Dispersionless Hirota equations and the genus 3 hyperelliptic divisor

Equations of dispersionless Hirota type have been thoroughly investigated in the mathematical physics and differential geometry literature. It is known that the parameter space of integrable Hirota type equations in 3D is 21-dimensional and the action of the natural equivalence group Sp(6, R) on the parameter space has an open orbit. However the structure of the `master-equation' corresponding to this orbit remained elusive. Here we prove that the master-equation is specified by the vanishing of any genus 3 theta constant with even characteristic. The rich geometry of integrable Hirota type equations sheds new light on local differential geometry of the genus 3 hyperelliptic divisor.


Introduction
A 3D dispersionless Hirota type equation is a second-order PDE of the form where u is a function of 3 independent variables (x 1 , x 2 , x 3 ). Equations of type (1.1) arise in numerous applications in non-linear physics, general relativity, differential geometry, theory of integrable systems, and complex analysis. For instance, the dispersionless Kadomtsev-Petviashvili (dKP) equation arises in non-linear acoustics [44], the theory of Einstein-Weyl structures [12] and in the context of conformal maps of simply-connected domains [43,45,31]. The Boyer-Finley (BF) equation describes a class of self-dual Ricci-flat 4-manifolds [1]. Let us review briefly some of the known properties of integrable Hirota type equations (see Section 2 for more details): Here U = Hess(u) is the Hessian matrix of the function u [14]. The classification of integrable equations is performed modulo equivalence (1.3).
• The parameter space of integrable equations (1.1) is 21-dimensional (see [14]). Furthermore, the action of the equivalence group Sp(6, R) on the parameter space is locally free. The fact that the group Sp(6, R) is 21-dimensional implies the existence of a generic Hirota master-equation generating an open 21-dimensional Sp(6, R)-orbit. The masterequation has no continuous symmetries from the equivalence group. Note that both dKP and BF equations have nontrivial symmetry groups and generate singular orbits of lower dimensions (14 and 15, respectively). • Geometrically, Hirota type equation (1.1) can be viewed as the defining equation of a 5-dimensional hypersurface M 5 in the 6-dimensional Lagrangian Grassmannian Λ 6 (locally identified with 3 × 3 symmetric matrices). Integrability of equation (1.1) has a clear interpretation in terms of differential geometry of the associated hypersurface M 5 ⊂ Λ 6 (see [14]). Although the structure of the Hirota master-equation has remained elusive, formula (1.3) suggests that this equation should have non-trivial modular properties. In this direction let us mention yet another example of integrable equation, which was discussed in [36,37]. Here the function h satisfies the Chazy equation whose general solution can be expressed in terms of the Eisenstein series of weight 2 on the modular group SL(2, Z): Note that h(t) is real for t > 0. This was one of the first examples where modular forms explicitly occurred as coefficients rather than as solutions of integrable systems, see also [30,13,2] (note that the above ODE admits non-trivial continuous symmetries and therefore does not generate an open orbit). We refer to Section 3 for other 'modular' examples. Before giving our main theorem, let us recall few facts about theta constants. All these facts can be found in [28]. Let H 3 be the Siegel upper half space of genus 3 i.e.
The theta constants with characteristics are defined by where µ, ν ∈ {0, 1} 3 and τ ∈ H 3 . The characteristic m = [ µ ν ] is called even if µν t is even. In genus 3, there are 36 such characteristics and they give rise to 36 theta constants that are modular forms of weight one-half on the theta group. Recall that there is an action of the group Sp(6, Z) on the set of even characteristics which is transitive. Taking the product of these 36 theta constants with even characteristics provides a cusp form of weight 18 on the full modular group Sp(6, Z), this form is classically denoted by χ 18 (see [27]). It is known that in genus 3 the vanishing of even theta constants characterises the hyperelliptic divisor.
Our main result is the following explicit formula for the master-equation: where ϑ m is any genus 3 theta constant with an even characteristic m. Therefore the corresponding hypersurface M 5 ⊂ Λ 6 is the genus 3 hyperelliptic divisor.
Remark 1.2. Note that although there are 36 even theta constants in genus 3, the corresponding equations (1.6) are equivalent due to the transitivity of the Sp(6, Z)-action on the set of even theta constants.
Using the cusp form χ 18 , the equations (1.6) can be compactly represented as Up to the multiplicative constant 2 28 , the Fourier expansion of χ 18 starts with The real-valued locus M 5 results on restriction of χ 18 to Λ 6 embedded into H 3 as the subset of purely imaginary 3 × 3 symmetric matrices. Theorem 1.1 will be proved in Section 4 by uncovering geometry behind the Odesskii-Sokolov construction [35] that parametrises broad classes of dispersionless integrable systems via generalised hypergeometric functions.
There exist several approaches to dispersionless integrability in 3D. Based on seemingly different ideas they however lead to equivalent classification results. These approaches are: • The method of hydrodynamic reductions based on the requirement that equation (1.1) has 'sufficiently many' special multiphase solutions. • Integrability on equation, meaning that the associated hypersurface M 5 ⊂ Λ 6 carries an integrable GL(2, R) geometry. • Integrability on solutions, based on the requirement that the characteristic variety of equation (1.1) defines Einstein-Weyl geometry on every solution. • Integrability via a dispersionless Lax representation.
The above properties of the master-equation shed new light on local differential geometry of the genus 3 hyperelliptic divisor, see Sections 2.1-2.4.

Four equivalent approaches to dispersionless integrability
In this Section we summarise the four different approaches mentioned above to dispersionless integrability in 3D.

Integrability via hydrodynamic reductions.
The method of hydrodynamic reductions applies to quasilinear systems of the form where v = (v 1 , ..., v m ) t is an m-component column vector of the dependent variables and A, B, C are l × m matrices, l ≥ m. Note that Hirota type equation (1.1) can be brought to quasilinear form (2.1) by representing it in evolutionary form, choosing the arguments of f as the new dependent variables v and writing out all possible consistency conditions among them. This results in the quasilinear representation (2.1) with m = 5, l = 8. Applied to system (2.1) the method of hydrodynamic reductions consists of seeking multiphase solutions in the form where the phases (Riemann invariants) R i (x), whose number N is allowed to be arbitrary, are required to satisfy a pair of commuting (1 + 1)-dimensional systems known as systems of hydrodynamic type [11,42]. The corresponding characteristic speeds µ i , λ i must satisfy the commutativity conditions [42] 3) are said to define an N -component hydrodynamic reduction of system (2.1). The following definition can be found in [22,13]. This requirement imposes strong constraints (integrability conditions) on the matrix entries of A, B and C. Applied to equation (2.2), the method of hydrodynamic reductions leads to an Sp(6, R)-invariant set of differential constraints for the function f expressing all third-order partial derivatives of f in terms of its first and second-order partial derivatives (35 relations that are rational in the partial derivatives of f ). The integrability conditions were first derived in [14]. Their involutivity implies that the parameter space of integrable Hirota type equations is 21-dimensional [14].
Note that (2.2) is the equation of the graph of the corresponding hypersurface M 5 ⊂ Λ 6 . Since the Hirota master-equation coincides with the equation of the genus 3 hyperelliptic divisor, the integrability conditions can be viewed as local differential constraints that characterise uniquely the hyperelliptic divisor up to Sp(6, R)-equivalence.

Integrability on equation
The Lagrangian Grassmannian Λ 6 (locally parametrised by 3 × 3 symmetric matrices) carries a flat generalised conformal structure defined by the family of degree 4 Veronese cones in T Λ 6 (identified with rank 1 symmetric matrices). Let M 5 be a hypersurface in Λ 6 . Taking a point s ∈ M 5 and intersecting the tangent space T s M 5 with the Veronese cone in T s Λ 6 one obtains a rational normal cone of degree 4 in T s M 5 . On projectivization, this results in a family of rational normal curves γ of degree 4 in PT M 5 . This structure is known as a GL(2, R) geometry on M 5 .

Definition 2.2.
A bisecant plane in T M 5 is a plane whose projectivisation is a bisecant line of γ.
A bisecant surface is a 2-dimensional submanifold Σ 2 ⊂ M 5 whose tangent planes are bisecant. A trisecant space in T M 5 is a 3-dimensional subspace whose projectivisation is a trisecant plane of γ.
To be more precise, we will need holonomic trisecant 3-folds which can be defined as follows. Note that each tangent space T Σ 3 carries 3 distinguished directions, namely those corresponding to the 3 points of intersection of PT Σ 3 with γ. These directions define a net on Σ 3 which we require to be holonomic i.e. a coordinate net.
It turns out that bisecant surfaces and holonomic trisecant 3-folds of a hypersurface M 5 correspond to 2-and 3-component hydrodynamic reductions of the associated dispersionless Hirota type equation. Furthermore, hypersurface M 5 corresponds to an integrable equation if and only if it has infinitely many holonomic trisecant 3-folds parametrised by 3 arbitrary functions of one variable [14]. Thus the existence of holonomic trisecant 3-folds is a geometric interpretation of the integrability property. The corresponding integrable GL(2, R) geometries were thoroughly investigated in [41].
Coming back to the Hirota master-equation, we can conclude that the genus 3 hyperelliptic divisor carries an integrable GL(2, R) geometry.

Integrability on solutions: Einstein-Weyl geometry.
Any solution of equation (1.1) carries a family of characteristic cones assumed to be non-degenerate. The inverse matrix of the associated quadratic form gives rise to the conformal structure [g] = g ij dx i dx j which depends on a solution due to non-linearity of (1.1). It was shown in [17] that integrability of equation ( is the symmetrised Ricci tensor of D, and ρ is some function [6]. It was shown in [17] that the covector ω can be expressed in terms of g by the universal explicit formula where D x k denotes the total derivative with respect to x k . We recall that in 3D the Einstein-Weyl equations (2.4) are integrable by twistor-theoretic methods [24]. Thus solutions of integrable PDEs carry 'integrable' conformal geometry. Combining 2.2 and 2.3, we can conclude that every trisecant 3-fold of the hyperelliptic divisor carries Einstein-Weyl geometry.
Cartan proved (see [6]) that a pair (D, [g]) defined on a 3-dimensional manifold satisfies Einstein-Weyl equations (2.4) if and only if there exists a 2-parameter family of surfaces which are totally geodesic with respect to the connection D and null with respect to the conformal structure [g]. Such surfaces come from the associated dispersionless Lax pairs [12,17,5].

Integrability via dispersionless Lax representation.
A pair of Hamilton-Jacobi type equations for an auxiliary function S, Here the functions f and g are defined by (we set S x 1 = x, u x 1 x 1 = t): where the function v is the Jacobi theta series ϑ 00 , see (3.3) for the definition, evaluated at (τ, z) = ( it π , x 2π ): v(x, t) = ϑ 00 (it/π, x/2π) = 1 + 2 ∞ n=1 e −n 2 t cos(nx).
In some cases it is more convenient to deal with parametric Lax pairs, where p is an auxiliary parameter. Such Lax pairs were used in the construction of the universal Whitham hierarchy [29]. Thus parametric form of the dKP Lax pair (2.7) is Note that the compatibility condition of equations (2.8) is It follows from [17,5] that Hirota type equation (1.1) is integrable if and only if it has a dispersionless Lax representation (satisfying a suitable non-degeneracy condition).

Examples
In this section, we give a few more examples of integrable equations of type (1.1) with nontrivial modular properties.

Equation of the form
This example has already been investigated [2,3,15]. For the reader's convenience, we explain it again. In this case, the integrability conditions obtained via the method of hydrodynamic reductions lead to the following system of third-order PDEs, we set b = u x 2 x 3 and c = u x 3 x 3 : (3.1) We will show that a generic solution of this system is given by the logarithmic derivative of any Jacobi theta series with characteristics. The first two equations lead to the Burgers' equation f f b − f c = νf bb where ν ∈ C * . Without any loss of generality, we set ν = −1 and using classical results about Burgers' equation, we can write the function f as f = 2 ψ b ψ where the function ψ satisfies the heat equation: ψ c = ψ bb . By substituting f = 2 ψ b ψ in the equations (3.1) and reducing modulo ψ c = ψ bb , the first two equations are automatically satisfied while the third one gives the following sixth-order ordinary differential equation Finally the last equation of (3.1) gives the derivative with respect to b of the latter one. We introduce the following Jacobi theta series with characteristics: ϑ αβ (τ, z) = n∈Z e πi((n+α/2) 2 τ +2(n+α/2)(z+β/2)) (3.3) where τ ∈ H 1 = {τ ∈ C | Im(τ ) > 0}, z ∈ C and (α, β) ∈ R 2 (characteristics). Recall that each Jacobi theta function satisfies the following PDE (heat equation): ) satisfies the equation ψ c = ψ bb . It remains to check that such a function ψ satisfies the equation (3.2). Note that if h is a solution of (3.2) then the following two functions h(b + g 1 (c), c), g 2 (c) e b g 3 (c) h(b, c) are also solutions of (3.2) for any function g i , the first statement is obvious while the second one requires some computations. Using the expression it is therefore sufficient to prove that the function ϑ 00 evaluated at (τ, z) = ( ic π , b 2π ) satisfies the ODE (3.2). The differential operator defined by the left hand side of (3.2), say D, has the following two properties: for α β γ δ in a suitable subgroup of SL(2, R) and (λ, µ) in a suitable sublattice of R 2 assume that a function h transforms as follows This indicates that the operator D sends, up to the condition on the Fourier expansion, Jacobi forms of weight one-half and index one-half to Jacobi forms of weight 12 and index 3 on a suitable subgroup of SL(2, R). Recalling that the Jacobi theta series ϑ 00 can be viewed as a Jacobi form of weight one-half and index one-half (with a multiplier system) on the principal congruence subgroup of level 4, checking that the function ϑ 00 evaluated at (τ, z) = ( ic π , b 2π ) satisfies the ODE (3.2) reduces to check it for sufficiently many Fourier coefficients (finite dimension of spaces of Jacobi forms) which can be done by any computer algebra systems. We conclude that any Jacobi theta series evaluated at (τ, z) = ( ic π , b 2π ) satisfies the ODE (3.2) and the PDE ψ c = ψ bb . Taking for example ψ(b, c) = ϑ 00 ( ic π , b 2π ), we get for the function f

Equations of the form
This case is obtained as a specialisation of the previous one: The integrability conditions in this case are obtained by substituting f (b, c) = b r(c) in the system (3.1). This leads to a single third-order ordinary differential equation for the function r, we set r (k) = d k r dc k : r ′′′ (r ′ − r 2 ) − (r ′′ ) 2 + 4r 3 r ′′ + 2(r ′ ) 3 − 6(rr ′ ) 2 = 0. (3.5) This case has already been investigated in section 4.5 of [3]. Let us briefly recall the results obtained there: we denote by D(r) the left hand side of (3.5). As noticed in [3], the differential operator D(r) has a SL(2, R)-invariance: for any α β γ δ ∈ SL(2, R), assume that the function r satisfies the following functional equation then we have Then modulo the SL(2, R)-action given by (3.6), the generic solution of (3.5) is given by (c < 0): Note that for τ ∈ H 1 , r( πiτ 2 ) = 1 − 8 n≥1 σ − 1 (n)e 2πinτ is a quasi-modular form of weight 2 on the congruence subgroup Γ 0 (2). In fact we have r( πiτ 2 ) = (4e 2 (2τ ) − e 2 (τ ))/3 where e 2 is the Eisenstein series of weight 2 on SL(2, Z), see (1.5).
We would like to give another description of the generic solution which will make a connection with the Legendre family of elliptic curves. We make the change of functions r = g ′ /g in (3.5), then the function g satisfies the following fourth-order ordinary differential equation: The section 5 of [7] proposed a method for linearising any equation of the form of (3.5) which has been carried out explicitly along the proof of the theorem 3 of [18]. This gives the following hypergeometric differential equation associated to the equation (3.5): Let w 1 and w 2 be two linearly independent solutions of this hypergeometric equation with the Wronskian normalised as w 2 w 1,s − w 1 w 2,s = 1 s(1−s) . These solutions can be written as periods of the holomorphic differential 1-form ω given by of the elliptic curve v 2 = t(t − 1)(t − s), we assume s ∈ (0, 1). Let us choose a basis of solutions such that w 1 has a logarithmic singularity at zero while w 2 is the standard hypergeometric series with parameters (1/2, 1/2; 1): This choice satisfies the Wronskian constraint. Noticing that the equation (3.7) has the following SL(2, R)-symmetry (this comes directly from (3.6)) c = αc + β γc + δg = (γc + δ)g, leads to a generic solution via parametric formulae (viewing s as a parameter) c = w 1 (s) w 2 (s) and g = w 2 (s). We refer to [4,26] for a review of these classical formulae.
Remark 3.1. Since the action of the group SL(2, Z) permutes the Jacobi theta series with characteristics in {0, 1} and the equation (3.7) admits a SL(2, R)-symmetry, another solution to the equation (3.7) can be chosen as g(c) = ϑ 2 10 (ic, 0). For τ ∈ H 1 , let f (τ ) = ϑ 2 10 (τ, 0). It is a well-known fact that the function f is modular form of weight 1 on Γ 0 (2) with a character. A direct computation shows that the logarithmic derivative of f transforms like a quasi-modular form of weight 2 on Γ 0 (2), note the cancellation of the character, and we have The latter formula connects the two descriptions presented in this section.

Hirota master-equation via the Odesskii-Sokolov construction:
Proof of theorem 1.1 It was proved in [35] that a generic integrable Hirota type equation (1.1) can be parametrised by generalised hypergeometric functions. Here we briefly summarise the construction. Consider the generalised hypergeometric system of Appell's type, Here s 1 , ..., s n+2 are arbitrary constants, σ = 1 + s 1 + · · · + s n+2 , and h is a function of n variables v 1 , . . . , v n . This system is involutive and has n + 1 linearly independent solutions known as generalised hypergeometric functions [21,35]. Introducing the differential solutions to (4.1) can be represented as the corresponding periods, q p ω where p, q ∈ {0, 1, ∞, v i }. This statement can be explicitly found in Mostow [32] who also noted that only n + 1 of these periods are linearly independent [33], see also [8,25]. In low dimensions, analogous observations were made by Schwarz in 1873 and Picard in 1883 [38]. With any generalised hypergeometric system (4.1) Odesskii and Sokolov associated a dispersionless integrable system in 3D having a dispersionless Lax representation [35].
We will need a particular case of the general construction where n = 5 and the constants have specific values s 1 = s 2 = s 3 = s 6 = s 7 = − 1 2 , s 4 = s 5 = 1 2 , see Example 5 of [35]. The first observation of [35] is that one can choose a basis of solutions h 1 , h 2 , h 3 , g 1 , g 2 , g 3 of the corresponding system (4.1) such that where we use the notation With these data Odesskii and Sokolov associated a dispersionless Hirota type equation (1.1) for an auxiliary function u(x 1 , x 2 , x 3 ) represented parametrically as The required equation results on the elimination of the 5 parameters v 1 , . . . , v 5 from the 6 relations (4.3). This equation was shown to be integrable via a dispersionless Lax representation. The geometry behind this construction is as follows. With the choice of constants s i specified above the differential ω takes the form which is a holomorphic differential 1-form on the genus 3 hyperelliptic curve C Choosing a system of cycles a i , b j on the curve C with the intersection matrix a i · b j = δ ij we denote by h 1 , h 2 , h 3 , g 1 , g 2 , g 3 the corresponding periods of ω which form a basis of solutions of the associated system (4.1). We will see that this basis automatically satisfies constraints (4.2) due to the Riemann relations. The basis of holomorphic differentials on C is given by Their periods over a and b cycles are given by the matrix where τ is the period matrix of the curve C. Explicitly we have

Multiplying this matrix by the inverse of
.

(4.4)
The symmetry of the period matrix τ is equivalent to constraints (4.2). Bases satisfying (4.2) are thus in 1-to-1 correspondence with canonical systems of cycles on C. Finally, relations (4.3) take the form τ ij = u x i x j . The elimination of the parameters v i yields a Hirota type equation for u which is the equation of the genus 3 hyperelliptic divisor, where ϑ m is a theta constant with an even characteristic m, the result that goes back to Schottky, 1880, see e.g. [39]. Although there are 36 even theta constants, the corresponding equations are equivalent due to the transitivity of the Sp(6, Z) action on the set of even theta constants. Note that if all parameters (branch points) v i of the hyperelliptic curve are real we can choose a basis of cycles such that the corresponding period matrix τ will be purely imaginary. Then u will also be purely imaginary, and the Hirota equation will be real. The uniqueness (over C) of the Hirota master-equation can be established as follows. Up to a suitable equivalence transformation, every equation (1.1) can be brought to a form where ∂F ∂ux 1 x 1 = 0. Thus, without any loss of generality, we can work with the evolutionary equation (2.2). The integrability conditions derived in [14] express all third-order partial derivatives of f as explicit rational functions of its first-order and second-order partial derivatives, symbolically, (4.5) Equations (4.5) define a rational connected 21-dimensional affine variety X in the affine space of third-order jets of f . The algebraic equivalence group Sp(6, C) (of dimension 21) acts on X in a locally free way [14], and therefore possesses a Zariski open orbit. The uniqueness of this orbit follows from the fact that any two Zariski open sets on a connected variety must necessarily intersect. Dispersionless Lax representation of the Hirota master-equation is given in parametric form (2.8), for i = 1, 2, 3 (4.6) where R(g i , t) are polynomials of degree 2 in t: see [35] for more details. Expressing the variables v 1 , . . . , v 5 in terms of the second derivatives u x i x j via formulae (4.3) one can rewrite the right hand sides of (4.6) in terms of u x i x j . Since integrands in (4.6) are holomorphic differentials on the hyperelliptic curve C, formula (4.6) is the corresponding Abel map. Equations (4.6) are naturally embedded in the universal Whitham hierarchy, see Theorem 7.8 of [29].

Concluding remarks
The attempts to generalise/specialise Theorem 1.1 lead to the following observations: • Compactification of the hyperelliptic divisor in genus 3 [19] is related to the classification of special integrable Hirota type equations (1.1) that have continuous symmetries from the equivalence group Sp(6, R). Such degenerations can be most interesting from the point of view of their potential applications. • The construction of Odesskii and Sokolov [35] can be adapted to show that for any higher genus g ≥ 4 the hyperelliptic locus defines a 3D integrable hierarchy of Hirotatype equations (in the notation of [35] this corresponds to the choice of constants s 1 = · · · = s g = s 2g = s 2g+1 = 1 2 , s g+1 = s 2g−1 = − 1 2 , n = 2g − 1, k = g). Here the equations of the hierarchy coincide with the defining equations of the hyperelliptic locus [34,39,40,23], which is known to be characterised by the vanishing of 1 2 (g − 1)(g − 2) theta constants (Hirota equations result on the substitution of τ = iHess(u) for the g ×g period matrix).
• Although it would be tempting to conjecture that for g=4 the Shottky divisor (image of the Torelli map M g ֒→ A g from the moduli space of curves of genus 4 to the moduli space of principally polarised Abelian varieties [9]) corresponds to an integrable Hirota type equation in 4D, this is no longer the case. Recent results of [16] show that in 4D the requirement of integrablity implies the symplectic Monge-Ampère property, which leads to a complete list of integrable heavenly-type equations classified in [10]. Thus, the occurrence of modular forms and theta functions in the classification of integrable Hirota type equations is the essentially 3-dimensional phenomenon. • Finally, if would be interesting to obtain a purely computational proof of Theorem 1.1 by directly demonstrating that even theta constants satisfy the integrability conditions discussed in Section 2.1.