Solutions Modulo p of Gauss Manin Differential Equations for Multidimensional Hypergeometric Integrals and Associated Bethe Ansatz

mathematics Article Solutions Modulo p of Gauss Manin Differential Equations for Multidimensional Hypergeometric Integrals and Associated Bethe Ansatz Alexander Varcheno Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA; anv@email.unc.edu Academic Editor: Laurentiu-George Maxim Received: 22 September 207; Accepted: 3 October 207; Published: 7 October 207 Abstract: We consider the Gauss Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p-analogs of the initial hypergeometric integrals. In some cases, we interpret the p-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field F p. This interpretation is similar to the classical interpretation by Yu. I. Manin of the number of points on an elliptic curve depending on a parameter as a solution of a Gauss hypergeometric differential equation. We discuss the associated Bethe ansatz. Keywords: Gauss Manin differential equations; multidimensional hypergeometric integrals; Bethe ansatz. Introduction We consider an arrangement of affine hyperplanes H j ) n j= in C. Let f j t,..., t ) be a first-degree polynomial on C whose ernel is H j. Let a j ) n j=, κ be nonzero complex numbers. An associated multidimensional hypergeometric integral is an integral of the form: I = γ n j= f jt,..., t ) aj κ dt dt where γ is a cycle in the complement to the union of the hyperplanes. We assume that the hyperplanes depend on parameters z,..., z n and move parallel to themselves when the parameters change. Then the integral extends to a multivalued holomorphic function of the parameters. The holomorphic function is called a multidimensional hypergeometric function and is associated with this family of arrangements. The simplest example of such a function is the classical hypergeometric function. The multidimensional hypergeometric functions can be combined into collections so that the functions of a collection satisfy a system of first-order linear differential equations called the Gauss Manin differential equations. If all polynomials f j ) n j= have integer coefficients and the numbers a j), κ are integers, then the Gauss Manin differential equations can be reduced modulo a prime integer p large enough. The goal of this paper is to construct polynomial solutions of the Gauss Manin differential equations over the field F p with p elements. Our solutions are p-analogs of the multidimensional hypergeometric integrals. The construction of the solutions is motivated by the classical paper [ by Yu. I. Manin cf. section Manin s Result: The Unity of Mathematics in [2; see also [3,4). The paper is organized as follows. In Section 2, we recall the basic notions associated with an affine arrangement of hyperplanes in C. In Section 3, we consider a family of arrangements of hyperplanes Mathematics 207, 5, 52; doi:0.3390/math5040052 www.mdpi.com/journal/mathematics

Mathematics 207, 5, 52 2 of 8 in C whose hyperplanes move parallel to themselves when the parameters of the family change. We introduce the Gauss Manin differential equations and multidimensional hypergeometric integrals. We show that the multidimensional hypergeometric integrals satisfy the Gauss Manin differential equations see Theorem 3). In Section 4, we consider the reduction of this situation modulo p and construct polynomial solutions of the Gauss Manin differential equations over F p see Theorem 5), which is the main result of this paper. We interpret our solutions as integrals over F p under certain conditions see Theorem 6). Such integrals could be considered as p-analogs of the multidimensional hypergeometric integrals. In Section 5, we consider examples. Under certain conditions, we interpret our polynomial solutions as sums over points on some hypersurfaces over F p see Theorem 0). This statement is analogous to the interpretation in Manin s paper [ of the number of points on an elliptic curve depending on a parameter as a solution of a Gauss hypergeometric differential equation. In Section 6, we briefly discuss the associated Bethe ansatz. We introduce a system of the Bethe ansatz equations and construct a common eigenvector to geometric Hamiltonians out of every solution of the Bethe ansatz equations see Theorem ). We show that the Bethe eigenvectors corresponding to distinct solutions are orthogonal with respect to the associated symmetric contravariant form see Corollary 3). 2. Arrangements We recall some facts about hyperplane arrangements, Orli Solomon algebras and flag complexes from [5. 2.. An Affine Arrangement Let and n be positive integers, where < n. Denote J = {,..., n}. Let C = H j ) j J, be an arrangement of n affine hyperplanes in C. Denote U = C j J H j as the complement. An edge X α C of the arrangement C is a nonempty intersection of some hyperplanes of C. Denote by J α J the subset of indices of all hyperplanes containing X α. Denote l α = codim C X α. We always assume that the arrangement C is essential; that is, C has a vertex, an edge that is a point. An edge is called dense if the subarrangement of all hyperplanes containing it is irreducible: the hyperplanes cannot be partitioned into nonempty sets, so that, after a change of coordinates, hyperplanes in different sets are in different coordinates. In particular, each hyperplane of C is a dense edge. 2.2. Flag Complex For l = 0,...,, let Flag l C) denote the set of all flags: C = L 0 L L l where each L j is an edge of C of codimension j. Let F l C, Z) denote the quotient of the free abelian group on Flag l C) by the following relations. For every flag with a gap: F = L 0 L L i L i+ L ), i < We impose: F F F = 0 ) in F l C, Z), where the sum is over all flags F = L 0 L... L l ) Flag l C), such that L j = L j for all j = i. The abelian group F l C, Z) is a free abelian group see [5, Theorem 2.9.2). There is an extension of flags differential d : F l C, Z) F l+ C, Z) defined by: dl 0 L L l ) = L l+l 0 L L l L l+ )

Mathematics 207, 5, 52 3 of 8 where the sum is over all edges L l+ of C of codimension l + contained in L l. It follows from Equation ) that d 2 = 0. Thus we have a complex, the flag complex, F C, Z), d). 2.3. Orli Solomon Algebra Define abelian groups A l C, Z), l = 0,,..., as follows. For l = 0, set A 0 C, Z) = Z. For l > 0, A l C, Z) is generated by l-tuples H,..., H l ) of hyperplanes H i C, subject to the relations: i) H,..., H l ) = 0 if H,..., H l are not in general position i.e., if codim H H l = l). ii) H σ),..., H σl) ) = ) σ H,..., H l ) for every permutation σ Σ l. iii) For any l + hyperplanes H,..., H l+ that have a non-empty intersection, H H l+ =, and that are not in general position: where Ĥ i denotes omission. l+ i= )i H,..., Ĥ i,..., H l+ ) = 0 The abelian group A l C, Z) is a free abelian group, see [6; [5, Theorem 2.9.2). The Orli Solomon algebra of the arrangement C is the direct sum A C, Z) = l=0 Al C, Z) endowed with the product given by H,..., H i ) H,..., H j ) = H,..., H i, H,..., H j ). It is a graded sew-commutative algebra over Z. 2.4. Orli Solomon Algebra as an Algebra of Differential Forms For each hyperplane H C, pic a polynomial f H of degree on C whose zero set is H; that is, let f H = 0 be an affine equation for H. Consider the logarithmic differential form: ιh) := d log f H = d f H f H on C. We note that ιh) does not depend on the choice of f H but only on H. Let A C, Z) be the Z-algebra of differential forms generated by and ιh), H C. The assignment H ιh) defines an isomorphism A C, Z) A C, Z) of graded algebras. Henceforth we shall not distinguish between A and A. 2.5. Duality See [5 cf. [7, Section 2.5) The vector spaces A l C, Z) and F l C, Z) are dual. The pairing A l C, Z) F l C, Z) Z is defined as follows. For H j,..., H jl in general position, set FH j,..., H jl ) = L 0 L l ) F l C, Z), where L 0 = C and: L i = H j H ji, i =,..., l For any F = L 0 L l ) F l C, Z), define H j,..., H jl ), F = ) σ, if F = FH jσ),..., H jσl) ) for some σ S l, and H j,..., H jl ), F = 0 otherwise. 2.6. Flag and Orli Solomon Spaces over a Field F For any field F and l = 0,...,, we define: 2.7. Weights F l C, F) = F l C, Z) Z F, A l C, F) = A l C, Z) Z F 2) An arrangement C is weighted if a map a : J C, j a j, is given; a j is called the weight of H j. For an edge X α, define its weight as a α = j Jα a j.

Mathematics 207, 5, 52 4 of 8 2.8. Contravariant Form and Map See [5 The weights determine a symmetric bilinear form S a) on F l C, C), given by: S a) F, F 2 ) = {j,...,j p } J a j a jl H j,..., H jl ), F H j,..., H jl ), F 2 where the sum is over all unordered l-element subsets. The form is called the contravariant form. It defines a homomorphism: S a) : F l C, C) F l C, C) A l C, C), L 0 L l ) a j a jl H j,..., H jl ) where the sum is taen over all l-tuples H j,..., H jl ), such that: H j L,..., H jl L l Theorem [5, Theorem 3.7). For l =,...,, choose a basis of the free abelian group F l C, Z). Then with respect to that basis, the determinant of the contravariant form S a) on F l C, Z) equals the product of suitable non-negative integer powers of the weights of all dense edges of C of codimension not greater than l. Corollary. If the weights of all dense edges of C are nonzero, then the contravariant map S a) : F l C, C) A l C, C) is an isomorphism for all l. 2.9. Aomoto Complex Define: νa) = j J a j H j ) A C, C) 3) Multiplication by νa) defines a differential: d a) : A l C, C) A l+ C, C), x x νa) on A C, C), d a) ) 2 = 0. The complex A C, C), d a) ) is called the Aomoto complex. The master function corresponding to the weighted arrangement C, a) is the function: Φ = Φ C,a = j J f a j H j 4) where each f Hj = 0 is an affine equation for the hyperplane H j. Then νa) = dφ/φ. Theorem 2 [5, Lemma 3.2.5; and [8, Lemma 5.). The Shapovalov map is a homomorphism of complexes: 2.0. Singular Vectors S a) : F C, Z), d) A C, Z), d a) ) An element v F C, C) is called singular if d a) A l C, C), v = 0. Denote by: the subspace of all singular vectors. 2.. Arrangements with Normal Crossings Only SingF C, C) F C, C) An essential arrangement C is with normal crossings only, if exactly hyperplanes meet at every vertex of C. Assume that C is an essential arrangement with normal crossings only. A subset {j,..., j l } J is called independent if the hyperplanes H j,..., H jl intersect transversally. A basis of A l C, C) is formed by H j,..., H jl ), where {j < < j l } are independent l-element

Mathematics 207, 5, 52 5 of 8 subsets of J. The dual basis of F l C, C) is formed by the corresponding vectors FH j,..., H jl ). These bases of A l C, C) and F l C, C) are called standard. In F l C, C), we have: FH j,..., H jl ) = ) σ FH jσ),..., H jσl) ) 5) for any permutation σ S l. For an independent subset {j,..., j l }, we have S a) FH j,..., H jl ), FH j,..., H jl )) = a j a jl and S a) FH j,..., H jl ), FH i,..., H il )) = 0 for any distinct elements of the standard basis. 3. A Family of Parallelly Transported Hyperplanes 3.. An Arrangement in C n C Recall that J = {,..., n}. Consider C with coordinates t,..., t, C n with coordinates z,..., z n, the projection C n C C n. Fix n nonzero linear functions on C, g j = b j t + + b j t, j J, where b i j C. Define n linear functions on Cn C : In C n C, define the arrangement: f j = z j + g j = z j + b j t + + b j t, j J 6) C = { H j f j = 0, j J} Denote Ũ = C n C j J H j. For every fixed z 0 = z 0,..., z0 n), the arrangement C induces an arrangement Cz 0 ) in the fiber over z 0 of the projection. We identify every fiber with C. Then Cz 0 ) consists of hyperplanes H j z 0 ), j J, defined in C by the same equations, f j = 0. Denote: UCz 0 )) = C j J H j z 0 ) 7) as the complement to the arrangement Cz 0 ). We assume that for any z 0, the arrangement Cz 0 ) has a vertex. This means that the span of g j ) j J is -dimensional. A point z 0 C n is called good if Cz 0 ) has normal crossings only. Good points form the complement in C n to the union of suitable hyperplanes called the discriminant. 3.2. Discriminant The collection g j ) j J induces a matroid structure M C on J. A subset C = {i,..., i r } J is a circuit in M C if g i ) i C are linearly dependent but any proper subset of C gives linearly independent g i s. For a circuit C = {i,..., i r }, let λ i C ) i C be a nonzero collection of complex numbers such that i C λ i C g i = 0. Such a collection is unique up to multiplication by a nonzero number. For every circuit C, we fix such a collection and denote f C = i C λ i C z i. The equation f C = 0 defines a hyperplane H C in C n. It is convenient to assume that λ i C = 0 for i J C and write f C = i J λ i C z i. For any z 0 C n, the hyperplanes H i z 0 )) i C in C have a nonempty intersection if and only if z 0 H C. If z 0 H C, then the intersection has codimension r in C. Denote by C C the set of all circuits in M C. Denote = C CC H C. The arrangement Cz 0 ) in C has normal crossings only, if and only if z 0 C n.

Mathematics 207, 5, 52 6 of 8 3.3. Good Fibers For any z, z 2 C n, the spaces F l Cz ), C), F l Cz 2 ), C) are canonically identified. Namely, a vector FH j z ),..., H jl z )) of the first space is identified with the vector FH j z 2 ),..., H jl z 2 )) of the second. Assume that weights a = a j ) j J are given. Then each arrangement Cz) is weighted. The identification of spaces F l Cz ), C), F l Cz 2 ), C) for z, z 2 C n identifies the corresponding subspaces SingF Cz ), C) and SingF Cz 2 ), C) and the corresponding contravariant forms. For a point z 0 C n, denote V C = F Cz 0 ), C), SingV C = SingF Cz 0 ), C). The triple V C, SingV C, S a) ) does not depend on z 0 C n under the above identification. 3.4. Geometric Hamiltonians cf. [9,0) For any circuit C = {i,..., i r }, we define a linear operator L C : V C V C in terms of the standard basis of V C see Section 2.). For m =,..., r, denote C m = C {i m }. Let {j < < j } J be an independent ordered subset and FH j,..., H j ) be the corresponding element of the standard basis. Define L C : FH j,..., H j ) 0 if {j,..., j } C < r. If {j,..., j } C = C m for some m r, then using the sew-symmetry property of Equation 5), we can write: FH j,..., H j ) = ± FH i, H i2,..., Ĥ im,..., H ir H ir, H s,..., H s r+ ) with {s,..., s r+ } = {j,..., j } C m. Define: L C : FH i,..., Ĥ im,..., H ir, H s,..., H s r+ ) ) m r l= )l a il FH i,..., Ĥ il,..., H ir, H s,..., H s r+ ) Lemma [9). The operator L C is symmetric with respect to the contravariant form. Consider the logarithmic differential one-forms: ω j = d f j f j, j J, ω C = d f C f C, C C C in variables t,..., t, z,..., z n. For any circuit C = {i,..., i r }, we have: ω i ω ir = ω C r l= )l ω i ω il ω ir Lemma 2 [9, Lemma 4.2; and [0, Lemma 5.4). We have: independent independent j J a j ω j ) ω j ω j FH j,..., H j ) = 8) C CC ω C ω j ω j L C FH j,..., H j ) Proof. The lemma is a direct corollary of the definition of the maps L C. The identity in Equation 8) is called the ey identity. Recall that ω C = d f C / f C and f C = j J λ i C z i. For i J, we introduce the End C V C )-valued rational functions in z,..., z n by the formula: K i z) = C CC λ i C f C z) L C, i J 9)

Mathematics 207, 5, 52 7 of 8 Then: The functions K i z) are called geometric Hamiltonians. C CC ω C L C = i J dz i K i z) 0) Corollary 2. The geometric Hamiltonians are symmetric with respect to the contravariant form, S a) K i z)x, y) = S a) x, K i z)y) for i J, x, y V C. 3.5. Gauss Manin Differential Equations The Gauss Manin differential equations with parameter κ C are given by the following system of differential equations on a V C -valued function Iz,..., z n ): κ I z i z) = K i z)iz), i J ) where K i z) are the geometric Hamiltonians defined in Equation 9). We introduce the master function: Φz, t, a) = j J f j z, t) a j 2) on Ũ C n C. The function Φz, t, a) /κ defines a ran-one local system L κ on Ũ, whose horizontal sections over open subsets of Ũ are univalued branches of Φz, t, a) /κ multiplied by complex numbers. For z 0 C n and an element γz 0 ) H UCz 0 )), L κ UCz 0 ))), we interpret the integration map A Cz 0 ), C) = V C, ω γz 0 ) Φz0, t, a) /κ ω as an element of F Cz 0 ), C) = V C. The vector bundle: z 0 C n H UCz 0 )), L κ UCz 0 )) ) Cn has a canonical flat Gauss Manin connection. A locally constant section γ : z γz) H UCz)), L κ UCz)) ) of the Gauss Manin connection defines a V C -valued function: The integrals: I γ) z,..., z n ) = ) independent Φz, t, γz) a)/κ ω j ω j FH j,..., H j ) 3) I γ) j,...,j z,..., z n ) = Φz, t, γz) a)/κ ω j ω j are called the multidimensional hypergeometric integrals associated with the master function Φz, t, a). Theorem 3 [0). The function I γ) taes values in SingV C and gives solutions of the Gauss Manin differential equations. The condition that the function I γ) taes values in SingV C may be reformulated as the system of equations: j J a j I γ) j,j 2,...,j = 0, for j 2,..., j J 4) 3.6. Proof of Theorem 3 We setch the proof following [3,5. The intermediate statements of this setch are used further when constructing solutions of the Gauss Manin differential equations over a finite field F p. The proof of Theorem 3 is based on the following cohomological relations, Equations 2) and 24).

Mathematics 207, 5, 52 8 of 8 For any j,..., j J, denote: d j,...,j = det i,l= bi j l ), W j,...,j z, t) = d j,...,j l= f j l z, t) 5) We have: ω j ω j = W j,...,j z, t) dt dt +... 6) where the dots denote the terms having differentials dz j,..., dz j. We note that the rational function W j,...,j z, t) has the form: P j,...,j z, t) j J f j z, t) 7) where P j,...,j z, t) is a polynomial with integer coefficients in variable z,..., z n, t,..., t and b i j, j J, i =,..., n see Equation 6)). For any j 2,..., j J, we write: ω j2 ω j = l= W j 2,...,j ;lz, t) dt dt l dt +... 8) where the dots denote the terms having differentials dz j2,..., dz j, and W j,...,j ;l are rational functions in z, t of the form: P j,...,j ;lz, t) j J f j z, t) 9) Here, P j,...,j ;lz, t) are polynomials with integer coefficients in variable z,..., z n, t,..., t and b i j, j J, i =,..., n see Equation 6)). The formula: νa) ω j2 ω j = j J a j ω j ω j2 ω j 20) implies the identity: κ d t Φz, t, a) /κ l= W j 2,...,j ;lz, t) dt dt l... dt ) = j J a j Φz, t, a) /κ W j,j2,...,j z, t) dt dt 2) where d t denotes the differential with respect to the variables t. Now we deduce a corollary of the ey identity, Equation 8). Choose i J. For any independent {j <... < j } J, we write: Φz, t, a) /κ ω j ω j = Φz, t, a) /κ W j,...,j z, t) dt dt 22) + dz i Φz, t, a) /κ l= W j,...,j ;i,lz, t) dt dt ) l dt +... where the dots denote the terms that contain dz j with j = i, and the coefficients W j,...,j ;i,lz, t) are rational functions in z, t of the form: b i j P j,...,j ;i,lt, z) j J f j z, t) 23) Here, P j,...,j ;i,lz, t) are polynomials with integer coefficients in variable z,..., z n, t,..., t and, j J, i =,..., n see Equation 6)). Equation 8) implies that for any i J, we have: κ independent ) Φz, t, a) /κ W z j,...,j z, t) dt dt 24) i + d t Φz, t, a) /κ n l= W j,...,j ;i,lt, z) dt dt l dt ))FH j,..., H j ) = K i z) independent Φz, t, a) /κ W j,...,j z, t) dt dt FH j,..., H j )

Mathematics 207, 5, 52 9 of 8 where d t denotes the differential with respect to the variables t. Integrating both sides of Equations 2) and 24) over γz) and using Stoes theorem, we obtain Equations 4) and ) for the vector I γ) z) in Equation 3). Theorem 3 is proved. 3.7. Remars It is nown from [5 that for generic κ, all SingV C -valued solutions of the Gauss Manin Equation ) are given by Equation 3). Hence, we have the following statement. Theorem 4 [0). The geometric Hamiltonians H i z), i J preserve SingV C and commute on SingV C, namely, [H i z 0 ) SingVC, H j z 0 ) SingVC = 0 for all i, j J and z 0 C n. 4. Reduction Modulo p of a Family of Parallelly Transported Hyperplanes 4.. An Arrangement in C n C over Z Similarly to Section 3., we consider C with coordinates t,..., t, C n with coordinates z,..., z n, the projection C n C C n. Fix n nonzero linear functions on C, g j = b j t + + b j t, j J, with integer coefficients b i j Z. Define n linear functions on Cn C : f j = z j + g j = z j + b j t + + b j t 25) where j J. Recall the matroid structure M C on J, the set C C of all circuits in M C, and the linear functions f C = i J λ i C z i labeled by C C C, where the functions are defined in Section 3.2. Each of these functions is determined up to multiplication by a nonzero constant. Definition. We fix the coefficients λ i C ) i J to be integers such that the greatest common divisor of λ i C ) i J equals. This is possible as all b i j are integers. This choice of the coefficients defines the function f C uniquely up to multiplication by ±. Let p be a prime integer and F p be the field with p elements. Let [ : Z F p be the natural projection. We introduce the following linear functions in z, t with coefficients in F p : [g j : = i= [bi j t i, [ f j := z j + [g j, j J 26) [ f C : = i J [λ i C z i, C C C The collection [g j ) j J induces a matroid structure M Fp on J. A subset C = {i,..., i r } J is a circuit in M Fp if [g i ) i C are linearly dependent over F p but any proper subset of C gives linearly independent [g i s. Definition 2. We say that a prime integer p is good with respect to the collection of linear functions g j ) j J if all linear functions in Equation 26) are nonzero and the matroid structures M C and M Fp on J are the same. In the following, we always assume that p is good with respect to the collection of linear functions g j ) j J. We have logarithmic differential forms: [ω j = d[ f j [ f j, j J, [ω C = d[ f C [ f C, C C Fp = C C

Mathematics 207, 5, 52 0 of 8 in variables t,..., t, z,..., z n with coefficients in F p. For any circuit C = {i,..., i r }, we have: [ω i [ω ir = [ω C r l= )l [ω i [ω il [ω ir Assume that the nonzero integer weights a = a j ) j J are given, where a j Z, a j = 0. The constructions of Section 3 give us the following: i) A vector space V Fp over F p with standard basis FH j,..., H j )) indexed by all independent subsets {j < < j } of J. ii) A vector subspace SingV Fp V Fp consisting of all linear combinations independent I j,...,j FH j,..., H j ) satisfying the equations: j J [a j I j,j2,...,j = 0, for j 2,..., j J 27) iii) A symmetric bilinear F p -valued contravariant form [S a) on V Fp defined by the formulas: [S a) FH j,..., H j ), FH j,..., H j )) = [a j [a j for any independent {j < < j } and [S a) FH j,..., H jl ), FH i,..., H il )) = 0 for any distinct elements of the standard basis. For any circuit C = {i,..., i r }, we define a linear operator [L C : V Fp V Fp by the formula of Section 3.4, in which the numbers a il are replaced with [a il. We have the ey identity: independent j J [a j [ω j ) [ω j [ω j FH j,..., H j ) = 28) independent C CF [ω C [ω j [ω j [L C FH j,..., H j ) p For i J, we define the End Fp V Fp )-valued rational functions in z,..., z n by the formula: [K i z) = C CF p [λ i C [ f C z) [L C, i J 29) We call the functions [K i z) the geometric Hamiltonians. The geometric Hamiltonians are symmetric with respect to the contravariant form [S a) [K i z)x, y) = [S a) x, [K i z)y) for i J, x, y V Fp. The Gauss Manin differential equations over F p with parameter [κ F p are given by the following system of differential equations: [κ I z i z) = [K i z)iz), i J 30) The goal of this paper is to construct polynomial SingV Fp -valued solutions of these differential equations. 4.2. Polynomial Solutions Let a prime integer p be good with respect to g j ) j J. Let a = a j ) j J be nonzero integer weights a j Z, a j = 0. Choose positive integers A = A,..., A n ), such that: [A j = [a j [κ 3)

Mathematics 207, 5, 52 of 8 in F p. Introduce the master polynomial: Φz, t, A) = j J f j z, t) A j Z[z,..., z n, t,..., t 32) where f j z, t) are defined in Equation 25). For any j,..., j, the function X j,...,j z, t, A) := d j,...,j Φz, t, A) l= f j l z,t) is a polynomial in z, t with integer coefficients. For fixed q = q,..., q ) Z, consider the Taylor expansion: X j,...,j z, t, A) = i,...,i 0 Ii,...,i j,...,j z, q, A) t q ) i... t q ) i 33) where I i,...,i j,...,j z, q, A) Z[z,..., z n for any i,..., i. We denote by [I i,...,i j,...,j z, q, A) the projection of I i,...,i j,...,j z, q, A) to F p [z,..., z n. Denote: [I i,...,i z, q, A) 34) = independent [I i,...,i j,...,j z, q, A) FH j,..., H j ) V Fp F p [z,..., z n Theorem 5. Let a prime integer p be good with respect to g j ) j J. Then for any integers A = A,..., A n ) satisfying Equation 3), any integers q = q,..., q ), and any positive integers ł = l,..., l ), the polynomial function Iz) = [I l p,...,l p z, q, A) satisfies the algebraic equations in Equation 27) and the Gauss Manin differential equations in Equation 30). The parameters A, q, l p,..., l p of the solution Iz) are analogs of the locally constant cycles γz) in Section 3.5. We note that the space of polynomial solutions of Equations 27) and 30) is a module over the ring F p [z p,..., zp n, as zp i z j = 0. Proof. To prove that Iz) satisfies Equations 27) and 30), consider the Taylor expansions at t = q of both sides of Equations 2) and 24) divided by dt dt. We note that the Taylor expansions are well defined as a result of Equations 7), 9) and 23). We project the Taylor expansions to V Fp F p [z,..., z n. Then the terms coming from the d t -summands equal zero, as dt l i p i )/dt i = l i pt l i p i 0 mod p). 4.3. Relation of Solutions to Integrals over F p For a polynomial Ft,..., t ) F p [t,..., t and a subset γ F p, we define the integral: γ We consider the vector of polynomials: Ft,..., t ) dt dt := t,...,t ) γ Ft,..., t ) Fx, t, A) = j J [ f j x, t) ) A j [d j,...,j independent l= [ f j l x, t) FH j,..., H j ) V Fp F p [t,..., t 35) Theorem 6. Let x = x,..., x n ) F p. Let [I p,...,p z, q, A) be the solution of Equations 27) and 30) considered in Theorem 5 for l,..., l ) =,..., ).

Mathematics 207, 5, 52 2 of 8 i) ii) If deg ti Fx, t, A) < 2p 2 for i =,...,, then: [I p,...,p ) x, q, A) = ) If the integers A = A j ) j J are such that: F p Fx, t, A) dt dt 36) A + + A n < + )p ) 37) then Equation 36) holds. Proof. Part follows from the statement that for a positive integer i, t F p t i equals if p ) i and equals zero otherwise 38) Part 2 also follows from Equation 38) by the following reason. The polynomial P = j J [ f j x, t) ) A j [d j,...,j l= [ f j l x,t) is a product of A + + A n linear functions in t. If Equation 37) holds, then t... t ) p is the only monomial t m... t m of the Taylor expansion of that polynomial, such that m l > 0 and p ) ml for l =,...,. The integral in Equation 36) is a p-analog of the hypergeometric integral of Equation 3) see also Section 5). 5. Examples 5.. Case = See [3 Let κ, a = a,..., a n ) be nonzero complex numbers. We consider the master function of complex variables: Φt, z,..., z n, a) = n i= t + z i ) a i 39) Let z 0 = z 0,..., z0 n) C n be a vector with distinct coordinates. We consider the n-vector I γ) z 0 ) = I z 0 ),..., I n z 0 )), where: I j = Φt, z 0 γz 0,..., ) z0 n, a) /κ dt t + z 0, j =,..., n 40) j The integrals are over a closed Pochhammer) curve γz 0 ) in C {z 0,..., z0 n} on which one fixes a uni-valued branch of the master function to mae the integral well-defined. Starting from such a curve chosen for a given {z 0,..., zn n} C, the vector I γ) z 0 ) 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 7. The vector I γ) z) satisfies the algebraic equation: a I z) + + a n I n z) = 0 4) and the differential equations: κ I z i Ω i,j = I, i =,..., n 42) z j =i i z j

Mathematics 207, 5, 52 3 of 8 where: Ω i,j = i. i a j a j.. j a i a i and all the remaining entries equal zero, see [5; [, Section.). Example. Let κ = 2, n = 3, and a = a 2 = a 3 =. Then I γ) z) = I z), I 2 z), I 3 z)), where: I j z) = γz). j.. dt 43) t + z )t + z 2 )t + z 3 ) t + z j In this case, the curve γz) may be thought of as a closed path on the elliptic curve: y 2 = t + z )t + z 2 )t + z 3 ) Each of these integrals is an elliptic integral. Such an integral is a branch of an analytic continuation of a suitable Euler hypergeometric function up to a change of variables. Example 2. Let p > 3 be a prime integer. Let κ = 2, and a = = a n = cf. Example ). For such κ and a j, the algebraic Equation 4) and the differential Equation 42) are well-defined when reduced modulo p. Choose the master polynomial: Φt, z,..., z n ) = n i= t + z i ) p 2 44) Consider the Taylor expansion of the polynomial see 35)): Ft, z) = n i= t + z i ) p 2 t + z,..., ) = t + z n i I i z)t i 45) Let [I i z) be the projection of I i z) to F p [z) n. Then the vector Iz) := [I p z) is a solution of the differential Equation 42) over F p [z and I z) + + I n z) = 0 see Theorem 5). If n 4 and x = x,..., x n ) F n p, then: Ix) = Ft, x) dt F p by Theorem 6. Let x = x,..., x n ) F p. Let Γx) be the affine curve: over F p. For a rational function h : Γx) F p, define the integral: as the sum over all points P Γx), where hp) is defined. y 2 = t + x )... t + x n ) 46) h = hp) 47) Γx) P Γx)

Mathematics 207, 5, 52 4 of 8 Theorem 8. Let n equal 3 or 4. Let [I p x) = [I p x),..., [I n p x)) be the vector of polynomials obtained from Equation 45). Then: Γx) t + x j = [I p j x), j =,..., n 48) Remar. Theorems 5 and 8 say that the integrals Γx,...,x n ) t +x are polynomials in x j,..., x n F p and the tuple of polynomials: ),..., Γx,...,x n ) t + x Γx,...,x n ) t + x n in these discrete variables satisfies the system of Gauss Manin differential equations cf. Example ). Remar 2. In [, Section 2 and in [2, an equation analogous to Equation 48) for n = 3 is considered, where the left-hand side is the number of points on Γx, x 2, x 3 ) over F p and the right-hand side is the reduction modulo p of a solution of a second-order Gauss hypergeometric differential equation. We note that the number of points on Γx, x 2, x 3 ) is the discrete integral over Γx, x 2, x 3 ) of the constant function h =. Proof of Theorem 8. The proof is analogous to the reasoning in [, Section 2 and [2. It is easy to see that: Γx,...,x n ) = t + x j t + F p, t =x j t + x j t F p t + x j n s= t + x s ) p 2 = t t F p + x j ) p 2 + t F p [I i j x,..., x n )t i = [Ip j x,..., x n ) i We note that t F p i [I i j x,..., x n )t i = [Ip j x,..., x n ), as for n = 3, 4, the degree of the left-hand side is less than 2p 2 see Theorem 6). See more examples with = in [3, Section ). 5.2. Counting on Two-Folded Covers As in Section 4., we consider n nonzero linear functions on C, g j = b j t + + b j t, j J, with integer coefficients b i j Z. Let a prime integer p > 3 be good with respect to g j) j J. Assume that all weights a j ) j J are equal to and κ = 2. Under these assumptions, consider the algebraic Equation 27) and differential Equation 30). To construct a solution of these equations, choose a master polynomial Φz, t) = j J f j z, t) p 2, and consider the Taylor expansion of the polynomial: at t = 0, to obtain the solution: d j,...,j Fz, t) = independent Φz, t) f j z, t)... f j z, t) FH j,..., H j ) 49) [I p,...,p z) = independent [I p,...,p j,...,j z) FH j,..., H j ) 50) of the algebraic Equation 27) and differential Equation 30) by taing the coefficient of t... t ) p of the Taylor expansion see Theorem 5). Let x = x,..., x n ) F n p. Let Γx) be the affine hypersurface: y 2 = j J [ f j x, t) 5) over F p. Recall that [ f j x, t) = [b j t + + [b j t + x j, where [ : Z F p is the natural projection.

Mathematics 207, 5, 52 5 of 8 For a rational function h : Γx) F p define the integral: as the sum over all points P Γx) with well-defined hp). Theorem 9. Let: h = hp) 52) Γx) P Γx) n p 2 < + )p ) 53) Let [I p,...,p x) be the vector in Equation 50) at z = x. Then for any independent {j < < j } J, we have: Γx) [d j,...,j l= [ f j l x, t) = ) [I p,...,p j,...,j x) 54) This theorem is a generalization of Theorem 8. Theorem 9 states that the integrals in the left-hand side of Equation 48) are polynomials in x,..., x n F p and satisfy the algebraic Equation 27) and differential Equation 30). Proof. It is easy to see the following cf. [,2: Γx) [d j,...,j l= [ f j l x, t) = t F p, l= [ f j l x,t) =0 [d j,...,j l= [ f j l x, t) + [d j,...,j t F p l= [ f j J [ f j x, t)) p 2 j l x, t) 55) Lemma 3. The first sum in the right-hand side of Equation 55) equals zero. Proof. Because p is good and {j,..., j } is independent, we have [d j,...,j = 0. Hence we may choose s l := [ f jl x, t), l =,...,, to be affine coordinates on F p. Then: t F p, l= [ f j l x,t) =0 [d j,...,j l= [ f j l x, t) = s F p, s...s =0 [d j,...,j s... s = s F p [d j,...,j s... s ) p 2 = 0 Lemma 4. The second sum in the right-hand side of Equation 55) equals ) [I p,...,p j,...,j x). Proof. The inequality of Equation 37) taes the form of Equation 53). Now the lemma follows from Theorem 6. Theorem 9 is proved. Remar 3. The inequality of Equation 53) holds if n 2 + 2 independently of p. 5.3. Counting on κ-folded Covers Let κ > 2 be a positive integer. As in Sections 4. and 5.2, we consider n nonzero linear functions on C, g j = b j t + + b j t, j J, with integer coefficients b i j Z. Let a prime integer p be good with respect to g j ) j J and κ p ). Assume that all weights a j ) j J are equal to κ ). Under these assumptions, consider the algebraic Equation 27) and differential Equation 30). To construct a solution of these equations, choose

Mathematics 207, 5, 52 6 of 8 κ ) p a master polynomial Φz, t) = j J f j z, t) κ, and consider the Taylor expansion at t = 0 of the polynomial Fz, t) in Equation 49) to obtain the solution [I p,...,p z) of the algebraic Equation 27) and differential Equation 30) by taing the coefficient of t... t ) p of the Taylor expansion see Theorem 5 and Formula 50)). Let x = x,..., x n ) F n p. Let Γx) be the affine hypersurface: over F p cf. Section 5.2). y κ = j J [ f j x, t) 56) Theorem 0. Let a prime integer p be good with respect to g j ) j J. Let κ p ) and κ = p. Let: nκ ) p κ < + )p ), nκ 2) p 2 κ Then for any independent {j < < j } J we have: < p ) 57) [d j,...,j Γx) l= [ f j l x, t) = ) [I p,...,p j,...,j x) 58) This theorem is a generalization of [3, Example.7 and Theorem 9. Proof. It is easy to see that: [d j,...,j Γx) l= [ f j l x, t) = t F p, l= [ f j l x,t) =0 [d j,...,j l= [ f j l x, t) + κ [d j,...,j p l κ l= t F p l= [ f j J [ f j x, t)) j l x, t) 59) The first sum on the right-hand side equals zero by Lemma 3. Consider the Taylor expansion of the polynomial [d j,...,j l= [ f j l x,t) j J [ f j x, t) ) p l κ. Consider the monomials of the form t l p )... t l p ), where l,..., l are positive integers. If l κ 2, the second inequality in Equation 57) implies that the coefficients of such monomials in the Taylor ) p [d j,...,j expansion are all equal to zero, and hence the sum l κ t F p j J [ f j x, t) equals zero l= [ f j l x,t) for l κ 2. If l = κ, the first inequality in Equation 57) implies that among the monomials of the form t l p )... t l p ), only t p... t p may appear with a nonzero coefficient in the Taylor expansion. Hence: [d j,...,j p κ ) κ t F p l= [ f j J [ f j x, t)) j l x, t) by Theorem 6. The theorem is proved. = ) [I p,...,p j,...,j x) Remar 4. The inequalities of Equation 57) are implied by the system of inequalities: κ κ + ) n, κ n 60) κ 2 κ independently of p. If κ 2, then the inequality κ + ) n implies inequality, enough for Theorem 0 to hold. In particular, if κ 2, then n = + 2 is admissible. κ κ 2 n, and is

Mathematics 207, 5, 52 7 of 8 6. Bethe Ansatz The goal of the Bethe ansatz is to construct mutual eigenvectors of the geometric Hamiltonians [K i x)) i J defined in Equation 29). As in Sections 4., 5.2 and 5.3, we consider n nonzero linear functions on C, g j = b j t + + b j t, j J, with integer coefficients b i j Z. Let a prime integer p be good with respect to g j ) j J. Let a = a j ) j J be nonzero integer weights a j Z, a j = 0. Recall the functions [ f C z) = i J [λ i C z i with C C Fp. Assume that x = x,..., x n ) F n p is such that f C x) = 0 for any C C Fp. Then [K i x)) i J are well-defined linear operators on V Fp. Introduce the system of the Bethe ansatz equations: j J b i [a j j [ f j x, t) = 0, i =,..., 6) with respect to the unnown t = t,..., t ) F p. Theorem. If t 0 F p is a solution of Equation 6), then the vector: Fx, t 0 [d j,...,j ) = independent [ f j x, t 0 )... [ f j x, t 0 ) FH j,..., H j ) 62) satisfies Equation 27) and is an eigenvector of the geometric Hamiltonians: [K i x)fx, t 0 ) = [a i [ f i x, t 0 ) Fx, t0 ), i =,..., n 63) Proof. Equation 27) for Fx, t 0 ) follows from Equations 20) and 2) reduced modulo p. Because t 0 is a solution of Equation 6), the left-hand side of Equation 2) equals zero. Equation 63) is a straightforward corollary of the ey identity of Equation 28) see the proof of [4, Theorem 2.). Corollary 3. Let t 0 and t be distinct solutions of the Bethe ansatz Equation 6); then [S a) Fx, t 0 ), Fx, t )) = 0. Proof. Because t 0 = t, there exists i such that [ f i x, t 0 ) = [ f i x, t ). Hence [K i x) has distinct eigenvalues on Fx, t 0 ), Fx, t ), but [K i x) is symmetric: [S a) [K i x)fx, t 0 ), Fx, t )) = [S a) Fx, t 0 ), [K i x)fx, t )) Acnowledgments: This wor was supported in part by NSF grants DMS-362924, DMS-665239. Conflicts of Interest: The author declares no conflict of interest. References. Manin, Y.I. The Hasse-Witt Matrix of an Algebraic Curve. Izv. Aad. Nau SSSR Ser. Mat. 96, 25, 53 72. In Russian) 2. Clemens, H.C. A Scrapboo of Complex Curve Theory, 2nd ed.; Graduate Studies in Mathematics; AMS: Providence, RI, USA, 2003. 3. Schechtman, V.; Varcheno, A. Solutions of KZ Differential Equations Modulo p. arxiv 207, arxiv:707.0265. 4. Varcheno, A. Remars on the Gaudin Model Modulo p. arxiv 207, arxiv:708.06264.

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