l = (4, 3). Then the vector (2.27) and similar other coordinates satisfies equations (2.12) and (2.5).

The system of equations (2.12) takes the form:

Let p = 4l + 3 for some l. We may choose

Notice that p+1 2 is even, the polynomial Φ (p) 2,2,M (t 1 , t 2 , z 1 , z 2 ) is symmetric with respect to permutation of t 1 , t 2 , and the solution

is nonzero. Here c J (z 1 , z 2 ) are the polynomials determined by the construction of Section 2.5. For example, for p = 3, Theorem 2.5. Fix x 1 , . . . , x n ∈ F p . Consider the vector of polynomials

of formula (2.25). Assume that deg t i F (t 1 , . . . , t k ) < 2p -2 for i = 1, . . . , k. Consider the solution I (p-1,...,p-1) (z, q) of equations (2.12) and (2.5), described in Theorem 2.4. Then

This integral is a p-analog of the hypergeometric integral (2.11).

Proof. Theorem 2.5 is a straightforward corollary of formula (1.17), cf. the proof of Theorem 1.3.

Example 2.4. The polynomial

2.7. Example of a p-analog of skew-symmetry. For J ∈ I k , the differential forms W J (t, z)dt 1 ∧ • • • ∧ dt k are skew-symmetric with respect to permutations of t 1 , . . . , t k . Here is an example of a p-analog of that skew-symmetry. Another demonstration of the skewsymmetry see in Example 2.5.

Consider the KZ differential equations with parameters n, k, κ, m 1 , . . . , m n ∈ Z >0 , where κ, m 1 , . . . , m n are even, κ = 2κ , m 1 = 2m 1 , . . . , m n = 2m n . Assume that κ is even and the prime p is such that κ (p -1) and (p -1)/κ is odd, cf. Example 2.5. Choose

as well as the product 1 i<j k (t i -t j ) are skew-symmetric with respect to permutations of t 1 , . . . , t k .

Let a be a generator of the cyclic group

The partition of F k p by subsets (γ (x)) κ =0 is invariant with respect to the action of the symmetric group S k of permutations of t 1 , . . . , t k . For every , the subset γ (x) is invariant with respect to the action of the alternating subgroup A k ⊂ S k . For J ∈ I k the restriction of the function W J (t, x) 1 i<j k (t i -t j ) to the set γ (x) is A k -invariant. We have

2.8. Relation of solutions to surfaces over F p .

Example 2.5. For distinct x 1 , x 2 ∈ F p let Γ(x 1 , x 2 ) be the closure in P 1 (F p ) × P 1 (F p ) of the affine surface (2.34) where P 1 (F p ) is the projective line over F p . For a rational function h : Γ(x 1 , x 2 ) → F p define the integral

as the sum over all points P ∈ Γ(x 1 , x 2 ), where h(P ) is defined.

Recall

Theorem 2.6. Let p = 4l + 3 for some l. Let

be the vector of polynomials appearing in the solution (2.28) of the KZ equations of Example 2.3. Then

Proof. The values of W J (t 1 , t 2 , x 1 , x 2 ) at infinite points of Γ(x 1 , x 2 ) equal zero, so the integrals are sums over points of the affine surface. We prove the first equality in (2.36). We have

Remark. Consider the projection Γ(x 1 , x 2 ) → F 2 p , (t 1 , t 2 , y) → (t 1 , t 2 ). For any distinct t 1 , t 2 ∈ F p exactly one of the two points (t 1 , t 2 ), (t 2 , t 1 ) lies in the image of the projection, since (t 1 -t 2 )(t 1 -x 1 )(t 2 -x 1 )(t 1 -x 2 )(t 2 -x 2 ) is skew-symmetric in t 1 , t 2 and -1 is not a square if p = 4l + 3. where µ n,k, ,Q (t, z) is a polynomial differential k -1-form in t, z with integer coefficients determined by n, k, , Q only, see pages 182-184 in [FSV1].

3.2. Resonances over F p . Given k, n ∈ Z >0 , m = (m 1 , . . . , m n ) ∈ Z n >0 , κ ∈ Z >0 , let p > 2 be a prime number such that p does not divide κ. Choose positive integers M s for s = 1, . . . , n, M i,j for 1 i < j n, M 0 and K such that

Fix integers q = (q 1 , . . . , q k ). As in Section 2.5 for any nonnegative integers l 1 , . . . , l k define the vector I (i 1 ,...,i k ) (z, q) ∈ (F p [z]) dim L ⊗m [|m|-2k] .

Theorem 3.3. Let ∈ Z >0 be such that

M s -(k -1)M 0 ≡ 1 (mod p). (3.3) Then for any integers q = (q 1 , . . . , q k ) and positive integers l = (l 1 , . . . , l k ), the vector of polynomials I (l 1 p-1,...,l k p-1) (z, q) satisfies the equation (ze) I (l 1 p-1,...,l k p-1) (z, q) = 0. Proof. The proof of Theorem 3.3 is similar to the proof of Theorem 2.1 and uses the universal differential k-1-forms η n,k, ,Q (t, z) of Section 3.1 instead of the differential k-1-forms η J (t, z) in (2.17 (z)f J v m of Example 2.2, which is a solution of (2.5) and (2.12). The resonance equation (3.3) in this case takes the form + 1 ≡ 0 (mod 3) and is satisfied for = 2. The condition (ze) 2 I (11,8)

The KZ equations are defined for any simple Lie algebra g or more generally for any Kac-Moody algebra, see for example [SV3]. Similarly to what was done in Sections 2 and 3, one can construct polynomial solutions of those KZ equations over F p as well as of the singular vector equations and resonance equations over F p .

The construction of the polynomial solutions over F p in the sl 2 case was based on the algebraic identities for logarithmic differential forms (2.14), (2.15) and the associated cohomological relations (2.20), (2.23) as well as on the cohomological relations associated with the differential forms η n,k, ,K (t, z) in (3.2). For an arbitrary Kac-Moody algebra the analogs of the algebraic identities in (2.14) and (2.15) are the identities of Theorems 6.16.2 and 7.5.2" in [SV3], respectively. For an arbitrary simple Lie algebra, the construction of analogs of the cohomological identities for the differential forms η n,k, ,K (t, z) is the main result of [FSV2].

Remark. The F p -analogs of multidimensional hypergeometric integrals associated with arrangements of hyperplanes see in [V4]. Remarks on the Gaudin model and Bethe ansatz over F p see in [V3].