Differentiation of Genus 4 Hyperelliptic Functions
aa r X i v : . [ n li n . S I] D ec DIFFERENTIATION OF GENUS HYPERELLIPTIC FUNCTIONS
V. M. BUCHSTABER, E. YU. BUNKOVA
Abstract.
In this work we give an explicit solution to the problem of differentiationof hyperelliptic functions in genus 4 case. It is a genus 4 analogue of the classical resultof F. G. Frobenius and L. Stickelberger [1] in the case of elliptic functions. An explicitsolution in the genus 2 case was given in [2]. An explicit solution in the genus 3 casewas given in [3].
Introduction
For a meromorphic function f in C g the vector ω ∈ C g is a period if f ( z + ω ) = f ( z )for all z ∈ C g . If the periods of a meromorphic function f form a lattice Γ of rank 2 g in C g , then f is called an Abelian function . Therefore an Abelian function is a mero-morphic function on the complex torus T g = C g / Γ. We denote the coordinates in C g by z = ( z , z , . . . , z g − ).We consider hyperelliptic curves of genus g in the model V λ = { ( x, y ) ∈ C : y = x g +1 + λ x g − + λ x g − + . . . + λ g x + λ g +2 } . The curve depends on the parameters λ = ( λ , λ , . . . , λ g , λ g +2 ) ∈ C g . Let B ⊂ C g be the subspace of parameters such that the curve V λ is nonsingular for λ ∈ B . Thenwe have B = C g \ Σ, where Σ is the discriminant hypersurface of the universal curve.A hyperelliptic function of genus g (see [2, 4, 5]) is a meromorphic function in C g × B ,such that for each λ ∈ B it’s restriction on C g × λ is an Abelian function, where thetorus T g = C g / Γ is the Jacobian J λ of the curve V λ . We denote by F the field ofhyperelliptic functions of genus g . For the properties of this field, see [4, 5].We consider the problem of constructing the Lie algebra of derivations of F , i.e.to find 3 g independent differential operators L such that LF ⊂ F . The expositionto the problem, as well as a general approach to the solution was developed in [6, 7].An overview is given in [5]. In [1, 2, 3] an explicit solution to this problem has beenobtained for g = 1 , ,
3. In the present work we give an explicit answer to this problemin the genus g = 4 case. It is based on the results of [8].Let U be the total space of the bundle π : U → B with fiber the Jacobian J λ ofthe curve V λ over λ ∈ B . Thus, we can say that hyperelliptic functions of genus g aremeromorphic functions in U . By Dubrovin–Novikov Theorem [9], there is a birationalisomorphism between U and the complex linear space C g .We use the theory of hyperelliptic Kleinian functions (see [4, 10, 11, 12], and [13] forelliptic functions). Take the coordinates ( z, λ ) = ( z , z , . . . , z g − , λ , λ , . . . , λ g , λ g +2 )in C g × B ⊂ C g . Let σ ( z, λ ) be the hyperelliptic sigma function (or elliptic function incase of genus g = 1). We set ∂ k = ∂∂z k . Following [2, 3, 5], we use the notation ζ k = ∂ k ln σ ( z, λ ) , ℘ k ,...,k n = − ∂ k · · · ∂ k n ln σ ( z, λ ) , (1)where n > k s ∈ { , , . . . , g − } . The functions ℘ k ,...,k n are hyperelliptic functions.The field F is the field of fractions of the ring of polynomials P generated by the functions ℘ k ,...,k n , where n > k s ∈ { , , . . . , g − } . We note that the derivations of F thatwe construct are derivations of P . Supported in part by RAS program ”Nonlinear dynamics: fundamental problems and applications”,RFBR project 17-01-00366 A, and Young Russian Mathematics award. . Lie algebra of vector fields in B Let g ∈ N . Following [14, Section 4], we consider C g +1 with coordinates ( ξ , . . . , ξ g +1 ).Let H be the hyperplane in C g +1 given by the equation P g +1 k =1 ξ k = 0. The permutationgroup S g +1 of coordinates in C g +1 corresponds to the action of the group A g on H .We associate a vector ξ ∈ H with the polynomial Y k ( x − ξ k ) = x g +1 + λ x g − + λ x g − + . . . + λ g x + λ g +2 , (2)where λ = ( λ , λ , . . . , λ g , λ g +2 ) ∈ C g . The orbit space H/A g is identified with C g .We denote the variety of regular orbits in C g by B . Thus, B ⊂ C g is the subspaceof parameters λ such that the polynomial (2) has no multiple roots, and B = C g \ Σ,where Σ is the discriminant hypersurface.The gradient of any A g -invariant polynomial determines a vector field in C g thatis tangent to the discriminant hypersurface Σ of the genus g hyperelliptic curve [14, 15].Choosing a multiplicative basis in the ring of A g -invariant polynomials, we can constructthe corresponding 2 g polynomial vector fields, which are linearly independent at eachpoint in B . These fields do not commute and determine a nonholonomic frame in B .In [14, Section 4] an approach to constructing an infinite-dimensional Lie algebra ofsuch fields based on the convolution of invariants operation is described. In the presentwork, we consider the fields L , L , L , . . . , L g − , corresponding to the multiplicative basis in the ring of A g -invariants, that is composed ofelementary symmetric functions. The structure polynomials of the convolution of invari-ants operation in this basis were obtained by D. B. Fuchs, see [14, Section 4]. Note thatthe nonholonomic frame in B corresponding to the multiplicative basis in the ring of A g -invariants composed of Newton polynomials is used in the works of V. M. Buchstaberand A. V. Mikhailov, see [16].We express explicitly the vector fields { L k } in the coordinates λ . For convenience,we assume that λ s = 0 for all s / ∈ { , , , . . . , g, g + 2 } and λ = 1.For k, m ∈ { , , . . . , g } , k m , we set T k, m = k − X s =0 k + m − s ) λ s λ k + m − s ) − k (2 g − m + 1)2 g + 1 λ k λ m , and for k > m we set T k, m = T m, k . Lemma 1.1.
For k = 0 , , , . . . , g − the formula holds L k = g +1 X s =2 T k +2 , s − ∂∂λ s . (3)The expressions for the matrix T = ( T k, m ) in (3) are taken from [3, § L is the Euler vector field. It determines the weights of the vectorfields L k . Namely, wt λ k = 2 k , wt L k = 2 k and for all k we have[ L , L k ] = 2 kL k . (4) he vector fields { L k } , where k = 0 , , , . . . , g −
1, generate a graded polynomial Liealgebra [15]. We denote it by L b . Denote by c s i, j ( λ ) the structure polynomials of L b , i.e.[ L i , L j ] = g − X s =0 c s i, j ( λ ) L s . (5)We have the relations c s j, i ( λ ) = − c s i, j ( λ ) , c s , k ( λ ) = 0 for s = k, c k , k ( λ ) = 2 k. Lemma 1.2.
In the genus g = 4 case we have the expressions (cid:0) c s , j ( λ ) (cid:1) = 29 λ − λ λ − λ λ − λ λ − λ λ − λ λ − λ for j = 2 , , , , , (cid:0) c s , j ( λ ) (cid:1) = 29 − λ λ − λ λ − λ λ − λ λ − λ λ − λ λ − λ λ − λ λ − λ λ − λ λ for j = 3 , , , , (cid:0) c s , j ( λ ) (cid:1) = 29 − λ − λ λ − λ λ λ − λ − λ λ − λ λ λ − λ − λ λ − λ λ λ − λ λ − λ λ for j = 4 , , , (cid:0) c s , j ( λ ) (cid:1) = 29 − λ − λ − λ λ − λ λ λ λ − λ − λ λ − λ λ λ − λ λ − λ λ for j = 5 , , (cid:0) c s , j ( λ ) (cid:1) = 29 (cid:18) − λ − λ λ − λ λ λ − λ λ − λ λ (cid:19) for j = 6 , (cid:0) c s , ( λ ) (cid:1) = 29 (cid:0) − λ λ − λ λ (cid:1) . The proof is a straightforward check using the explicit expressions (3).
Corollary 1.3.
In the genus g = 4 case for k = 3 , , , , we have the expression L k = 12( k −
2) [ L , L k − ] + 2( k − k −
2) ( λ k L − λ L k − ) . The proof is obtained from (5) for i = 1 using the explicit expression for c s , j ( λ )from Lemma 1.2. . Lie algebra of derivations
The following explicit form for the operators follows from the general theory developedin [18]. It is based on the results of [8]. We will give independent proofs.In the genus g = 4 case we set: L = ∂ , L = ∂ , L = ∂ , L = ∂ , L = L − z ∂ − z ∂ − z ∂ − z ∂ ; L = L − ζ ∂ + 43 λ z ∂ − (cid:18) z − λ z (cid:19) ∂ − (cid:18) z − λ z (cid:19) ∂ − z ∂ ; L = L − ζ ∂ − ζ ∂ − + 2 λ z ∂ − (cid:18) λ z − λ z (cid:19) ∂ − (cid:18) z + 3 λ z − λ z (cid:19) ∂ − (3 z + 5 λ z ) ∂ ; L = L − ζ ∂ − ζ ∂ − ζ ∂ ++ 53 λ z ∂ + 169 λ z ∂ − (cid:18) λ z + 2 λ z − λ z (cid:19) ∂ − ( z + 3 λ z + 4 λ z ) ∂ ; L = L − ζ ∂ − ζ ∂ − ζ ∂ − ζ ∂ + (cid:18) λ z − λ z (cid:19) ∂ ++ 209 λ z ∂ − (cid:18) λ z − λ z (cid:19) ∂ − ( λ z + 2 λ z + 3 λ z ) ∂ ; L = L − ζ ∂ − ζ ∂ − ζ ∂ ++ ( λ z − λ z − λ z ) ∂ + 53 λ z ∂ + 43 λ z ∂ − ( λ z + 2 λ z ) ∂ ; L = L − ζ ∂ − ζ ∂ ++ (cid:18) λ z − λ z − λ z (cid:19) ∂ + (cid:18) λ z − λ z (cid:19) ∂ + 149 λ z ∂ − λ z ∂ ; L = L − ζ ∂ + (cid:18) λ z − λ z (cid:19) ∂ + (cid:18) λ z − λ z (cid:19) ∂ + 79 λ z ∂ . We denote the Lie algebra generated by the vector fields L , L , L , L , L , L , L , L , L , L , L , and L by L . Theorem 2.1.
The Lie algebra L is the Lie algebra of derivations of the field F ,i.e. L k ϕ ∈ F for ϕ ∈ F .The proof of this Theorem will be given in Section 4. Lemma 2.2.
For the commutators in the Lie algebra L we have the relations: [ L , L k ] = k L k , k = 0 , , , , , , , , , , , L k , L m ] = 0 , k, m = 1 , , , . Proof.
We use the explicit expressions for L k and the fact that L is the Euler vectorfield, thus L ζ k = kζ k . (cid:3) emma 2.3. For the commutators in the Lie algebra L we have the relations: [ L , L ][ L , L ][ L , L ][ L , L ][ L , L ][ L , L ][ L , L ] = ℘ , − ℘ , ℘ , − ℘ , ℘ , ℘ , − ℘ , ℘ , ℘ , ℘ , ℘ , ℘ , ℘ , ℘ , ℘ , ℘ , L L L L ; [ L , L ][ L , L ][ L , L ][ L , L ][ L , L ][ L , L ][ L , L ] = ℘ , − λ − ℘ , ℘ , − λ − ℘ , ℘ , ℘ , − λ ℘ , ℘ , ℘ , ℘ , − λ ℘ , ℘ , ℘ , ℘ , ℘ , ℘ , L L L L + 13 λ λ λ λ λ λ λ L ; [ L , L ][ L , L ][ L , L ][ L , L ][ L , L ][ L , L ][ L , L ] = ℘ , − ℘ , ℘ , − λ ℘ , ℘ , ℘ , − λ − λ ℘ , − λ ℘ , ℘ , − λ ℘ , − λ − λ ℘ , − λ ℘ , ℘ , − λ − λ − λ ℘ , ℘ , − λ − λ ℘ , L L L L + 49 λ λ λ λ λ λ λ L ; [ L , L ][ L , L ][ L , L ][ L , L ][ L , L ][ L , L ][ L , L ] = ℘ , ℘ , ℘ , − λ ℘ , ℘ , ℘ , − λ ℘ , ℘ , ℘ , ℘ , − λ − λ ℘ , ℘ , ℘ , − λ − λ − λ ℘ , ℘ , − λ − λ − λ ℘ , L L L L + 29 λ λ λ λ λ λ λ L . The proof follows from the explicit expressions for L k . Lemma 2.4.
For the commutators in the Lie algebra L we have the relations: [ L , L ][ L , L ][ L , L ][ L , L ][ L , L ][ L , L ] = (cid:0) c s , j ( λ ) (cid:1) (cid:0) L k (cid:1) + 12 − ℘ , , ℘ , , − ℘ , , − ℘ , , ℘ , , ℘ , , − ℘ , , − ℘ , , ℘ , , ℘ , , ℘ , , − ℘ , , − ℘ , , ℘ , , ℘ , , ℘ , , − ℘ , , ℘ , , ℘ , , − ℘ , , ℘ , , L L L L . Proof.
From [18] we obtain the expressions for L k ζ s , k = 1 , , , , , , s = 1 , , , L k . (cid:3) . Polynomial Lie algebra in C Following [3, Chapter 5], we consider the diagram U π (cid:15) (cid:15) ϕ / / ❴❴❴ C p (cid:15) (cid:15) B (cid:31) (cid:127) / / C .Here π : U → B is the bundle described above,
B ⊂ C is the embedding given by thecoordinates λ for g = 4. We determine the map ϕ by the set of generators of F . Denotethe coordinates in C by x i,j , where i ∈ { , , } and j ∈ { , , , } , and x i +1 = x i, .Then ϕ is determined by the map (see the notation in (1))( z, λ ) x x , x , x , x x , x , x , x x , x , x , = ℘ , ℘ , ℘ , ℘ , ℘ , , ℘ , , ℘ , , ℘ , , ℘ , , , ℘ , , , ℘ , , , ℘ , , , . The polynomial map p is determined by the relations λ = − x + 12 x − x , ,λ = 2 x + 14 x − x x − x x , + 12 x , − x , ,λ = (4 x + x , ) x , − x x , + 12 x , − x , −
12 ( x x , − x x , + x x , ) ,λ = 2 x x , + 14 x , − x , x , + 2 (cid:0) x + x , (cid:1) x , − x x , + 12 x , −−
12 ( x x , − x x , + x x , ) ,λ = 4 x x , x , + x , + (cid:0) x + 2 x , (cid:1) x , −−
12 ( x , x , − x , x , + x , x , + x x , − x x , + x x , ) ,λ = 2 x x , + 14 x , − x , x , + (4 x x , + 2 x , ) x , −−
12 ( x , x , − x , x , + x , x , ) ,λ = 4 x x , x , + x , −
12 ( x , x , − x , x , + x , x , ) ,λ = 2 x x , + 14 x , − x , x , . We also have [3, Proof of Theorem 5.3] the formulas for w i,j = ℘ i,j , where i, j ∈ , , w , = 3 x x , − x , + 3 x , ,w , = 3 x x , − x , + 3 x , ,w , = 3 x x , − x , ,w , = 12 ( x x , − x x , + x x , ) − (cid:0) x + x , (cid:1) x , + 5 x x , − x , ,w , = 12 ( x x , − x x , + x x , ) − (cid:0) x + x , (cid:1) x , ,w , = 12 ( x , x , − x , x , + x , x , ) − (4 x x , + x , ) x , . ext we introduce explicitly a set of polynomial vector fields in C .We have the polynomial vector fields (see [3, Lemma 6.2]): D = 2 x ∂∂x + 3 x ∂∂x + 4 x ∂∂x + 4 x , ∂∂x , + 5 x , ∂∂x , + 6 x , ∂∂x , ++ 6 x , ∂∂x , + 7 x , ∂∂x , + 8 x , ∂∂x , + 8 x , ∂∂x , + 9 x , ∂∂x , + 10 x , ∂∂x , ; D = x ∂∂x + x ∂∂x + 4(3 x x + x , ) ∂∂x ++ X s =3 , , x ,s ∂∂x ,s + x ,s ∂∂x ,s + 4( x x ,s + 2 x x ,s + x ,s +2 ) ∂∂x ,s ;where x , = 0. We denote by x ′ ,k the expression D ( x ,k ), by w ′ k,s the expression D ( w k,s ),by w ′′ k,s the expression D ( D ( w k,s )), and by w ′′′ k,s the expression D ( D ( D ( w k,s ))). Thenwe have the polynomial vector fields (see [3, Lemma 6.3]) for k = 3 , , D k = x ,k ∂∂x + x ,k ∂∂x + x ′ ,k ∂∂x + X s =3 , , w ′ k,s ∂∂x ,s + w ′′ k,s ∂∂x ,s + w ′′′ k,s ∂∂x ,s . Consider the vector fields D and D determined by the conditions D x = − x + 12 x + 2 x , + 79 λ ; D x , = − x x , + 12 x , + 3 x , − λ x + w , ; D x , = − x x , + 12 x , + 5 x , − λ x , + w , ; D x , = − x x , + 12 x , − λ x , + w , ; D x = − x x , + x , + 2 x , + 23 λ ; D x , = − x , + 3 x , + λ x , − λ x − λ − x w , + w ′′ , + w , ; D x , = − x , x , + 3 λ x , − λ x , − x w , + w ′′ , + w , ; D x , = − x , x , + 5 λ x , − λ x , − x w , + w ′′ , + w , ;and by the relations (cid:18) [ D , D ][ D , D ] (cid:19) = (cid:18) x − x , x − (cid:19) D D D . We set (see Corollary 1.3 and Lemma 2.4): D = 14 (2[ D , D ] + x , D − x D ) −
43 ( λ D − λ D ) ; D = 18 (cid:0) D , D ] + ( w ′ , + x , ) D − x , D − x D ) (cid:1) −
59 ( λ D − λ D ) ; D = 112 (cid:0) D , D ] + (2 w ′ , + x , ) D − x , D − x , D − x , D (cid:1) −
827 ( λ D − λ D ) ; D = 116 (cid:0) D , D ] + (2 w ′ , + w ′ , ) D − x , D − x , D − x , D (cid:1) −
16 ( λ D − λ D ) ; D = 120 (cid:0) D , D ] + 2 w ′ , D − x , D − x , D (cid:1) −
445 ( λ D − λ D ) . enote the ring of polynomials in λ ∈ C by P . Let us consider the polynomialmap p : C → C . A vector field D in C will be called projectable for p if there existsa vector field L in C such that D ( p ∗ f ) = p ∗ L ( f ) for any f ∈ P. The vector field L will be called the pushforward of D . Lemma 3.1.
The vector fields D s , s = 1 , , , are projectable for p with trivial push-forwards, i.e. D s ( f ) = 0 , s = 1 , , , , for any f ∈ P. The vector fields D k , k = 0 , , , , , , , , are projectable for p and their pushforwardsare L k , i.e. D k ( p ∗ f ) = p ∗ L k ( f ) , k = 0 , , , , , , , , for any f ∈ P. Proof.
The formulas above define all the expressions involved explicitly, so the check is adirect calculation. (cid:3)
We denote the polynomial Lie algebra generated by the vector fields D , D , D , D , D , D , D , D , D , D , D , and D by D . Lemma 3.2.
For the commutators in the Lie algebra D we have the relations: [ D , D ][ D , D ][ D , D ][ D , D ][ D , D ][ D , D ][ D , D ] = x − x , x − x , x , x − x , x , x , x x , x , x , x , x , x , D D D D ; [ D , D ][ D , D ][ D , D ][ D , D ][ D , D ][ D , D ][ D , D ] = x , − λ − w , x , − λ − w , w , x , − λ w , w , w , x , − λ w , w , w , w , w , w , D D D D + 13 λ λ λ λ λ λ λ D ; [ D , D ][ D , D ][ D , D ][ D , D ][ D , D ][ D , D ][ D , D ] = x , − w , x , − λ w , w , x , − λ − λ w , − λ w , w , − λ x , − λ − λ w , − λ w , w , − λ − λ − λ w , w , − λ − λ w , D D D D + 49 λ λ λ λ λ λ λ D ; [ D , D ][ D , D ][ D , D ][ D , D ][ D , D ][ D , D ][ D , D ] = x , w , x , − λ w , w , x , − λ w , w , w , x , − λ − λ w , w , w , − λ − λ − λ w , w , − λ − λ − λ w , D D D D + 29 λ λ λ λ λ λ λ D . The proof is a direct calculation. . Proof of Theorem D is a solution to [3, Problem 6.1] for g = 4, namely thepolynomial vector fields D s , s = 0 , , , , , , , , , , ,
14, are projectable for p andindependent at any point in p − ( B ). In [3, Chapter 6], the connection of this problemwith the problem of constructing the Lie algebra of derivations of F is described. Namely,a solution is given by the differential operators L s such that L s ϕ ∗ x i,j = ϕ ∗ D s x i,j for the coordinate functions x i,j in C , see Section 3.By the construction (see [3]), we have L s = ∂ s for s = 1 , , , L is the Euler vectorfield. This coincides with the operators presented in Section 2. The vector fields L k ,where k = 2 , , , , ,
7, are determined by the conditions:(1) L k = L k + f k, ( u, λ ) ∂ + f k, ( z, λ ) ∂ + f k, ( z, λ ) ∂ + f k, ( z, λ ) ∂ ,(2) the fields L k satisfy commutation relations obtained by ϕ ∗ from the commutationrelations of Lemma 3.2.By comparing Lemma 3.2 with Lemma 2.3 we see that for the operators from Section 2these conditions are satisfied. The condition (2) determines the coefficients f k,j ( u, λ ),where j = 1 , , ,
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Steklov Mathematical Institute of Russian Academy of Sciences
E-mail address : [email protected], [email protected]@mi-ras.ru, [email protected]