Inversion of a general hyperelliptic integral and particle motion in Hořava Lifshitz black hole space-times

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1 Inversion of a general hyperelliptic integral and particle motion in Hořava Lifshitz black hole space-times Victor Enolski, Betti Hartmann, Valeria Kagramanova, Jutta Kunz, Claus Lämmerzahl et al. Citation: J. Math. Phys. 53, (01; doi: / View online: View Table of Contents: Published by the American Institute of Physics. Related Articles On the stability of the massive scalar field in Kerr space-time J. Math. Phys. 5, 1050 (011 Lensing by Kerr black holes. II: Analytical study of quasi-equatorial lensing observables J. Math. Phys. 5, (011 Topics in cubic special geometry J. Math. Phys. 5, 0830 (011 The Bañados, Teitelboim, and Zanelli spacetime as an algebraic embedding J. Math. Phys. 5, (011 Past horizons in twisting algebraically special space times J. Math. Phys. 51, 1501 (010 Additional information on J. Math. Phys. Journal Homepage: Journal Information: Top downloads: Information for Authors:

2 JOURNAL OF MATHEMATICAL PHYSICS 53, (01 Inversion of a general hyperelliptic integral and particle motion in Hořava Lifshitz black hole space-times Victor Enolski, 1,,5,a Betti Hartmann, 3,b Valeria Kagramanova, 4,c Jutta Kunz, 4,d Claus Lämmerzahl, 5,4,e and Parinya Sirimachan 3,f 1 Hanse-Wissenschaftskolleg (HWK, 7733 Delmenhorst, Germany Institute of Magnetism, 36-b Vernadsky Blvd., Kyiv 0314, Ukraine 3 School of Engineering and Science, Jacobs University Bremen, 8759 Bremen, Germany 4 Institut für Physik, Universität Oldenburg, 6111 Oldenburg, Germany 5 ZARM, Universität Bremen, Am Fallturm, 8359 Bremen, Germany (Received 11 July 011; accepted 1 December 011; published online 31 January 01 The description of many dynamical problems such as the particle motion in higher dimensional spherically and axially symmetric space-times is reduced to the inversion of hyperelliptic integrals of all three kinds. The result of the inversion is defined locally, using the algebro-geometric techniques of the standard Jacobi inversion problem and the foregoing restriction to the θ-divisor. For a representation of the hyperelliptic functions the Klein Weierstraß multi-variable σ -function is introduced. It is shown that all parameters needed for the calculations such as period matrices and abelian images of branch points can be expressed in terms of the periods of holomorphic differentials and θ-constants. The cases of genus two, three, and four are considered in detail. The method is exemplified by the particle motion associated with genus one elliptic and genus three hyperelliptic curves. Applications are for instance solutions to the geodesic equations in the space-times of static, spherically symmetric Hořava Lifshitz black holes. C 01 American Institute of Physics. [doi: / ] I. INTRODUCTION A. The mathematical problem Various problems of physics are reduced to the inversion of a hyperelliptic integral. Namely, let y = 4x g+1 + λ g x g λ 0,λ i C (1.1 be a hyperelliptic curve X g of genus g with one branch point at infinity realized as a two-sheeted covering over the extended complex plane. A point P X g has coordinates P = (x, y, where the sign of the second coordinate y indicates the chosen sheet on a Riemann surface. Let R(x, y be a rational function of its arguments x and y. We consider here the problem of the inversion of an abelian integral x x 0 R(x, ydx = t (1. a Electronic mail: V.Z.Enolskii@ma.hw.ac.uk. b Electronic mail: b.hartmann@jacobs-university.de. c Electronic mail: va.kagramanova@uni-oldenburg.de. d Electronic mail: jutta.kunz@uni-oldenburg.de. e Electronic mail: laemmerzahl@zarm.uni-bremen.de. f Electronic mail: p.sirimachan@jacobs-university.de /01/53(1/01504/35/$ , C 01 American Institute of Physics

3 Enolski et al. J. Math. Phys. 53, (01 resulting in a function x(t which is a function of the complex variable t. The integral (1. can be decomposed by routine algebraic operations to E(x E(x 0 + g x a k x 0 du k + g x b k x 0 dr k + n x c k x 0 d αk,β k = t, (1.3 where E(x is an elementary function including logarithms and rational functions; a k, b k, c k are certain constants; and du k,dr k, and d αk,β k are differentials of the first, second, and third kind, respectively. Namely, du k are holomorphic differentials, dr k are meromorphic differentials of the second kind with a unique pole of the order g k +, and d αk,β k are meromorphic differentials of the third kind with first order poles in the points α k and β k and residues ± 1 in the poles. We suppose that in the case considered there are n 0 differentials of the third kind. It is well known that only in the case of elliptic curves, i.e., for g = 1, the correspondence x t is one-to-one and the aforementioned inversion problem can be solved in terms of single-valued elliptic functions. In the case of higher genera, g > 1, a one-to-one correspondence is achieved between the symmetrized products of curves X g... X g and a multi-dimensional complex space, the Jacobi variety. Single-valued functions in this case appear to be multi-periodic functions of many complex variables, called abelian functions. These ideas already developed by Jacobi and Abel had led Riemann to the concept of the Riemann surfaces, to the introduction of multi-dimensional θ-functions, and the formulation of his celebrated theorems. Moreover, it is also known that the function x(t becomes single-valued on the infinitely sheeted Riemann surface. In particular, when abelian integrals reduce to elliptic integrals, this Riemann surface becomes finitely sheeted. 7 For the special case of a genus two hyperelliptic curve a detailed construction of such an infinitely sheeted Riemann surface was given by Fedorov and Gómes- Ulate. 0 It also follows from Ref. 0 that x(t is well defined on the complex plane from which an infinite lattice of polygons with (4g 4-edges called windows is extracted. In our investigation we are considering a function x(t defined on the complement to these windows and suppose that the integration paths in (1.3 never intersect these windows. Our approach to the inversion of hyperelliptic integrals is based on the well developed theory of hyperelliptic abelian functions and on various relations between the θ-functions and θ-constants. We are describing the function x(t as the restriction of an abelian function, which can be expressed in terms of symmetric functions of the divisor in the associated Jacobi inversion problem, to the one-dimensional stratum of the θ-divisor. We are implementing the Klein-Weierstraß realization of hyperelliptic functions in terms of multi-variative σ -functions that represent a natural generalization of the standard Weierstraß σ -function to hyperelliptic curves of higher genera. This paper continues our recent work 15 where the inversion of hyperelliptic holomorphic integrals was considered. The novelty of our approach is the simultaneous consideration of all three kinds of abelian integrals from a unified viewpoint. At the heart of our method lie algebraic expressions of the symmetric bi-differential of the second kind involving the explicitly given Kleinian -polar. The inversion procedure involves the theta-constant relations (the Thomae and Bolza formulae that manifest the link between branch points and Riemann period matrices. We are also developing a computer algebra procedure that allows to use Maple/algcurves software without explicit knowledge of a homology basis encrypted in the Tretkoff-Tretkoff algorithm. This will enable us to find all needed quantities like the vector of Riemann constants, the period matrices of the first and second kind, as well as the correspondence between branch points and θ-characteristics. We emphasize that in this investigation we concentrate on the algebraic side of the derivation. The function x(t obtained in that way is defined only locally, while its analytic continuation never intersects the infinite set of cuts introduced in Ref. 0. Another approach to the problem of inversion of integrals of the second and third kind that is based on the generalized θ-function goes back to Clebsch and Gordan 11 and was developed in Refs. 7 and 19. Here we do not discuss generalized Jacobians, and we plan to make a comparison between these two methods in another publication.

4 Hyperelliptic integrals Hořava Lifshitz BH ST J. Math. Phys. 53, (01 B. Physical motivation The mathematical results described in this paper have direct applications to the solution of the geodesic equation in certain Hořava Lifshitz black hole space-times. The Hořava Lifshitz theory 34, 35 is an alternative gravity theory that is powercountable renormalizable. The basic idea is that only higher spatial derivative terms are added, while higher temporal derivatives which would lead to ghosts are not considered. This leads unavoidably to the breaking of Lorentz invariance at short distances. Static and spherically symmetric black hole solutions have been studied in this theory. 38, 40, 46 Considering the Hořava Lifshitz theory as a modification of general relativity, one can study the solutions of the geodesic equation in the Hořava Lifshitz black hole space-times. In this paper, we are mainly interested in one of the black hole space-times given in Ref. 40. The mathematical techniques described in this paper cannot only be used to solve analytically the geodesic equation in the Hořava Lifshitz space-time considered here. The differentials of the first and third kind with underlying polynomial curves of arbitrary genus appear in the geodesic equations in many general relativistic space-times. The holomorphic differentials appear in the equations for the r- and ϑ-coordinates, while the differentials of the third kind appear in the equations for the ϕ- and t-coordinates. This is also the case, e.g., in the geodesic equations for neutral particles in Taub-NUT (Ref. 37 space-times and in the space-times of Schwarzschild and Kerr black holes pierced by a cosmic string, 8 as well as for charged particles in the Reissner- Nordström 5 space-time, where elliptic integrals of the first and third kind appear. They also appear in the Schwarzschild-de Sitter 9, 30 and Kerr-de Sitter space-times 33 as well as in generalized black hole Plebański Demiański space-times in 4 dimensions 3 with underlying hyperelliptic curves of genus two in the geodesic equations. Also in the higher dimensional space-times of Schwarzschild, Schwarzschild-de Sitter, Reissner-Nordström, and Reissner-Nordström-de Sitter, 15, 31 this powerful mathematics of the theory of hyperelliptic functions of higher genera is successfully applicable. Geodesics in higher dimensional axially symmetric space-times, the Myers-Perry space-times, are integrated by the hyperelliptic functions of arbitrary genus as well. 15 In Ref. 15 the integration of holomorphic integrals for any genus of the underlying hyperelliptic polynomial curve has been presented. Here we expand our considerations and present the solution for the integrals of the third kind for arbitrary genus as well. C. Outline of the paper The paper is organized as follows. Section II represents a short introduction to the theory of hyperelliptic functions that is adjusted to the aim of the paper. In this section we develop the σ -functional realizations of hyperelliptic functions. The key formula in Lemma.5 relates the integral of the second kind to the ζ -function and the Z-vector. In Sec. III we consider the inversion of the holomorphic integral, integrals of the second kind and third kind, and also their arbitrary combination. Section IV shows that already developed means of computer algebra, like Maple/algcurves, are sufficient to compute all period matrices relevant to the σ -functional approach. As a particular feature the explicit knowledge of the homology basis ciphered in the software is not necessary. Sections V-VIIexemplify the developed method in the case of genus two, three, and four, correspondingly. Section VIII is devoted to the application of the developed method to the problem of geodesic motion in Hořava Lifshitz black hole space-times. We conclude in Sec. IX. II. HYPERELLIPTIC ABELIAN FUNCTIONS A. Abelian differentials and their periods We first introduce a canonical homology basis of cycles (a 1,...,a g ; b 1,...,b g, a i a j = b i b j =, a i b j = b i a j = δ ij, where δ ij is the Kronecker symbol and denotes the intersection of cycles. We denote by du(p = (du 1 (P,...,du g (P T the basis set of holomorphic differentials

5 Enolski et al. J. Math. Phys. 53, (01 (of the first kind: du i = xi 1 dx, i = 1,...,g, (.1 y and by dr(p = (dr 1 (P,...,dr g (P T kind with a unique pole at infinity, the associated meromorphic differentials (of the second dr i = g+1 i k=i x k (k + 1 iλ k+1+i dx, i = 1,...,g, (. 4y where the coefficients λ i are as in Eq. (1.1. These canonical holomorphic differentials du and associated meromorphic differentials dr are chosen in such a way that their g g period matrices in the fixed homology basis ω = ( du i a k i,,...,g ω = ( du i b k i,,...,g η = ( satisfy the generalized Legendre relation ( dr i, a k i,,...,g η = dr i b k i,,...,g (.3 MJM T = iπ J (.4 with ( ( ω ω 0g 1 M = η η, J = g, (.5 1 g 0 g where 0 g and 1 g are the zero and unit g g matrices. The differential of the third kind P1,P (P with poles at finite points P 1 = (a 1, y 1 and P = (a, y and residues + 1 and 1, respectively, can be given in the form (One can add an arbitrary combination of holomorphic differentials. However, we take this form as the most simple one which is sufficient for the following derivations. P1,P (P = y + y 1 dx (x a 1 y y + y dx (x a y. (.6 If the poles P 1, P have the same x-coordinate and lie on different sheets, i.e., P 1 = (a, y(a and P = (a, y(a, then (.6 takes the form P1,P (P = y(a dx x a y. (.7 The differentials du k,dr k, and P1,P (P given above describe the entries in relation (1.3. We introduce the fundamental bi-differential (Q, SonX g X g which is uniquely defined by the following conditions: 1. It is symmetric, (Q, S = (S, Q.. It has the poles along the diagonal Q = S, namely, if ξ(q and ξ(s are local coordinates of the points Q and S in the vicinity of the point P (ξ(p = 0 then the following expansion is valid: (Q, S = dξ(qdξ(s (ξ(q ξ(s + where mn (P are holomorphic in P. 3. It is normalized such that m,n 1 mn (Pξ(Q m 1 ξ(s n 1 dξ(qdξ(s, (.8 a j (Q, S = 0, j = 1,...,g. (.9

6 Hyperelliptic integrals Hořava Lifshitz BH ST J. Math. Phys. 53, (01 From (.9 and the bilinear Riemann relation (see, e.g., Refs. 4 and for details it follows that the b-periods of (Q, S are (Q, S = iπdv j (S, j = 1,...,g, (.10 b j where dv(s = (dv 1 (S,...,(dv g (S T = (ω 1 du(s is the vector of normalized holomorphic differentials. We present here the algebraic construction of the fundamental bi-differential (Q, S. To do that we will construct at first a non-normalized bi-differential Ɣ(Q, S subject to the first two items in the definition of (Q, S above. Lemma.1: A symmetric bi-differential with the only second order pole along the diagonal defined up to a bilinear symmetric form in holomorphic differentials is given by Ɣ(P, Q = y + w dxdz + dr(z,w T du(x, y, (.11 z (x z y where P = (x, y and Q = (z, w, or, equivalently, F(x, y + yw dx dz Ɣ(P, Q = 4(x z y w. (.1 Here, dr is the vector of meromorphic differentials (. and F(x, z is a so-called Kleinian -polar given by g F(x, z = x k z k (λ k + λ k+1 (z + x (.13 k=0 such that F(x, x = y and F(x, z = F(z, x. Proof: Consider the differential of the third kind (.6 Q,Q (P = y + w dx (x z y y + w dx (.14 (x z y depending on the variable P = (x, y and possessing poles in the points Q = (z, w and Q = (z, w. The bi-differential z Q,Q (Pdz = y + w dxdz (.15 z (x z y as a form in P has a second order pole along the diagonal P = Q, but as a form in Q it has unwanted poles at z =. This -form can be symmetrized (Ɣ(P, Q = Ɣ(Q, P by adding an additional term dr(z,w T du(x, y that annihilates the aforementioned poles. The differential (.11 is defined up to a holomorphic -form du T (z,wκdu(x, y, where κ is a symmetric g g-matrix, κ T = κ. This fact will be used for the symmetrization and normalization of the bi-differential. Thus, the bi-differential Ɣ(P, Q turns into (P, Q = y + w dxdz + dr(z,w T du(x, y + du T (z,wκdu(x, y (.16 z (x z y or, equivalently, F(x, y + yw dx dz (P, Q = 4(x z y w + dut (z,wκdu(x, y. (.17 The matrix κ is chosen so that it normalizes (P, Q according to (.9 and (.10 and the factor in the second term of (.17 is chosen to get precisely the Weierstraß definitions in the case g = 1. This defines the matrix κ in terms of the η- and ω-periods as κ = η(ω 1. (.18

7 Enolski et al. J. Math. Phys. 53, (01 We denote by Jac(X g the Jacobian of the curve X g, i.e., the factor C g /Ɣ, where Ɣ = ω ω is the lattice generated by the periods of the canonical holomorphic differentials. Any point u Jac(X g can be represented in the form u = ωε + ω ε, (.19 where ε, ε R g. The vectors ε and ε combine to a g matrix and form the characteristic ε of the point u, ( ( ε T ε [u] := ε T = 1... ε g =: ε. (.0 ε 1... ε g If u is a half-period, then all entries of the characteristic ε are equal to 1 or 0. Beside the canonic holomorphic differentials du we will also consider the normalized holomorphic differentials defined by dv = (ω 1 du. (.1 Their corresponding holomorphic periods are 1 g and τ, where the Riemann period matrix τ := ω 1 ω is in the Siegel upper half space S g of g g matrices (or half space of degree g, S g = { τ g g matrix τ T = τ, Im(τ positive definite }. (. The corresponding Jacobian is introduced as Jac(X g := (ω 1 Jac(X g = C g /1 g τ. (.3 We will use both versions: the first one (ω, ω in the context of the σ -functions, and the second one (1 g, τ in the case of the θ-functions. The Abel map A :(X g n C g with the base point P 0 relates the set of points (P 1,...,P N which are called the divisor D, with a point in the Jacobian Jac(X g A(P 1,...,P N := N Pk P 0 du. (.4 The divisor D in (.4 can also be denoted as P P N NP 0. More generally the divisor D is the formal sum D = n 1 P n N P N with integers n j, j = 1,...,N and N N. The degree of the divisor, deg(d, is the sum deg(d = n n N.The divisor of a meromorphic function is of degree zero. Two divisors D and D are linearly equivalent if D D is the divisor of a meromorphic function. Linearly equivalent divisors constitute a class. In particular, the canonical class K Xg is the divisor class of abelian differentials of degree g. The divisor is positive if all n j 0. Let l(d be the dimension of the space of meromorphic functions that have poles in the points of D of multiplicities not higher than the multiplicity of these points in D.Ifdeg(D g for a divisor in general position the dimension l(disgivenby l(d = deg(d g + 1. (.5 Such divisors are called non-special. All remaining divisors with deg(d are called special. For a detailed explanation see, e.g., Ref. 17. Analogously we define Ã(P 1,...,P n = n Pk P 0 dv = (ω 1 A(P 1,...,P n. (.6 In the context of our consideration we choose P 0 at infinity, P 0 = (,.

8 Hyperelliptic integrals Hořava Lifshitz BH ST J. Math. Phys. 53, (01 B. θ- andσ -functions The hyperelliptic θ-function with characteristic ε is a mapping θ : Jac(X g S g C defined through the Fourier series θ[ε](v τ := m Z g e πi{(m+ε T τ(m+ε +(v+ε T (m+ε }. (.7 It possesses the periodicity property θ[ε](v + n + τ n τ = e iπn T (v+ 1 τ n e iπ(nt ε n T ε θ[ε](v τ. (.8 For vanishing characteristic we abbreviate θ(v := θ[0](v τ. In the following, the values ε k, ε k are either 0 or 1. The property (.8 implies θ[ε]( v τ = e 4πiεT ε θ[ε](v τ, (.9 so that the function θ[ε](v τ with characteristic ε of only half-integers is even if 4ε T ε is an even integer, and odd otherwise. Correspondingly, ε is called even or odd, and among the 4 g half-integer characteristics there are 1 (4g + g even and 1 (4g g odd characteristics. The nonvanishing values of the θ-functions with half-integer characteristics and their derivatives are called θ-constants and are denoted as θ[ε] := θ[ε](0; τ, θ ij [ε] := θ[ε](z; τ z i z, etc., for even [ε]; j z=0 θ i [ε] := θ[ε](z; τ 3 z, θ ijk [ε] := θ[ε](z; τ i z=0 z i z j z, etc., for odd [ε]. k z=0 Even characteristics ε are called nonsingular if θ[ε] 0, and odd characteristics ε are called nonsingular if θ i [ε] 0 for at least one index i. We identify each branch point e j of the curve X g with a vector A j := (e j,0 du =: ωε j + ω ε j Jac(X g, j = 1,...,g +, (.30 which defines the two vectors ε j and ε j. Evidently, [A g+] = [0] = 0. In terms of the g + characteristics [A i ]all4 g half-integer characteristics ε can be constructed as follows. There is a one-to-one correspondence between these ε and the partitions of the set Ḡ ={1,...,g + } of indices of the branch points (Ref. 18,p.13,Ref.3 p. 71. The partitions of interest are where m is any integer between 0 and vector I m J m ={i 1,...,i g+1 m } {j 1,..., j g+1+m }, (.31 ]. The corresponding characteristic ε m is defined by the m = [ g+1 g+1 m à ik + K =: ε m + τε m, (.3 where K Jac(X g is the vector of Riemann constants with base point, which will always be used in the argument of the θ-functions, and which is given as a vector in Jac(X g by K := à j (.33 all odd [A j ] (see, e.g., Ref. 17, p. 305, for a proof. It can be seen that characteristics with even m are even, and with odd m are odd. There are ( 1 g+ ( g+1 different partitions with m = 0, g+ g 1 different partitions with m = 1, and, in general, ( g+ ( g+1 m down to g+ ( 1 = g + partitions if g is even and m = g/, or g+ 0 = 1 partitions if

9 Enolski et al. J. Math. Phys. 53, (01 b 1 b e 1 e a 1 e 3 e 4 a b g e g 1 e g e g+1 e g+ = a g FIG. 1. A homology basis on a Riemann surface of the hyperelliptic curve of genus g with real branch points e 1,...,e g + =(upper sheet. The cuts are drawn from e i 1 to e i for i = 1,...,g + 1. The b-cycles are completed on the lower sheet. g is odd and m = (g + 1/. One may check that the total number of even (odd characteristics is indeed g 1 ± g 1. According to the Riemann theorem for the zeros of θ-functions, 18 θ( m + v vanishes to order m at v = 0 and, in particular, the function θ(k + v vanishes to the order [ g+1] at v = 0. Let us demonstrate, following, Ref. 17 p. 303, how the set of characteristics [A k ] [à k ], k = 1,..., g + looks like in the homology basis shown in Fig. 1. Using the notation f k = 1 (δ 1k,...,δ gk t and τ k for the kth column vector of the matrix τ, we find à g+1 = à g+ à g = à g+1 à g 1 = à g g (e k,0 (e k 1,0 (e g,0 (e g+1,0 (e k,0 (e k 1,0 dv = dv = dv = g 1 g f k, [à g+1 ] = 1 ( , g f k + τ g, [à g ] = 1 ( , ( f k + τ g, [à g 1 ] = 1 Continuing in the same manner, we get for arbitrary 1 k < g ( [à k+ ] = 1 ({}}{ [à k+1 ] = 1 ({}}{ k k , , (.35 and finally [à ] = 1 ( , [à ] = 1 ( ( The characteristics with even indices, corresponding to the branch points e n, n = 1,...,g, are odd (except for [A g+ ] which is zero; the others are even. Therefore, in the basis drawn in Fig. 1 we get K = g à k. (.37

10 Hyperelliptic integrals Hořava Lifshitz BH ST J. Math. Phys. 53, (01 The formula (.37 is in accordance with the classical theory where the vector of Riemann constants is defined as (see Fay, Ref. 18 Eq. (14 Divisor K P0 = (g 1P 0, (.38 where is the divisor of degree g 1 that is the Riemann divisor. In the case considered P 0 =and = e + e e g. The calculation of the divisor of the differential g (x e kdx/y leads to the required conclusion = K Xg, where K Xg is the canonical class. The Kleinian σ -function of the hyperelliptic curve X g is defined over the Jacobian Jac(X g as σ (u; M := Cθ[K ]((ω 1 u; τ e ut κu, (.39 where the symmetric g g matrix κ is defined in (.18. Here, K Jac(X g and D g Pk u = du du, (.40 g where D = P P g is a divisor in the general position. The constant π C = g 1/4 (e i e j, (.41 det(ω 1 i< j g+1 and M defined in (.5 contains the set of all moduli ω, ω and η, η. In the following we will use the shorter notation σ (u; M = σ (u. Sometimes the σ -function (.39 is called fundamental σ -function. The multi-variable σ -function (.39 represents a natural generalization of the Weierstraß σ - function given by σ (u = π ω g ɛ ( u 4 (e1 e (e 1 e 3 (e e 3 ϑ 1 exp ω { ηu ω }, ɛ 8 = 1, (.4 where ϑ 1 is the standard θ-function. We note that (.39 differs in the case of genus one from the Weierstraß σ -function by an exponential factor that appears when the shift on a half-period in the θ-argument is taken into account in the θ-characteristics. The fundamental σ -function (.39 possesses the following properties: It is an entire function on Jac(X g, It satisfies the two sets of functional equations σ (u + ωk + ω k ; M = e (ηk+η k T (u+ωk+ω k σ (u; M, (.43 σ (u;(γ M T T = σ (u; M, where γ Sp(g, Z, that is, γ Jγ T = J, and M T is the matrix M with interchanged submatrices ω and η. The first of these equations displays the periodicity property, and the second one displays the modular property. In the vicinity of the origin the power series of σ (u isoftheform σ (u = S π (u + higher order terms, (.44 where S π (u are the Schur Weierstraß functions associated to the curve X g and defined on C g (u 1,...,u g by the Weierstraß gap sequence at the infinite branch point. For g > 1it is always a Weierstraß point. The partition π = (π g,...,π 1 is defined by the Weierstraß gap sequence w = (w 1,...,w g asfollows:π i = w g i i g. Details of the definition are given in Ref. 9, see also Ref. 15. As an example we will present here the first few functions S π (u : g = 1: S 1 (u 1 = u 1, (.45 g = : S,1 (u 1, u = 1 3 u3 u 1, (.46

11 Enolski et al. J. Math. Phys. 53, (01 g = 3: S 3,,1 (u 1, u, u 3 = 1 45 u u u 3 3 u + u 1u 3, (.47 g = 4: S 4,3,,1 (u 1, u, u 3, u 4 = u u7 4 u u u 5 4 (.48 u 4 u u3 4 u 1 + u u 3 u 4 u + u 1u 3. The partitions constructed by the Weierstraß gap sequences are denoted in the subscripts. In particular, in the case of genus g = 4 the partition π = (1,, 3, 4 corresponds to the gap sequence 0, 1,, 3, 4, 5, 6, 7, 8, 9, 10,..., where orders of existing functions are overlined. These are so-called non-gap numbers, in contrast to the gap-numbers. The genus is defined by the number of gaps or, equivalently, the first number starting from which no gap appears equals g (in this example 8 = g. The Kleinian ζ and -functions are a natural generalization of the Weierstraß ζ and -functions and are given by the logarithmic derivatives of σ, ζ i (u = u i ln σ (u, ij (u = ln σ (u, u i u j 3 ijk (u = ln σ (u, u i u j u k etc., (.49 where i, j, k {1,...,g}. In this notation the Weierstraß -function is 11 (u. For convenience, we introduce the vector of ζ -functions ζ (u = (ζ 1 (u,...,ζ g (u T and also denote the derivatives of the σ -function by σ i (u = u i σ (u, σ ij (u = u i u j σ (u, etc. (.50 C. Main formula We consider now the integration of the differentials of the second and third kind. Proposition.: Let X g be a hyperelliptic curve of genus g with branch point at infinity. Let D = (P 1,...P g, D = (P 1,...P g, P k = (Z k, W k, P k = (Z k, W k be non-special divisors of degree g. Let P = (x, y and P = (x, y be two arbitrary points of X. Then ( P g P Pk F(x, z + yw dx dz σ P P k 4(x z y w = ln P 0 du ( D g du P σ ( P σ P 0 du P 0 du D g du ln ( D g du P σ P 0 du. D g du (.51 Proof: We introduce non-special divisors D = ((Z 1, W 1,...,(Z g, W g, D = ((Z 1, W 1,...,(Z g, W g as well as two arbitrary points P = (x, y, P = (x, y. The integration of (P, Q given by (.16, P g (Zk,W k (P, Q (.5 P (Z k,w k

12 Hyperelliptic integrals Hořava Lifshitz BH ST J. Math. Phys. 53, (01 yields, according to the Riemann vanishing theorem, ( P θ P 0 dv g ( (Zk,W k P P 0 dv + K P0 θ P ln ( P θ P 0 dv 0 g ln ( (Z k,w k P dv + K P0 θ P 0 P 0 dv g (Zk,W k dv g P 0 dv + K P0, (.53 (Z k,w k P 0 dv + K P0 where P 0 is the base point of the Abel map (that we suppose to be infinity and K P0 is the vector of Riemann constants with the base point P 0. Using the definition of the fundamental σ -function (.39 we get (.51. The following corollaries follow from the main formula (.51. Corollary.3: Let P = (x, y, P k = (x k, y k, k = 1,...,g. Then ( g P g Pk ij du du x i 1 k x j 1 = F(x, x k + yy k, k = 1,...,g. (.54 4(x x k i, j=1 P 0 r=1 P 0 Proof: Take the partial derivative / x x k on both sides of (.51. Corollary.4: Let D = P P g be a positive divisor of degree g. Then the standard Jacobi inversion problem given by the equations g Pk du = u (.55 is solved in terms of Kleinian -functions as (In the previous work 15 in this formula numbered as (3.44 as well in its particular cases (5.5 and (6.5 the sign - was misplaced. x g gg (ux g 1 g,g 1 (ux g... g,1 (u = 0, (.56 y k = ggg (ux g 1 k + gg,g 1 (ux g k gg,1 (u, k = 1,...,g. (.57 Proof: To prove (.56 consider x. Substitute x = 1/ξ into (.54. Comparison of the coefficients of 1/ξ (g 1 on the RHS and LHS of (.54 gives relation (.56. Formula (.57 follows from (.56 and (.54. For details see Ref. 8. For the meromorphic differentials of the second kind the following relation was proved by Buchstaber and Leykin. 10 Lemma.5 (ζ -formula: Let D 0 be a divisor supported by g branch points e i1,...,e ig such that g (eik,0 ( ε T du = K, [K ]:= ε T, (.58 and D is a non-special divisor of degree g, D = P P g. Then for any vector u = D g du Jac(X, (.59 the following relation is valid: D dr = ζ (u + (η ε + ηε + 1 Z(u, (.60 D 0 where the components Z j (u of the vector Z(u are Z g (u = 0, Z g 1 (u = ggg (u,

13 Enolski et al. J. Math. Phys. 53, (01 and the other components at 1 j < g 1 are given by the j j determinants: gg (u g 1,g (u gg (u Z j (u = (g k k+1,g (u k+,g (u ( (g j 1 j+,g (u j+3,g (u gg (u 1 (g j j+1,g,g (u j+,g,g (u g 1,g,g (u ggg (u The corresponding formula in Buchstaber and Leykin 10 coincides with those given by Baker, Ref. 3 p. 31, only for j = g and j = g 1 and differs for 1 j < g 1. Our further consideration is based on (.61. For our purposes it is necessary to rewrite the expressions for Z j in terms of symmetric functions of the divisor points P 1 = (x 1, y 1,...,P g = (x g, y g. With the solution of the Jacobi inversion problems (.56 and (.57, the components Z j (u yield Z g = 0, Z g 1 = y 1 (x 1 x (x 1 x g + permutations, Z g = y 1 x 1 (x x g (x 1 x (x 1 x g + permutations, Z g 3 = y 1 x 1 x 1(x x g + x x x g 1 x g (x 1 x (x 1 x g + permutations.. Note that the characteristics in formula (.60 are not reduced. From (.60 one obtains the relation between the periods of holomorphic and meromorphic integrals η ik = (ζ i (ω k + K i,,...,g, η ik = ( ζ i (ω k + K (.6 i,,...,g with ω k being the kth column of the matrix ω. Proposition.6: Using (.17 formula (.51 can be rewritten in the form suitable for the inversion of the integral of the third kind P g [ y + Wk y + W ] k dx P x Z k x Z k y ( P (.63 P = du T (x, y P D D σ dr(z,w + ln σ ( P du D g du ( P du D σ ln g du σ where the expression D D dr(z,w can be calculated by (.60. D. Stratification of the θ-divisor and inversion du D g du ( P du, D g du The θ-divisor is defined as the subset of Jac(X g that nullifies the θ-function and, therefore, the σ -function, i.e., = { v Jac(X g θ(v 0 }. (.64

14 Hyperelliptic integrals Hořava Lifshitz BH ST J. Math. Phys. 53, (01 TABLE I. Orders m( k ofzerosθ( k + v atv = 0onthestrata k. g m( 0 m( 1 m( m( 3 m( 4 m( 5 m( The subset k,0 k < g, is called kth stratum if each point v admits a parametrization { k Pj } k := v v = dv + K, (.65 where 0 ={K } and g 1 =. We furthermore denote g = Jac(X g and we have the natural embedding g 1 g = Jac(X g. (.66 j=1 We define the θ-function to be vanishing to the order m( k along the stratum k if for all sets α j, j = 1,...,g with 0 α α g < m holds: α α g u α θ(v τ 0, v uα g k, (.67 g and there is a certain set of α j, with α α g = m such that (.67 does not hold. The orders m( k of the vanishing of θ( k + v along the stratum k for some genera are given in Table I. In the following we focus on the stratum 1 corresponding to the variety 1 Jac(X g, which is the image of the curve inside the Jacobian, { 1 := v v = P dv + K }. (.68 We remark that another stratification was introduced in Ref. 48 for hyperelliptic curves of even order with two infinite points + and that was implemented for studying the poles of functions on Jacobians of these curves. The same problem relevant to strata of the θ-divisor was studied in Ref. 1. III. INVERSION OF HYPERELLIPTIC INTEGRALS OF HIGHER GENERA A. Inversion of holomorphic integrals For the case of genus two the inversion of a holomorphic hyperelliptic integral by the method of restriction to the θ-divisor was obtained independently by Grant 3 and Jorgenson 36 in the form x = σ 1(u σ (u, u = (u 1, u T. (3.1 σ (u=0 This result was implemented in Ref. 14, and explicitly worked out in the series of publications, 9 33 and others. The case of genus three was studied by Ônishi, 45 where the inversion formula is given in the form x = σ 13(u, u = (u 1, u, u 3 T. (3. σ 3 (u σ (u=σ3 (u=0

15 Enolski et al. J. Math. Phys. 53, (01 Formula (3. is based on the detailed analysis of the genus three KdV hierarchy and its restriction to the θ-divisor. Below we will present the generalization of (3.1 and (3. to higher genera. For doing this we first analyze the Schur Weierstraß polynomials that represent the first term of the expansion of σ (u in the vicinity of the origin u 0. The θ-divisor and its strata k in the vicinity of the origin u 0 are given as polynomials in u. An analysis of the Schur Weierstraß polynomials leads to the following. Proposition 3.1: The following statements are valid for the Schur Weierstraß polynomials S π (u associated with a partition π: 1. In the vicinity of the origin, an element u of the first stratum 1 is singled out by S π (u = 0, j u j g S π (u = 0 j = 1,...,g. (3.3. The derivatives fulfill j ug j { 0 if 1 j < g(g 1 S π (u 0 if j g(g 1 with u 1. ( The following equalities are valid for u 1 x = 1 u g = M u 1 u M 1 g M u u M 1 g M+1 S π (u S u 1 u M π (u g = M+1, (3.5 S π (u S u ug M π (u where M = 1 (g (g The order of vanishing of S π restricted to 1 is the rank of the partition π. It was noted in Ref. 9 that the Schur Weierstraß polynomials respect all statements of the Riemann singularity theorem. In particular, if ( z g 1 Z = zk 1 z3,...,..., g 1 k 1 3, z and if π is the partition at the infinite Weierstraß point of the hyperelliptic curve X g of genus g, then the function G(z := S π (Z u (3.7 either has g zeros or vanishes identically. Moreover, we will conjecture here that the properties of the Schur Weierstraß polynomials given in Proposition 3.1 can be lifted to the fundamental σ -function (.39. The above analysis permits to conjecture the following inversion formula for the general case of hyperelliptic curves of genus g > : 15 and x = M+1 σ (u u 1 ug M M+1 σ (u u ug M u 1 { 1 = u Jac(X g σ (u = 0,, M = j u j g (g (g 3 ( (3.8 } σ (u = 0 j = 1,...,g. (3.9 The analog of this formula for strata k,1< k < g and (n, s-curves in the terminology of 4, 43 Ref. 9 was recently considered by Matsutani and Previato.

16 Hyperelliptic integrals Hořava Lifshitz BH ST J. Math. Phys. 53, (01 B. Inversion of the integrals of the second kind Also the meromorphic integrals can be expressed in terms of the σ -functions restricted to the stratum 1 of the θ-divisor. To do that we consider the ζ -formula (.60 and move to infinity the g 1 points of the divisor D on both sides of the equality. The poles on both sides cancel and the nonvanishing terms yield the required formula. We demonstrate this procedure here for the example of the genus three case given in Ref. 13 (the analysis of the genus two case can be found in Ref. 14. For that we use the following proposition. Proposition 3.: Let X 3 be the genus three curve y = 4x 7 + λ 6 x λ 0. Let u 1 = {u σ (u = σ 3 (u = 0}. Then the following formulae are valid P P 0 P dr 3 = σ 3(u σ (u + c 3, P 0 dr = 1 P P 0 dr 1 = 1 σ (u σ (u + c, σ 1 (uσ (u σ (u σ 1(u σ (u + c 1, (3.10 where P = (x, y is an arbitrary point of the curve, and P 0 = (x 0, y 0 (, is any fixed point. It is convenient to choose, in particular, P 0 = (e g,0.the constants c i are fixed by the requirement that the right hand side vanishes at P = P 0. Proof: In the case of g = 3 the vector Z(u (.61 yields Z 1 (u 33 (u 333 (u + 33 (u Z(u Z (u = Z 3 (u 333 (u 0. (3.11 First we restrict relation (.60 tothestratum. To do that we use the following expansions: P3 1 5 ξ 5 + O(ξ 7 P3 ξ 5 + O(ξ 3 du = 1 3 ξ 3 + O(ξ 5, dr = ξ 3 + O(ξ 1 (3.1 x3 =1/ξ ξ + O(ξ 3 (e 6,0 x 3 =1/ξ ξ 1 + O(ξ taking into account the condition σ (u = 0 in the expansions. Then we restrict the resulting formulae to the stratum 1. To do that we expand the holomorphic and meromorphic integrals P du and P (e 4,0 dr in the vicinity of P = (, taking into account the condition σ 3 (u = 0. From that relations (3.10 follow. For the simplification of the obtained relations we used the formulae from Ref. 45 below, which result from the restriction to 1 of the KdV hierarchy associated with the genus three curve: {σ 333 (u σ (u} 1 = 0, {σ 33 (u σ 3(u + σ 1 (u} = 0, (3.13 σ (u 1 {σ 133 (u σ 1(uσ 3 (u σ 1(u } = 0. σ (u σ (u 1 Cases of higher genera can be considered in an analogous way though it is not possible to present a general formula.

17 Enolski et al. J. Math. Phys. 53, (01 C. Inversion of the integral of the third kind We consider (.63 when all points P k, P k have the same coordinate Z k = Z k but W k = W k and choose the divisors It is evident that D ={(Z, W, (e 4, 0,...,(e g, 0}, D ={(Z, W, (e 4, 0,...,(e g, 0}. (3.14 D dr(z,w = D From the ζ -formula (.60 we get (Z,W (Z, W dr(z,w = ζ dr(z,w = ζ (Z,W ( (Z,W ( (Z,W where Z g (Z, W = 0 and for 1 j < g we have Z j (Z, W = dr(z,w (Z,W dr(z,w. (3.15 du + K + (η ε + ηε + 1 Z(Z, W, du + K (η ε + ηε 1 Z(Z, W, g j 1 W g k= (Z e ( 1 g k+ j+1 Z k S g k j 1 (e. (3.16 k k=0 The S k (e are elementary symmetric functions of order k built on g 1 branch points e 4,...,e g : S 0 = 1, S 1 = e e g, etc. Then the solution of the inversion problem for the integral of the third kind (.63 takes the form P 1 dx P [ ( (Z,W W P x Z y = du T (x, y ζ du + K (η ε + ηε 1 ] P Z(Z, W ( P σ du ( (Z,W P du K σ + ln ( P σ du + du (Z,W du K ln ( (Z,W P du K σ du +. (Z,W du K (3.17 Note that (3.17 represents a natural generalization of the known antiderivative for elliptic functions (v du = uζ (v + ln σ (u v ln σ (u + v. (3.18 (u (v The inversion procedure for the integral of the third kind P 1 dz W = t, (3.19 x Z w is described as follows. Here for simplicity the lower bound is fixed at a branch point, say at P =. In this case formula (3.17 takes the form where W P v = 1 dx x Z y = (A u [ζ (v + K (η ε + ηε 1 Z(Z, W ] +ln σ (u v K σ (u + v K lnσ (A v K σ (A + v K, (3.0 (Z,W du, A = (e,0 du and u 1, u = (x,y du.

18 Hyperelliptic integrals Hořava Lifshitz BH ST J. Math. Phys. 53, (01 D. Solving the inversion problem Now we are in a position to describe the inversion procedure for integral (1.. The formulae given above permit to represent Eq. (1. intheform F(x, u 1,...,u g, constants = t. (3.1 The function F depends on x via the g variables u 1,...,u g that are abelian images of (x, y and various constants given by the poles of the integrals of the third kind and the coefficients a k, b k, c k, and possibly via the elementary function E(x in Eq.(1.3. This relation is complemented by the g 1 conditions u 1 σ (u = 0, j u j g σ (u = 0 j = 1,...,g. (3. From these relations one can find (numerically the functions u 1 = u 1 (t,...,u g = u g (t and plug these into (3.8 to find x = x(t. IV. COMPUTER ALGEBRA SUPPORTING THE METHOD Presently, effective means of computer algebra are developed to execute the above claimed program of integral inversion. We will consider for this, e.g., the Maple/algcurves code. A. Riemann period matrix and winding vectors For a given curve of genus g we compute first the period matrices (ω,ω and τ = ω 1 ω by means of the Maple/algcurves code. From that we determine the winding vectors, i.e., the columns of the inverse matrix, (ω 1 = (U 1,...,U g. (4.1 B. Homology basis In our analysis we used a specific homology basis for a hyperelliptic curve (see Fig. 1. That was done just to clarify the approach. But the result should be independent of the choice of the homology basis. It is possible to perform all the calculations without making an explicit plot of the homology basis and to use that one given in the Tretkoff-Tretkoff construction that is programmed in the Maple/algcurves. Nevertheless, in certain cases we should know the correspondence between the branch points and half-periods in the homology basis used by Maple/algcurves. In particular, the proposed method of inversion supposes the knowledge of this correspondence. One can find this correspondence using the generalized Bolza formulae. That can be done as follows. We first find all nonsingular odd characteristics by direct computation of all odd θ-constants. According to Table I we have two sets B 1 g 1 and B g of nonsingular odd half-periods. For each element of b 1 B 1 there are e i1,...,e ig 1 such that b 1 = (ei1,0 dv (eig 1,0 and for each element of b B there are e i1,...,e ig such that b = (ei1,0 dv (eig,0 dv + K g 1 (4. dv + K g. (4.3 Using the known values of the winding vectors (Ref. 15, Proposition 4.3 one can find the correspondence between the sets {e i1,...,e ig 1 } and {e i1,...,e ig } of branch points and the nonsingular odd characteristics [(ω 1 (A i1,...,i g 1 + K ] and [(ω 1 (A i1,...,i g + K ]. Then one can add these

19 Enolski et al. J. Math. Phys. 53, (01 characteristics and find the one-to-one correspondence (eig 1,0 dv [A ig 1 ], i = 1,...,g +. (4.4 C. Second period matrix We present the formula that permits to express the second period matrices η, η, and κ in terms of the first period matrices ω, ω, the branch points, and the θ-constants Proposition 4.1: Let A I0 + ωk be an arbitrary even nonsingular half-period corresponding to the g branch points of the set of indices I 0 ={i 1,...,i g }. We define the symmetric g g matrices P(A I0 := ( ij (A I0 i, j=1,...,g (4.5 and the g g matrix H which is expressible in terms of the even nonsingular θ-constants (In the previous work 15 this formula numbered as (3.50 was written in an equivalent but more complicated form with misplaced sign -. 3 H(A I0 = 1 θ[ε] (θ ij[ε] i, j=1,...,g, where ε = [(ω 1 A I0 + K ]. (4.6 Then the κ-matrix is given by κ = 1 P(A I 0 1 ((ω 1 T H(A I0 (ω 1. (4.7 The half-periods η and η of the meromorphic differentials can be represented as η = κω, η = κω iπ (ω 1 T. (4.8 We remark that (4.7 represents the natural generalization of the Weierstraß formulae (The correspondence between the numeration of branch points e i and θ-constants ϑ j (0 depends on the chosen homology basis. 4 ηω = e 1 ω 1 ϑ (0 ϑ (0, ηω = e ω 1 ϑ 3 (0 ϑ 3 (0, ηω = e 3ω 1 ϑ 4 (0 ϑ 4 (0, (4.9 see, e.g., the Weierstraß Schwarz lectures, Ref. 49 p. 44. From (4.9 follows ωη = 1 ( ϑ (0 1 ϑ (0 + ϑ 3 (0 ϑ 3 (0 + ϑ 4 (0. (4.10 ϑ 4 (0 Therefore, Proposition 4.1 allows the reduction of the variety of moduli necessary for the calculation of the σ - and -functions to the first period matrix. The generalization of (4.10 to arbitrary higher genera algebraic curves has recently been discussed in Ref. 39 where its role in the construction of invariant generalizations of the Weierstraß σ -function to higher genera is elucidated. For the following Lemma we introduce g vectors: E j = (1, e j,...,e g 1 j T, j I 0. Lemma 4.: Let S (ij k be elementary symmetric functions of order k built on the g 1 elements {e 1,...,e g + 1 }/{e i, e j }. Then the following relations are valid Here, F(x, z is the Kleinian -polar (.13. Ei T P(A I 0 E j F(e g 1 i, e j 4(e i e j e n i en j S(ij g n 1. (4.11 n=0

20 Hyperelliptic integrals Hořava Lifshitz BH ST J. Math. Phys. 53, (01 Proof: The LHS of the relation (4.11 follows from (.54: ( g g Pk ij du xr i 1 xs j 1 = F(x r, x s y r y s, r s, P i, j=1 P 0 4(x r x s k = (x k, y k, (4.1 where the argument of the -function is a nonsingular even half-period given as an abelian image of g branch points P k = (e ik, 0, i 1,...,i g {1,...,g + 1}. The branch points e r and e s on the right hand side of (4.1 belong to the chosen subset of g branch points. The RHS of relation (4.11 can be checked directly for small genera and presents a conjecture for higher genera. Varying the integers i, j along the set I 0 one can obtain from (4.10 g(g 1/ equations with respect to g(g 1/ components, g 1, g 1,..., 11 of the matrix P(A I0 and then solve them by the Kramer rule. Indeed, since the components gi (I 0 are already known from the solution of the Jacobi inversion problem (II.58 then Eq. (IV.10 can be simplified for every pair i j I p as follows: Ê T i P(A I0 Ê j = e g 1 i e g 1 j S (ij 1 + g e k + ei n en j S(ij g n 1 (4.13 k I 0 /{i, j} n=0 where P(A I0 = ( ij (A I0 i, j=1,...,g 1, Ê i = (1, e i,...,e g i T. (4.14 Examples for genus two, three, and four in Sec. V VII show that the entries in P are polynomials in the branch points e i (see Eqs. (5.14, (6.19, and (7.1. We infer that this is true, in general, but do not present here the general expression of these polynomials. D. Characteristics We explained above how to find the correspondence between the half-periods and the abelian images of the branch points on the basis of Bolza type formulae. This correspondence is necessary for the description of the real evolution of the system that corresponds to the motion of the divisor points, i.e., functions on the upper bound in the Abel map, over the Riemann surface between certain branch points or between the corresponding half-periods in the Jacobi variety. We intend to reduce a number of cases when complete hyperelliptic integrals are calculated. Finding the above correspondence allows to present initial/final points of the evolution in terms of a half-period. E. Vector of Riemann constants The vector of Riemann constants K enters the derived inversion formulae and can be computed as follows. It was proved that the vector of Riemann constants with the base point fixed in a branch point, e.g., infinity in our considerations, is a half-period. From Table I for the stratum 0 follows whether K is even or odd (parity of m and whether it is singular (m > 1. In accordance with the definition of the θ-divisor it is sufficient to find the even or odd half-period K which satisfies the condition θ ( g 1 Pk dv + K = 0, where P 1,...,P g 1 are g 1 arbitrary points on the curve X g. Alternatively, one can use the correspondence between the branch points and the half-periods. Among the g + characteristics (4.4 there should be precisely g odd and g + even characteristics. The sum of all odd characteristics gives the vector of Riemann constants with the base point at infinity. This characteristic will then be singular of the order of [ g+1 ].

21 Enolski et al. J. Math. Phys. 53, (01 b 1 b e 1 e e 3 e 4 e 5 e 6 = a 1 a FIG.. Homology basis on the Riemann surface of the curve X with real branch points e 1 < e <... < e 6 =(upper sheet. The cuts are drawn from e i 1 to e i, i = 1,, 3. The b-cycles are completed on the lower sheet (dotted lines. V. HYPERELLIPTIC CURVE OF GENUS TWO We consider a hyperelliptic curve X of genus two w = 4(z e 1 (z e (z e 3 (z e 4 (z e 5 = 4z 5 + λ 4 z 4 + λ 3 z 3 + λ z + λ 1 z + λ 0. From (.1 and (. the basic holomorphic and meromorphic differentials are du 1 = dz w, du = zdz w, Then the Jacobi inversion problem for the equations (z1,w 1 dz w + (z1,w 1 zdz w + is solved according to (.56 and (.57 in the form (5.1 dr 1 = 1z3 +λ 4 z +λ 3 z 4w dz, (5. dr = z dz. (5.3 w (z,w (z,w dz w = u 1, zdz w = u z 1 + z = (u, z 1 z = 1 (u, w k = (uz k + 1 (u, k = 1,. (5.4 (5.5 A. Characteristics in genus two The homology basis of the curve is fixed by defining the set of half-periods corresponding to the branch points. The characteristics of the abelian images of the branch points are defined as [ (ei,0 ] ( T ( ε i ε [A i ] = du = = i,1 ε i,, (5.6 ε i,1 ε i, which can be also written as [A 1 ] = 1 A i = ωε i + ω ε i, i = 1,...,6. In the homology basis given in Fig. the characteristics of the branch points are ( 1 0 ( 1 0 ( , [A ] = 1 ε T i, [A 3 ] = 1, (5.7

22 Hyperelliptic integrals Hořava Lifshitz BH ST J. Math. Phys. 53, (01 [A 4 ] = 1 ( 0 1, [A ] = 1 The characteristics of the vector of Riemann constants K yield [K ] = [A ] + [A 4 ] = 1 ( 0 0, [A ] = 1 ( ( 0 0. ( (5.9 From the above characteristics we build 16 half-periods. Denote 10 half-periods for i j = 1,...,5thatareimages of two branch points as ij = ω(ε i + ε j + ω (ε i + ε j, i = 1,...,5. (5.10 Then the characteristics of the 6 half-periods [ (ω 1 ] A i + K =: δi, i = 1,...,6 (5.11 are nonsingular and odd, whereas the characteristics of the 10 half-periods [ (ω 1 ij + K ] =: εij, 1 i < j 5 (5.1 are nonsingular and even. Odd characteristics correspond to partitions {6} {1,...,5} and {k} {i 1,...,i 4,6} for i 1,...,i 4 k. The first partition from these two corresponds to 0 and the second corresponds to 1. From the solution of the Jacobi inversion problem we obtain for any i, j = 1,...,5,i j, From the relation one can also find e i + e j = ( ij, e i e j = 1 ( ij. ( (u = F(x 1, x y 1 y 4(x 1 x, e i e j (e p + e q + e r + e p e q e r = 11 ( ij, (5.14 where i, j, p, q, and r are mutually different. From (5.13 and (5.14 we obtain an expression for the matrix κ (4.7 that is useful for numeric calculations because it reduces the second period matrix to an expression in the first period matrix and θ-constants, namely, in the case e i = e 1, e j = e, κ = 1 ( e1 e (e 3 + e 4 + e 5 + e 3 e 4 e 5 e 1 e 1 1 e 1 e e 1 + e (ω 1T θ[ε] where the characteristic ε in the fixed homology basis reads ε = [A 1 ] + [A ] + [K ] = ( ( θ11 [ε] θ 1 [ε] θ 1 [ε] θ [ε] (ω 1, (5.15 B. Inversion of a holomorphic integral Taking the limit z in the Jacobi inversion problem (5.4 we obtain The same limit in the ratio (z,w dz w = u 1, (z,w zdz w = u. ( (u (u = z 1z z 1 + z (5.17