Genus 4 trigonal reduction of the Benney equations

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1 Genus 4 trigonal reduction of the Benney equations Sadie Baldwin and John Gibbons Imperial College 8 Queen s Gate London SW7 BZ sadie.baldwin@imperial.ac.uk, j.gibbons@imperial.ac.uk Abstract. It was shown by Gibbons and Tsarev 996 Phys. Lett. A 9, 999 Phys. Lett. A that N-parameter reductions of the Benney equations correspond to particular N-parameter families of conformal maps. In recent papers J. Phys. A: Math. Gen. 36 No 3 8 August , J. Phys. A: Math. Gen. 37 No May , the present authors have constructed examples of such reductions where the mappings take the upper half p-plane to a polygonal slit domain in the λ-plane. In those cases the mapping function was expressed in terms of the derivatives of Kleinian σ functions of hyperelliptic curves, restricted to the -dimensional stratum Θ of the Θ-divisor. This was done using an extension of the method given in Enolskii et al 3 J. Nonlinear Sci extended to a genus 3 curve V Z Enolski and J Gibbons, Addition theorems on the strata of the theta divisor of genus three hyperelliptic curves, in preparation. Here, we use similar ideas, but now applied to a trigonal curve of genus 4. Fundamental to this approach is a family of differential relations which σ satisfies on the divisor. Again, it is shown that the mapping function is expressible in terms of quotients of derivatives of σ on the divisor Θ. One significant by-product is an expansion of the leading terms of the Taylor series of σ for the given family of 3, 5 curves; to the best of the authors knowledge, this is new. PACS numbers:.3.ik,.3.jr,.3.zz. Introduction.. Reductions of the Benney Moment Equations The Benney equations [] are an example of a system of hydrodynamic type with infinitely many degrees of freedom. These can be written as a Vlasov equation [], [3], f p f t x A f x p =. Here f = fx, p, t is a distribution function. The moments A n are defined by A n = p n f dp.

2 Genus 4 trigonal reduction of the Benney equations Following [4], we let λ R x, p, t, be given by the integral λ R = p P fx, p, t p p dp 3 where P denotes the principal value. Comparing the first derivatives of λ R x, p, t, we obtain the PDE λ R p λ R t x = λ R p p t p p x A. 4 x If we hold λ R constant in 4, then this gives the conservation equation p t x p A =. 5 Alternatively, if we now hold p constant in 4, we obtain λ R t p λ R x A x λ R p = 6 which is a Vlasov equation of the same form as. Thus and 6 have the same characteristics. Any function of λ R and f must satisfy the same equation. Suppose further that for some point p = ˆp i x, t, λ R ˆp i = ˆλ i x, t we have: λ R p =, p=ˆpi then substituting into 4 gives λ R t = ˆλ i p=ˆpi t and λ R x = ˆλ i p=ˆpi x ˆλ i t ˆp ˆλ i i x =. We say that ˆλ i is a Riemann invariant with characteristic speed ˆp i. We are interested in the case where the function λ R p, x, t is such that only N of the moments A n are independent. Then it was shown in [5] that there are N characteristic speeds, assumed real and distinct, and N corresponding Riemann invariants ˆp i, ˆλ i. Then Benney s equations reduce to a diagonal system of hydrodynamic type with finitely many, N, dependent variables ˆλ i, satisfying: ˆλ i t ˆp iˆλ ˆλ i = i =,,..., N. 7 x Such a system is called a reduction of Benney s equations. The construction of a general family of solutions for equations of this type was outlined in [5] and [6]. Instead of considering the principal value integral 3, we now define a new function λ x, p, t : fx, p, t λ x, p, t = p dp 8 p p Υ

3 Genus 4 trigonal reduction of the Benney equations 3 where Υ is an indented contour passing below the point p. This function has the same asymptotics as λ R x, p, t, provided all the moments A n exist, and it can be analytically continued throughout the upper half p-plane, provided that f is Hölder continuous. We now suppose that the relation f = F λ R holds in some region of the x, p- plane at some time t, and that f = outside this region. Then since both and 6 have the same characteristics, the relation will be preserved by the dynamics. In this case the definition for λ 8 becomes a nonlinear singular integral equation: λ x, p, t = p Υ F λ R x, p, t p p dp. 9 Some solutions to 9 can be described in terms of a conformal mapping of a slit domain. We take the upper half λ-plane, Γ, and draw a Jordan arc c in Γ starting from a point, λ, on the real axis. We then fix an arbitrary point on this arc, ˆλ, and make a slit γ running along the arc from λ to ˆλ. ˆλ c γ Γ λ Figure. The slit γ on the Jordan arc c = γ c. Note that the slit γ in figure is given by the relation Imλ = πf Re λ and so, for consistency, F must be continuous with F. The function p λ, ˆλ is then determined uniquely by the following properties. i p λ, ˆλ has a branch point at ˆλ, that is p ˆp cλ ˆλ Oλ ˆλ. ii p λ, ˆλ is real on the real λ -axis and on both sides of γ. iii p λ, ˆλ is analytic in the cut half plane Γ. iv As λ, with Imλ, p λ, ˆλ has the expansion p λ, ˆλ λ O. The evolution of p is then given by 5; expanding near ˆλ gives: ˆλ t ˆp ˆλ x =. λ

4 Genus 4 trigonal reduction of the Benney equations 4 Thus ˆλ is a Riemann invariant with characteristic speed ˆp = p ˆλ. It is possible to generalize this construction to N non-intersecting slits. Here, each of the slits γ i is made along a fixed path starting on the real λ -axis and ending in a branch point ˆλ i. Again, ˆλ i are the Riemann invariants of the system with associated characteristic speeds pˆλ i and the slits γ i are given by Imλ = πf i Re λ where F i are continuous functions. In the particular case that the slits are all straight line segments, making angles with the real λ axis which are rational multiples of π, the usual Schwartz-Christoffel construction gives a mapping function of the form: λ = p p [φp ]dp, where φp is some algebraic function. In this case it is natural to consider this expression as an integral of a second-kind differential on the corresponding algebraic curve. This approach was used in [7], [8] and [9], where the slits were all at right angles to the real axis, and the corresponding curves were then elliptic or hyperelliptic. The question thus arises whether a similar approach is equally useful for a curve which is not of this type; here we look at a particular example, where the underlying family of curves are trigonal. While the resulting formula 3 is clearly highly transcendental, it is remarkable that all known examples of such explicit representations of Schwarz-Christoffel slit mappings may be written as rational functions of derivatives of σ-functions for the corresponding algebraic curve. The principal advantage of such an approach is that the original Schwartz-Christoffel integral depends on many parameters, which must satisfy integral constraints. In the present trigonal case for example, it depends on parameters, satisfying 6 constraints - it is thus very hard to use the integral representation to calculate the properties of the reduced system, without evaluating the integral anyway. Calculating the Hamiltonian structure, for example, in terms of the σ function representation may well be more tractable; work on this is continuing. Some recent closely related work on expressing the analogous hyperelliptic mappings in terms of automorphic functions, by Crowdy [], [] suggests further generalisations may be possible. In that representation, the constraints are satisfied automatically, and the mapping no longer contains spurious parameters, depending only upon the dynamical variables. In a series of papers by Wiegmann, Krichever, Mineev-Weinstein, Zabrodin and co-workers, see, e.g. [],[3], a related problem, in a sense inverse to this one, is addressed. There, families of conformal maps are constructed in terms of the solutions of dispersionless integrable hierarchies, and these are further related to the solutions of random matrix models. The detailed connections between that work and this still remain to be clarified. It is clear, however, that the topics of conformal mappings and of dispersionless integrable hierarchies are intimately connected.

5 Genus 4 trigonal reduction of the Benney equations 5. A Trigonal Reduction To motivate the calculations which follow, we consider reductions where the slits are straight line segments making angles of π/3 or π/3 with the real axis, leading to a trigonal curve. There is one elementary example with this slit geometry, leading to the dispersionless Boussinesq hierarchy [4]. Here the mapping is λ = p 3 3A p 3A /3 = p P p P p P 3 /3, and the two slits have fixed base point at the origin, which is the image of the three points {P, P, P 3 }. See figures and 3. We should point out that although the mapping itself is written in elementary functions, the curve is non-trivial, having genus. P Figure. The p-plane. P ˆp P ˆp P 3 ˆλ ˆλ L ˆλ = Figure 3. The λ-plane associated with figure. Let us now consider a similar reduction, but instead with pairs of slits, as shown in Figure4 where the conformal mapping λ : P L is constructed as follows. We define P to be the upper half p-plane with points marked on the real axis. These satisfy P < ˆp < P < ˆp < P 3 < P 4 < ˆp 3 < P 5 < ˆp 4 < P 6. The domain L is the upper half λ-plane with pairs of slits on it, as in figure 5. The first pair of slits radiate at 6 degree angles from the fixed real point λ ; The end points of these slits move along the radial lines and are labelled ˆλ and ˆλ. A second pair of slits is arranged similarly, radiating at 6 degree angles from the fixed real point λ 4. Here, the variable end points are labelled ˆλ 3 and ˆλ 4. As in the hyperelliptic cases, the

6 Genus 4 trigonal reduction of the Benney equations 6 point ˆλ i is the Riemann invariant associated with the characteristic speed ˆp i. By setting λp i = λ i and imposing the conditions λ = λ = λ 3 = λ, λ 4 = λ 5 = λ 6 = λ 4, it follows that L is a slit domain of the form shown in figure 5. P P ˆp P ˆp P 3 P 4 ˆp 3 P 5 ˆp 4 P 6 Figure 4. The p-plane. λˆp λp λˆp λˆp 3 λp 5 λˆp 4 L λp λp 3 λp 4 λp 6 λ λ 4 Figure 5. The λ-plane associated with figure 4. where The mapping λ : P L can then be given in the Schwartz-Christoffel form: where λp = p ϕp = p ϕp dp. 3 4 p ˆp i [ 6 p P i ] /3 = 4 p ˆp i y, 4 6 y 3 = p P i. 5 From the conditions, we see that once the base points λ and λ 4 have been fixed, the mapping λp is a function of 4 independent real parameters. As in the hyperelliptic case, it is natural to take these to be the variable imaginary parts of the slit ends: Iλˆp i, i =,, 3, 4. From the construction of the conformal mappings, we also have the property lim p ϕp O p.

7 Genus 4 trigonal reduction of the Benney equations 7 This provides a relation between the characteristic speeds ˆp j and the fixed points P i. We have lim ϕp = 6 4 P i ˆp i p 3 p O p and so 4 ˆp i = 3 6 P i. 6 Following the process used in the hyperelliptic cases, we now define the Riemann surface Γ: { 6 } Γ = p, y C : y 3 = p P i 7 We will then be able to use the properties of this surface to evaluate the integral λp. This is a 3, 6 curve and so relates each point p, except the branch points P i, to three values in the complex plane and so the Riemann surface for 5 consists of three sheets, triply branched at the points P i. For all p in the finite plane, other than the branch points, each branch of the function yp is finite and so the curve is regular here, and p is a good local parameter at such points. However, if we evaluate y along a contour encircling the point P i, the values at the end points differ by a factor of ω = exp iπ/3. Hence, the P i are regular branch points of order 3. The local co-ordinates at the branch points are ξ = p P i /3 i =,..., 6. In the neighbourhood of P i, y is an analytic function of the corresponding local coordinate ξ. We may describe the Riemann surface more precisely, and label the different sheets, by noting that: y p exp πi k as p, 3 where k =, or 3. The different sheets are joined along the real intervals the cuts [P, P ], [P, P 3 ], [P 4, P 5 ] and [P 5, P 6 ]. Specifically, as p passes from a point on the upper side of [P, P 3 ] or [P 5, P 6 ] to the lower side, y moves from sheet k to sheet k mod 3, and as p passes from a point on the upper side of [P, P ] or [P 4, P 5 ] to the lower side, y moves from sheet k to sheet k mod 3. The branch cuts, and the connections between the different sheets, are shown in figure 6. The k-th sheet is completed by adding a point at infinity, denoted k, where a good local co-ordinate is ξ = /p. Expanding yp in terms of this local co-ordinate gives yp expπik /3 ξ 3 6 P i ξ O

8 Genus 4 trigonal reduction of the Benney equations 8 and so at each of the 3 points at infinity the function y has poles of order. Definition. Any Riemann surface R given by y n = Q m x where n is an integer and Q m is a polynomial of order m, is called a cyclic n, m Riemann surface. Since all the n sheets have common branch points, at the zeroes of Q m x, and all branch points are ramified in the same way, these curves are much simpler than more general examples. Thus in our example, the curve Γ: 6 y 3 = p P i is a cyclic 3, 6 curve. R P ˆp P ˆp P 3 P 4 ˆp 3 P 5 ˆp 4 P 6 Figure 6. The cyclic trigonal Riemann surface Γ. The bold lines are cuts on the surface. The solid curve is on sheet, the dashed curve on sheet and the dotted curve on sheet 3... Properties of the cyclic trigonal Riemann surface We now investigate some of the properties of the Riemann surface Γ. Some key results for trigonal Riemann surfaces have been found by Eilbeck, Enolski and Leykin, [5], by Buchstaber, Enolski and Leykin, [6], which consider 3, 4 surfaces in detail, and recently by Ônishi [7], who finds formulae holding on cyclic 3, 4 surfaces, while Matsumoto [8] has looked at trigonal curves with 6 branch points, as in our case. Our approach follows the method of [5] and [6] closely. First it is necessary to calculate the genus of the curve, and to define a basis of a and b cycles. From the Riemann-Hurwitz theorem, the genus of a cyclic n, l curve is given by g = n ln giving in this case, with n = 3, l = 6, g = 3 6./ = 4.

9 Genus 4 trigonal reduction of the Benney equations 9 We can thus define a basis of cycles on R consisting of four a and four b cycles. These must have intersection index given by: a i a j =, a i b j = δ ij, b i b j =, where δ ij is the Kronecker delta. A suitable set of cycles for this first homology basis, H R, Z = {a, a, a 3, a 4 ; b, b, b 3, b 4 }, is shown in [8]. To identify a basis of holomorphic differentials on Γ we need to calculate the Weierstrass gap sequence [9] for the curve. This process is simplified if the orders of y and p are co-prime, that is, if the curve is in canonical form. To achieve this we transform the curve by sending one of the branch points, P 6, to infinity, using the invertible rational map: p = P 6 t, 8 P i = P 6 T i i =... 5, 9 s = yt K, 5 K 3 = P 6 P i. For the curve Γ this canonical form is then given by 5 s 3 = t T i t= = λ λ t λ t λ 3 t 3 λ 4 t 4 t 5. We will call the Riemann surface for this cyclic 3, 5 curve T 4. This surface is made from three sheets of the complex plane. It has branch points of order 3 at T,..., T 5, T 6 = and so there is just one infinite point. The local co-ordinate near t = is then t = /ξ 3. It follows that each sheet of T 4 has branch cuts along the closed intervals [T, T ], [T, T 3 ], [T 4, T 5 ], [T 5, ] and is regular elsewhere. We note that if P i < P 6 for i 5, then T i > for i 5. The three sheets are then connected in the same way as the Riemann surface Γ, replacing P i by T i for i =,..., 5 and P 6 by. The Weierstrass non-gap sequence for this cyclic 3, 5 curve is the set of all positive integers expressible as sums 3 α i 5 β i = W NG i 3 for non-negative integers α i, β i. These numbers are: {, 3, 5, 6, 8,...}. We use the notation λ i for the moduli of the curve, following e.g. [5]. We should emphasise that these bear no direct relation to the function λp used above.

10 Genus 4 trigonal reduction of the Benney equations Only the first g terms of this are of interest here; all integers g are trivially members of the sequence. The complement of this set is the Weierstrass gap sequence; here it is given by W G = {γ 4, γ 3, γ, γ } = {,, 4, 7}. Following [5], we can now define a set of holomorphic differentials on T 4 by du T t, s = U T dt f y = {U, U, U 3, U 4 } dt 3s, where U i = t α i s β i and the exponents α i, β i are as above. The analyticity of these differentials follows from direct expansion in terms of the local parameters at branch points. Solving equation 3 for α i, β i Z, as shown in table, we see du T =, t, s, t dt 3s. 4 U i W NG i α i β i U U 3 U 3 5 U 4 6 Table. A list of positive integers P and Q satisfying 3 α i 5 β i = W NG i where W NG i is the ith Weierstrass non-gap number. These differentials may be re-expressed in terms of p and y, to construct a set of holomorphic differentials on the original curve, but instead we will work with the canonical form of the curve, and transform the integral 3 into the variables t and s. The integrand ϕp dp 4 becomes ϕp dp = 4 [P 6 ˆp i t ] { 6 [P 6 P i t ] } /3 = 3K = 3K t A t dt t A t A t A 3 t 3 A 4 t 4 [t 5 λ 4 t 4 λ 3 t 3 λ t λ t λ ] /3 dt 3t A A 3 t A 4 t dt 3s 5 where A i are constants, and K is defined as above. We note that we can write the constant A in terms of the curve moduli, λ i, as follows. Evaluating 5 explicitly we find 4 A = P 6 ˆp i

11 Genus 4 trigonal reduction of the Benney equations and λ = 5 3 K3/, λ = P 6 P i. 6 K 3/ If we now use identity 6, which relates the branch points P i to the ˆp j, we see 4 A = 4 P 6 ˆp i = 4 P 6 6 P i 3 = 3 5 P 6 5 P i = 3 5 P 6 P i = λ. 3 λ It follows that ϕt dt = k A A 3 t A 4 t dt 3s k t λ dt 3 λ t 3s, 7 where the first term is a sum of holomorphic differentials and the second term is a second kind differential. To identify functions on the surface T 4, we first define the period matrices: ω ij = du i, ω ij = du i 8 a j b j where ω and ω are 4 4 matrices. The lattice of points generated by these periods is given by Λ = { m ω n ω : m, n Z 4}. 9 We define Abelian functions on C 4 as meromorphic functions which are invariant under translations by this period lattice Λ; that is, they satisfy fp n ω m ω = fp for n, m Z. We now define the Jacobian of T 4 by JacT 4 = C 4 /Λ. As in the hyperelliptic cases, we can map T 4 into JacT 4 using the Abel map. For any point t and base point t a T 4 this is given by t At = dut mod Λ 3 t a = ut 3 The map At forms a dimensional image of the T 4, a subset of the 4 dimensional JacT 4. We denote this one dimensional stratum of JacT 4 by { t } Θ = u : u = du mod Λ. t a Henceforth we will always choose the base point t a =. Since λt is given by a single integral with respect to one parameter, a point t, s T 4, it makes sense to rewrite the integral 5 as an integral on the one-dimensional stratum Θ of JacT 4. Thus, we need to understand how meromorphic functions on T 4 correspond to the restrictions of Abelian functions to this subspace of the Jacobi variety. Similar such problems of inverting meromorphic differentials on lower-dimensional strata of the Jacobi variety of a curve have been studied for example, by Alber and Fedorov, [], and Enolski, Pronine and Richter, [].

12 Genus 4 trigonal reduction of the Benney equations 3. Abelian differentials and the sigma function It is possible to construct the correspondence between meromorphic functions on T 4 and the restrictions of Abelian functions on JacT 4 to Θ, using the Kleinian σ function. Key to this construction was the definition of the associated second kind differentials and the set of normalized holomorphic differentials. Thus we will begin by constructing a set of associated second-kind differentials on the Riemann surface T 4 and then recall the main properties of the normalized differentials. 3.. Differentials We recall that T 4 has the set of holomorphic differentials 4: du T = U T dt 3 s = U, U, U 3, U 4 dt 3 s =, t, s, t dt 3 s. To evaluate a set of associated second kind differentials dr T = R, R, R 3, R 4 dt 3 s. we use the procedure described in [5] and [6]. Klein s fundamental second kind -form, dωt, z on T 4 is defined as the unique -form, depending symmetrically on two distinct points t, s and z, w on T 4 : which satisfies: s 3 = t 5 λ 4 t 4 λ 3 t 3 λ t λ t λ, w 3 = z 5 λ 4 z 4 λ 3 z 3 λ z λ z λ t, s z, w, dωt, z t z Odtdz with no singularities except on the diagonal t, s = z, w. It may be constructed by setting dωt, z = d Ψ T z, w Φt, s dz dt dz t z 3 s RT z, w Ut, s dz dt 3 w 3 s and where Ψ T z, w =, w, w, Φ T t, s = s, s,. To identify the unknown polynomials R T t, s = R, R, R 3, R 4, we impose the symmetry condition dωt, z dωz, t =.

13 Genus 4 trigonal reduction of the Benney equations 3 We find: s s w w dz dt dωt, z = d dz If we multiply this by and then set and 3s dt 3w dz t z 3 s R z, w t R z, w s R 3 z, w t R 4 z, w dz dt [ ] 3 w 3 s swz ww z = s sw w dz dt t z t z 3 s R z, w t R z, w s R 3 z, w t R 4 z, w dz dt 3 w 3 s. Q t, z = 3w s w z 6 w 3 w z t z 3w s 3w 3 s 3 w 4 t z Q t, z = R z, w t R z, w s R 3 z, w t R 4 z, w, then the symmetry condition is equivalent to Q t, z Q z, t = Q z, t Q t, z. 3 We simplify the left hand side of 3 using and This gives w 3 = z 5 λ 4 z 4 λ 3 z 3 λ z λ z λ, 3 w w z = d dz Q t, z = w 3 = 5 z 4 4 λ 4 z 3 3 λ 3 z λ z λ. [ s d w 3 w d ] w 3 dz dz t z [ 3 w s 3 sww 3 3 w w 3 ] t z = s w t z 5 z 4 4 λ 4 z 3 3 λ 3 z λ z λ 3 w s 3 s w t z z 5 λ 4 z 4 λ 3 z 3 λ z λ z λ, t z with Q z, t evaluated in a similar way. If we now expand the expression Q t, z Q z, t and then rearrange, we obtain Q t, z Q z, t = 4 t z 3 t λ 3 tz λ 4 λ tz λ 5 z 3 λ 5 z λ 4 5 t λ 4 7 t 3 λ 5 s 33 4 z t 3 z λ 3 tz λ 4 λ tz λ 5 t 3 λ 5 t λ 4 5 z λ 4 7 z 3 λ 5 w

14 Genus 4 trigonal reduction of the Benney equations 4 Recalling that the right hand side of 3 is Q z, t Q t, z = R t, s z R t, s w R 3 t, s z R 4 t, s R z, w t R z, w s R 3 z, w t R 4 z, w, we can now evaluate the polynomials R i t, s by matching coefficients of s and w. We note that the R i t, s are not defined uniquely, but one such set is given by R t, s = s t 3 λ 3 7 t 5 t λ 4, R t, s = s t t λ 4, R 3 t, s = t 3 t λ 4 λ, R 4 t, s = s t. 34 The second kind differentials dr associated to the set of first kind differentials du are then given by dr T = R, R, R 3, R 4 dt 3s, we can now define the corresponding two 4 4 period matrices η and η : η ij = dr i, η ij = dr i, i, j =,..., g. 35 a j b j By construction, and the use of Riemann s bilinear identity, these period matrices η, η and ω, ω must satisfy the generalized Legendre relation; if the g g matrix M is defined by then: M = M ω ω η η g g and so M belongs, up to a factor of [], p. 37]. 3.. The σ function, M T = iπ g g iπ, to the Symplectic group Sp8, C see, e.g. If we introduce the normalized holomorphic differentials dv on T 4 by setting a i dv j = δ i,j, i, j =,..., 4, then their periods around the b cycles are given by b i dv j = ω ω i,j = τ i,j, i, j =,..., 4, where, as usual, the matrix τ must be symmetric, with positive definite imaginary part. Following [9], we can now define the Kleinian σ function on T 4 :

15 Genus 4 trigonal reduction of the Benney equations 5 Definition 3. Let t a be any regular point on the Riemann surface T 4, and let {t,..., t 4 } T 4 4 ; the Abel map of the divisor t t t 3 t 4 is defined by: 4 ti u = du. t a Then the fundamental Abelian σ-function on C 4, the covering space of JacT 4, is given by σu; M = π exp 4 Dv detω where Dv is the discriminant of the curve T 4 Dv = T i T j, i<j 5 ut η ω u m Z 4 expiπm T τm m T ω u Δ ta and Δ ta is the Riemann constant with base point t a ; if we fix t a =, and choose the homology basis as in [8] then the corresponding Riemann constant Δ was shown there to be Δ =,,, T. The fundamental properties of σ are: it is an entire function on C 4, the covering space of JacT 4, it is quasi-periodic: σu ωk ω k ; M = exp{ηk η k T u ωk ω k } σu; M 36 it is invariant under changes in the basis of cycles - it is a modular invariant: σu; γm = σu; M, γ Spg, Z 37 the first term of the of the σ-series is the Schur-Weierstrass polynomial which is defined as follows. If e k is the elementary symmetric function of weight k with respect to the variables z,..., z g, then the determinant dete gj k j,k=,...,g, can necessarily be expressed as a polynomial in terms of Newton polynomials p k = z k... zg k, k =,..., g. The substitution p k = u k, k =,..., g defines the required Schur-Weierstrass polynomial in JacT 4 ; for the curve T 4 this polynomial has weight 8 in the Sato-Weierstrass grading, where we assign weights: u = 7, u = 4, u 3 =, u 4 =. the higher order terms in the Taylor expansion of σ with respect to u, u, u 3, u 4 are also all isobaric polynomials of weight 8 in these variables and the curve moduli λ, λ, λ, λ 3, λ 4, where the weights of the moduli are assigned as follows: λ = 5, λ =, λ = 9, λ 3 = 6, λ 4 = 3.

16 Genus 4 trigonal reduction of the Benney equations 6 This is used to define a higher genus analogue of the ζ function and Weierstrass function. We have, analogously to the elliptic case, and where we denote: ζ i u = [log σu] = σ i u, i =,..., 4 u i σ ij u = [log σu] = σ ij u i u j σ u σ i σ j u, i, j =,..., 4 σ σ i = σ u i, σ ij = σ u j u i,. Higher order logarithmic derivatives are written, for example, 3 ijk u = [log σu], i, j, k =,..., 4. u i u j u k The periodicity properties of these functions are as follows: i ζ i u ωm ω m = ζ i u ηm η m i i, j =,..., 4. where the subscript i on the last term denotes the ith component of this vector; ii ij u ωm ω m = ij u i, j =,..., 4. Thus the Kleinian ij functions and all their derivatives are Abelian functions on JacT Genus 4 Trigonal curve: Jacobi s inversion theorem and some relations between the ijkl We begin by rewriting Klein s theorem Thm. 3.4 in [5], for the case of the curve T 4. Theorem 3. For arbitrary distinct t, s, and base point t a, st a on T 4 and an arbitrary set of g = 4 distinct points {t, s,..., t 4, s 4 } T 4 it follows that 4 t 4 tk i,j du du U i t, s U j t r, s r = F t, s; t r, s r r =,..., 438 t a t t r i,j= k= t a U T t, s =, t, s, t and F is the symmetric function F t, s; t r, s r = 3s rs [ t 3 r t t 4 rt 3μ μ t r t μ t r tt r μ 3 3t rt μ 4 t 3 rt t t r ] s [ t 3 t r t 4 t r 3μ μ t t r μ t tt r μ 3 3t t r μ 4 t 3 t r t t r ] s r appearing in the numerator of the second kind fundamental -form: F t, s; t r, s r dt dt r t t r 3s 3s r = dωt, s; t r, s r

17 Genus 4 trigonal reduction of the Benney equations 7 This result allows us to write down the Jacobi inversion formula on T 4 explicitly, and also to find some PDE satisfied by the derivatives on Θ 4. To do this we follow the procedure outlined in [5] for a 3, 4 curve. We fix the base point t a of the Abel map at infinity. We then let t. Expanding equation 38 in the local co-ordinate t = /ξ 3 gives the following Taylor series expansion: and RHS = 3 LHS = t r s r 44 3 t r s r λ 4 t r 3 s r t 3 r s r 4 t r s r 3 s r [ [ ξ ξ O ξ s r 3 44 t r 3 44 t r s r t r s r s r ] ξ 3 34 t r s r ] ξ O ξ 3 If we multiply both sides by 3 s r and subtract the right hand side from the left, then the coefficient of ξ i, C i, is: C = 4 4 t r 34 s r 44 t r t r s r ; 39 C = t r s r λ 4 t r t 3 r; 4 C = [ t r s r t r]. 4 It follows that C i must be zero for any u Θ 4 and some t r, s r T 4. In the hyperelliptic case, the first term, analogous to C here, defines the inversion equation - there it is a polynomial of order g in the unknown t r. Here, however, equation 39, however, contains both t r and s r and so we simplify this further by eliminating s r. Evaluating the resultant of C and C with respect to s r gives the quartic D = t 4 r λ t 3 r 34 λ t r t r This forms the key equation in Jacobi s inversion theorem for the curve T 4. For each of the four roots {t,... t 4 } of this equation, the corresponding point s i can be found from C =. The g = 4 points t i, s i T 4 form the Abel preimage of u. If we similarly eliminate s r from the pair of equations C and C then we obtain D = t 4 r

18 Genus 4 trigonal reduction of the Benney equations t 3 r t r t r , another quartic in t r which must have the same roots {t i } as D,. Multiplying D by and subtracting this from D,, we have a cubic equation in z. Since this must be satisfied by four distinct t i T 4 the coefficients in this cubic must equal zero. This allows us to rewrite the highest order derivative in each coefficient in terms of lower order derivatives. For example, from this pair of equations, D, and D,, we obtain an expression for 4444 from the coefficient of z 3, 3444 from z, 444 from z and 444 from the constant term. Simplifying these, we find, for example, = By looking at higher order terms of ξ in the expansion of theorem 3. and eliminating s r and t r as shown above, we may obtain relations for more of the derivatives. It is not yet clear how a fundamental set of such relations might be constructed. Theorem 3. Jacobi inversion for genus 4 Trigonal curve [[9], p 3] Let T 4 be the genus 4 cyclic 3, 5 curve defined by 4 s 3 = t 5 λ i t i, let u i = 4 k= i= tk du i, where t t t 3 t 4 is a non-special divisor and du is the vector of holomorphic differentials. The Abel preimage of the point u T 4 is then given by the set {t, s,..., t 4, s 4 } T 4 4, where {t,..., t 4 } are the zeros of the polynomial Pt; u = t 4 λ t 3 34 λ t

19 Genus 4 trigonal reduction of the Benney equations t , and the pairs {t r, s r } 4 each satisfy: Qt r, s r ; u =, where Qt r, s r ; u = 4 4 t r 34 s r 44 t r t r s r Strata of the Jacobian and the inversion theorem on Θ Consider T 4 k, the k-fold symmetric product of T 4, containing divisors of the form k D k = t i, s i and define the Abel map of such a divisor with base point : k ti u = ut,..., t k = du modλ. If we set Θ k = { u : u = k ti du then evidently we have the stratification modλ JacT 4 = Θ 4 Θ 3 Θ Θ Θ =. }, k 4 43 We may let a point in Θ k descend towards Θ k by allowing t k, s k to tend to. From the Jacobi inversion theorem, we know that one root of P must tend to infinity as u descends to Θ 3, implying that σ is zero there, so we can therefore define Θ 3 equivalently by Θ 3 = {u : σu = }. 44 In principle this approach could be used to descend successively to lower strata, as was done in the hyperelliptic case, but this approach requires detailed knowledge of the partial differential equations satisfied by the ij. Instead, we use a theorem from a paper of Jorgenson [3] but see also Fay [4], p.3 for a closely related result to identify an alternative expression for Θ more directly. This result is the following: Let k t k be a divisor of degree k < g on C g and define its Abel map in the usual way: k tk u = du. t a

20 Genus 4 trigonal reduction of the Benney equations Then the following equation holds: g j= σ j u a j g j= σ = det [ a dut dut k dut dut g k ] j u b j det [b dut dut k dut dut g k ] where dut i denotes the column of i-th derivatives of the holomorphic differentials dut. For the genus 4 trigonal curve T 4 the set of holomorphic differentials is given by 4 du T =, t, s, t dt 3s and so we can construct the strata Θ k successively as follows. We have already noted that on Θ 3, σu =. In that case 45 reduces to: 4 j= σ j u a j 4 j= σ j u b j = det [a dut dut dut 3 ] det [b dut dut dut 3 ]. 46 Now as u in Θ 3 approaches Θ, t 3. We can therefore express the fourth column of both determinants in terms of du i for t 3 near infinity. The local co-ordinate is t 3 = /ξ 3 and so substituting this into 4 we find du dξ = ξ6 3 λ 4 ξ 9 Oξ 47 du dξ = ξ3 3 λ 4ξ 6 Oξ 9 48 du 3 dξ = ξ 3 λ 4 ξ 4 Oξ 7 49 du 4 dξ = 3 λ 4 ξ 3 Oξ 6. 5 Letting ξ tend to zero, the determinant in the numerator of 46 becomes a a C t t. a 3 s s a 4 t t The denominator is of the same form but with b i instead of a i. Evaluating the determinants gives 4 j= σ j u a j 4 j= σ = a t s s t a s s a 3 t t j u b j b t s s t b s s b 3 t t. 5 This condition holds for ut, t Θ. Since a 4 and b 4 do not appear in the right hand side, we must set their coefficients to be zero and so the stratum Θ is characterised by Θ = {u : σu = σ 4 u = }. 45

21 Genus 4 trigonal reduction of the Benney equations In Θ, 45 reads 4 j= σ j u a j 4 j= σ j u b j = det [a dut dut dut ] det [b dut dut dut ]. 5 Now, as before, we let t. The third column of the two determinants can be expanded, as before, in powers of ξ = /t /3. The fourth column is given by the derivatives of these expressions: d u dξ = 6 ξ5 6 λ 4 ξ 8 Oξ d u dξ = 3 ξ 4 λ 4 ξ 5 Oξ 8 d u 3 dξ = 4 3 λ 4 ξ 3 Oξ 6 d u 4 dξ = λ 4 ξ Oξ 5. Letting ξ tend to zero, the numerator of 5 now becomes a a C t a 3 s a 4 t and again the matrix in the denominator is of the same form but with a i replaced by b i. Hence, equation 45 gives the relation 4 j= σ j u a j 4 j= σ j u b j = a a t b b t. Since a 3, a 4 and b 3, b 4 do not appear in right hand side of this equation we must set their coefficients equal to zero. It follows that the stratum Θ can therefore be characterised by Θ = {u : σu = σ 4 u = σ 3 u = } 53 see [5] for an analogous result for a, 5 curve, and we obtain the relation a σ u a σ u b σ u b σ u = a a t b b t. If we now set a =, a = and b =, b =, we find t = σ u ; 54 σ which gives the inversion of the restriction of the Abel map to T 4 : u = t du 55 for u Θ, the one-dimensional stratum of the 4-dimensional Jacobian JacT 4.

22 Genus 4 trigonal reduction of the Benney equations 4. Evaluation of ϕ We can now transform integrand 7 ϕt dt using the inversion formula 55 t = σ u for u Θ σ and the expressions for the holomorphic differentials du = dt 3s, du = t dt 3s, du 3 = s dt 3s, From equation 7 we have ϕt dt = k A A 3 t A 4 t dt 3s k du 4 = t dt 3s. 56 t λ 3 λ t dt 3s. Separating this into the holomorphic and meromorphic parts ϕ respectively by ϕ = A A 3 t A 4 t dt 3s and ϕ = t λ dt 3 λ t 3s we see and and ϕ, given ϕ t dt = k [A du A 3 du A 4 du 4 ] 57 [ σ ] ϕ t dt = k u λ σ u du. 58 σ 3 λ σ Thus ϕ is a sum of holomorphic Abelian differentials on the Riemann surface T 4. Since ϕp has zero residue at p = on all three sheets, and residues are invariant under conformal maps, we know that the second term ϕ must have a double pole with zero residue when σ u = ; modulo periods; there are three such points, denoted u, ωu, and ω u, where ω = i 3, corresponding to one point on each sheet of T 4. ϕ is regular everywhere else on Θ. Thus as in the hyperelliptic cases we must construct a function Ψ which satisfies d [Ψu] = ϕ u, u Θ. du As the integral of a second kind differential, it can have at worst simple poles at the three points ω i u. From the holomorphic differentials 56, we have = t, = s, = t u u u 3 u u 4 u and so the differential operator D = d/du Θ is given by D = d Θ du = t s t u u u 3 u 4 = σ s σ u σ u u 3 σ u 4

23 Genus 4 trigonal reduction of the Benney equations 3 where s 3 = [ ] t 5 λ 4 t 4 λ 3 t 4 λ t λ t λ [ ] σ σ σ σ σ = λ 4 λ 3 λ λ λ. σ σ σ σ σ Since this has 3 distinct roots for s, we cannot compare D Ψ and ϕ directly. This problem is therefore approached in a similar way to the higher genus hyperelliptic cases. We begin by identifying a function Ψ whose derivative has the same expansion as ϕ near each of the poles ω i u. We then verify that this function has no other poles. The solution can then be obtained by using an extension of Liouville s theorem on the stratum Θ of the Jacobi variety. 4.. Expansion near the pole u. We will begin by expanding the function ϕ near the point u = u and then compare this with the expansion of a suitable function Ψ. We note that because of the cyclic automorphism of T 4, the expansions at the other points ω n u are not essentially different; the conditions to be imposed at the three points ω n u all hold or fail together. Let u = u,, u,, u,3, u,4, then the Taylor series of the terms in ϕ are given as follows. Writing w i = u i u,i we have λ σ = [ ] λ σ σ w σ 3 w 3 / σ w..., 3 λ σ 3 λ σ w σ 3 w 3 / σ w σ 3 w w 3 σ w / σ 33 w3... [ ] σ σ σ σ w σ σ 3 w 3... =. σ w σ 3 σ w w 3 σ3 w3... σ Since this expansion is on Θ, we can rewrite it in terms of the single parameter t. On Θ we know t = σ /σ and so σ u = corresponds to the points t =. This means that the u,i are given by the integrals u,i = du i i =,..., 4. As the Riemann surface T 4 has no singularities and the branch points satisfy T i >, t = is a regular point and so we write the w i = u i u i in terms of the local parameter t. This gives, on the sheet on which s λ /3 as t, w = = t t du dt s /3 du = t du = t λ t Ot 3, 59 3λ /3 9 λ 5/3 with similar formulae on the other sheets. Similarly, we see that w = 6 λ /3 t Ot 3, 6

24 Genus 4 trigonal reduction of the Benney equations 4 and w 3 = 3 w 4 = 9 λ /3 λ /3 t Ot 6 t 3 Ot 4. 6 We can thus rewrite the series 6-6 in terms of the parameter w. This gives w = u u, = 3 λ/3 w O w 3, 63 w 3 = u 3 u,3 = λ /3 w O w, 64 w 4 = u 4 u,4 = 3λ 4/3 w 3 O w Substituting in the expressions for w i in terms of w we obtain the following expansions: [ ] λ σ = λ σ O ; 3 λ σ 3 λ σ λ /3 σ 3 w σ σ = σ σ λ /3 w σ 3 C O w where the coefficient C is given by σ C = [ ] 4 66 σ λ /3 σ 3 [ σ σ λ /3 σ σ3 4 λ /3 σ σ 3 σ λ /3 σ 3 σ λ σ 3 σ3 4 λ /3 σ 3 σ 3 σ σ σ σ λ /3 σ σ σ 33 σ σ σ 3 σ σ 3 σ 3 λ /3 λ σ σ 3 σ λ σ σ 3 σ 33 3 σ σ σ λ /3 3σ σ 3 σ ]. λ /3 λ λ /3 σ σ3 The second and third order sigma derivatives in this term are and σ, σ, σ 3, σ 3, σ, σ 3, σ 33. Thus to check that our integrand ϕ has zero residue, as it must, we need to find lower order expressions for these derivatives at the point u Θ. First we will find some relations holding throughout Θ, and then specialise to the 3 points ω n u.

25 Genus 4 trigonal reduction of the Benney equations Relations between the σ-derivatives holding throughout Θ. We start in Θ 3. The point at infinity is a branch point of period 3 and so the expansion for t 3 is given in terms of the local parameter t 3 = /ξ 3. Substituting this into the definitions of u i from the Abel map, we find and u i t, t, /ξ 3 u i t, t, = /ξ 3 u t, t, /ξ 3 u t, t = 7 ξ7 5 λ 4 ξ O ξ 3, 67 u t, t, /ξ 3 u t, t = 4 ξ4 λ 4 ξ 7 O ξ, 68 u 3 t, t, /ξ 3 u 3 t, t = ξ 5 λ 4 ξ 5 O ξ 8 69 u 4 t, t, /ξ 3 u 4 t, t = ξ 6 λ 4 ξ 4 O ξ 7. 7 If we now calculate the Taylor series for σu =, which holds identically in Θ 3, in terms of ξ using 7, and then substitute in identities 67-7 we obtain du i = σ ut, t, [ut, t, ut, t, t 3 ] σut, t, σ 4 ut, t, ξ 7 σ 3ut, t, σ 44ut, t, ξ Oξ 3 7 = σ 3ut, t, σ 44ut, t, ξ Oξ 3, 73 since σ 4 = σ = on Θ. The terms in the right hand side are evaluated at the point u = ut, t, Θ. Setting the coefficients of ξ equal to zero, we find σ 44 σ 3 =, u Θ. 74 on Θ. If we repeat this process, expanding 74 as u Θ obtain the relation t, we similarly σ 33 σ =, u Θ Relations between the σ-derivatives holding at u Θ. At the point u = u we have the additional restriction σ u =. This will yield enough relations to evaluate the expansion of ϕ. We have Θ = {u : σu = σ 4 u = σ 3 u = } and the terms we need to express in terms of lower derivatives are σ, σ, σ 3, σ 3,

26 Genus 4 trigonal reduction of the Benney equations 6 and σ, σ 3, σ 33. If we were to expand σ 4 u = u Θ for u near u, then all of its terms would contain derivatives with respect to u 4, which we do not require. Therefore we will just consider the identities σ = and σ 3 =, both valid throughout Θ, and in particular near u. The stratum Θ 3 is given by the set of points u JacT 4 such that 3 ti ut, t, t 3 = du where the divisor t t t 3 3 has dimension 3. This can also be defined by condition 44: Θ 3 = {u : σu = }. We begin by calculating the Taylor series of σu = for u Θ near the point u. This gives = σ u u u = σ σ w σ 3 w 3 σ w σ 3 w w 3 σ w σ 33 w 3 where w i = u i u,i i =,..., 4. If we now substitute in the expressions for w i as functions of w 6-6 and use the results valid at u Θ : we find σ u = σ 3 u = σ 4 u = σu =, = σ = [ σ 3 λ /3 σ ] w [ σ 3 λ /3 σ 3λ 3 λ/3 σ 33 σ 3 σ λ /3 ] w Ow 3 [ 3 σ λ /3 σ σ 3 λ /3 ] λ/3 σ 33 w Ow 3 76 where each term on the right hand side is evaluated at u. Setting the leading coefficient, that of w, in 76 to be zero gives σ = 3λ /3 σ λ /3 σ 3 λ /3 σ 33 at u Θ. An analogous relation for σ can be found from the coefficient of w. 3 The substitutions for σ 3 and σ 3 are then obtained from the Taylor series of σ 3 u = for u Θ near u.

27 Genus 4 trigonal reduction of the Benney equations 7 Now these still involve σ 33 and σ 33. The former was found above 75, which holds throughout Θ. Expanding near u = u gives the required result. To summarise, the full list of substitutions required to evaluate C is σ = λ /3 σ, 77 σ 3 = λ /3 σ, 78 σ 33 = σ, 79 σ = 6 λ /3 σ 3 λ /3 λ 3 σ, 8 σ 3 = λ /3 σ 3 λ /3 σ λ 3 σ, 8 σ 33 = λ /3 σ 3 σ. 8 valid for u = u Θ. If we substitute these into the coefficient C 66, then this term vanishes. Thus the expression for ϕ dt is indeed a second kind differential on Θ : Let where ϕu u u = We now consider the function Substituting into ψu we see ψ = σ 9 λ 4/3 w Ow. 83 Ψu = σ 3 σ u. 84 ψu = d [Ψu] = t s t du u u u 3 s 3 = t 5 λ 4 t 4 λ 3 t 3 λ t λ t λ. t = σ σ u s σ 3 σ 3 σ σ σ 3 s σ 33 σ 3 σ σ 3 σ u 4 Ψu σ σ 34 σ 3 σ 4.85 σ Since s is regular at u = u, all singularities must come from the coefficients of σ and σ. Thus for u near u we will write s = st t and define [ ψ u = s σ σ 3 σ 3 σ σ 3 s σ 33 σ ] 3 σ O. σ σ To expand this near u = u we first need to evaluate s. Using Maple, we calculate the Taylor series of s for t near zero. The first few terms are s = λ /3 λ λ t λ t Ot λ /3 λ /3 λ 5/3 σ

28 Genus 4 trigonal reduction of the Benney equations 8 We can then invert the series for w t 59 to rewrite this as s = λ /3 λ 3 w 3λ /3 λ w Ow Using expression 87 for s, we can now expand ψ for u near u in the same manner as ϕ. We obtain ] σ λ ψ = [ σ /3 λ σ 3 3 Ow σ 3 σ λ σ 3 λ 4/3 σ λ /3 w. 88 This can be simplified by writing σ and σ 3 in terms of σ. From identities 77 and 78, we see [ ] ψ = Ow 3 89 λ /3 and so the function ψu = d [ ] σ 3 u 3 λ du 3 λ σ has same principal part as ϕ near u = u see equation 83. w 5. The expansion of σu near u = We now need to verify that the function [ d ] σ 3 u du 3 λ σ like ϕ, is regular at u =. To do this we need the expansion of σ for u near. Such an expansion for a σ- function was first found in the elliptic case by Weierstrass [6], [7]. A similar expansion was found for the genus hyperelliptic case by Baker [8], and was recently generalised to arbitrary genus hyperelliptic curves by Buchstaber and Leykin [9]. In this case the leading terms of σ can be evaluated as follows. We know that the first term in the Taylor expansion of σ is the Schur-Weierstrass polynomial. To calculate this polynomial we proceed as follows. For a general point on JacT 4 we have 4 u i t, t, t 3, t 4 = u i t k i =,..., 4. k= Let us now study the stratum Θ 3 near u =. We let t, t, and t 3 approach infinity, and set t 4 =. We replace u i t, t, t 3,, by the leading term in the expansion of the Abel map: u t, t, t 3, = 7 ξ 7 7 ξ 7 7 ξ 3 7, u t, t, t 3, =, u 3 t, t, t 3, = 4 ξ 4 4 ξ 4 4 ξ 3 4 ξ ξ ξ 3 u 4 t, t, t 3, = ξ ξ ξ 3.,

29 Genus 4 trigonal reduction of the Benney equations 9 If we evaluate the resultants of these equations, successively with respect to ξ 3, ξ and then ξ, we obtain the polynomial S = C u u 56 u 3 u 4 4 u u u 3 u 4 u u 4 u where C R is an irrelevant constant. The Schur-Weierstrass polynomial is the leading term in σ, which must vanish on Θ 3. Now the factor u 4 is non-vanishing except at the origin. The Schur-Weierstrass polynomial for T 4 is thus given by the factor SW = 448u 56u 3 u 4 4 u u u 3 u 4 u u 4 u. Now the weights of the terms u i are given by the Weierstrass gap sequence. These are 7, 4, and respectively and so this polynomial has weight 8. We now use this as the starting point for the Taylor series expansion of σ near u =. From 67-7 we see that the powers of ξ in the Taylor series expansion of u i near u = increase by steps of 3. Thus the total weights of the successive terms in the Taylor series for σ near u = will increase by weights of 3. We look for a series for σ as a sum of monomials of weights 8 3n with n, with coefficients of u n u n u n 3 3 u n4 4 which are isobaric polynomials in the λ i - they have weight of λ i is 3i 5. To include all the information about the curve, this series needs to contain all of the curve moduli λ i. The lowest weight at which all λ i appear is 3, so we must calculate at least to this order. There is a unique series, σ, including terms of weight 3, which satisfies the conditions: The leading term, of Ou 8 4 of σ, is the Schur-weierstrass polynomial. σ = Ou 6 4 on Θ, Θ, and Θ 3. On JacT 4, the Jacobi inversion formulae are satisfied to sufficiently high order: and 4 4 t 34 s 44 t t s =, t s t = t 3. Here the Kleinian -functions are to be replaced by the corresponding logarithmic derivatives of σ, and evaluated at ut, s, t, s, t 3, s 3, t 4, s 4, where all points t i, are allowed to tend to, so that u. This series is given in full in the appendix. Using this Taylor series expansion for σu as u we can quickly compare the properties of σ ϕ u = λ 3 σ 3 λ and σ d du Ψu = d du σ 3 σ σ

30 Genus 4 trigonal reduction of the Benney equations 3 for u near. We represent the sigma derivatives in terms of the known Taylor series σ valid near u =. The derivatives of σ are determined up to order: σ i = σ i Ou 6 γ i 4. The leading terms are of order Since u has weight SW i = Ou 8 γ i 4. W u = γ = 7, it follows that the series for ϕu is valid up to and including terms of weight 6. It is given by λ lim ϕu = lim ξ 3 ξ 6 Oξ 7. u ξ 3 λ We can calculate the expansion for σ3 D Ψu = D = σ σ s u σ u u 3 σ σ = σ 3 σ σ 3 σ σ σ 3 σ σ 3σ σ σ s σ 33 s σ 3 σ σ σ σ 34 σ 3σ 4 σ σ in the same way as for ϕ although, here, we also need to use the series expansion for s as t. The highest order derivative in this expression is σ 3 which has weight 7 7 = 6 and so the Taylor series we obtain is valid to order 7. We find [ ] d [ lim Ψu = lim λ λ ξ 3 3λ ξ 6 Oξ 8 ] u du ξ which is regular as u tends to. Thus the functions ϕ and d Ψu du 3 λ have the same series expansion near the pole u = u and are regular everywhere else on Θ. It follows that we can write [ d [ϕ ] du A du A 3 du A 4 du 4 = ] Ψu du B T du 9 du 3 λ for some vector of constants B T = B, B, B 3, B 4. u 4 σ 3 σ

31 Genus 4 trigonal reduction of the Benney equations Evaluation of the vector B We can now evaluate the vector B for the trigonal case using the same technique as for the higher genus hyperelliptic reductions. Consider the Abelian differential [ ϕ d ] Ψu du. du 3 λ By definition du is a first kind Abelian differential and so has zeros of degree g = 6 and no poles on T 4. From the calculations above we know that [ F = ϕ d ] Ψu du 3 λ is regular on T 4 and so F must be a constant. If we compare the expansions of ϕ and d Ψu du 3 λ near u =, 83 and 89, we see that F = λ 3 λ and so identity 9 can be written λ du = [ B A B A 3 t B 3 s B 4 A 4 t ] dt 3 λ 3s. 9 To evaluate the vector B we look at the expansion of 9 as t tends to infinity. The values of du i are given by equations The left hand side is therefore λ ξ 6 Oξ 9 3 λ and the right hand side becomes A 4 B 4 B 3 ξ 3 λ 4 B 4 A 4 A 3 B ξ 3 [ 3 λ λ 4 Matching coefficients of ξ, we find B 4 = A 4, B 3 =, B = A 3, λ B = A 3 λ and so equation 9 becomes 3 λ 4B 3 ξ 4 B 4 A 4 3 λ 4B A 3 A B ] ξ 6 Oξ 7. [ϕ ] du A du A 3 du A 4 du [ 4 d = ] σ 3 du A λ u du A 3 du A 4 du 4. du 3 λ σ 3 λ

32 Genus 4 trigonal reduction of the Benney equations Explicit formula for the trigonal reduction From the definition of λ 6, we set Substituting K = 3 λ /3. p = P 6 t = P 6 σ u u Θ σ into 3, we have λp = p = = p P 6 σ 3 λ /3 P 6 σ 3 λ /3 ϕp dp u σ P 6 p 3 λ /3 P 6 p [ d [ A 3 ] u σ λ λ ] du A 3 du A 4 du 4 P 6 p σ 3 du du 3 λ u σ [ A λ u A 3 u A 4 u 4 ] σ 3 u 9 3 λ 3 λ σ ] C 93 [ σ σ u where the A i are defined in equation 5. To calculate the constant C we recall that the expansion of λp as p tends to infinity is lim λp = p O. p p From the identity p = P 6 σ σ u u Θ, we see that p is equivalent to σ u. The Taylor series for equation 93 is thus calculated in terms of w i = u u e i. We can then use the substitutions to rewrite w j j =, 3, 4 in terms of w i. We have lim p [λp p] = 94 { [ lim 3 λ /3 A λ u A 3 u A 4 u 4 ] σ 3 u σ } u. 95 u u 3 λ 3 λ σ σ Calculating the Taylor series for the first terms gives { [ lim 3 λ /3 A ] } λ u A 3 u A 4 u 4 σ 3 u u u 3 λ λ /3 σ [ ] = 3 λ /3 w [ 3 λ /3 A ] λ u, A 3 u, A 4 u,4 c Ow 3 λ dt t

33 Genus 4 trigonal reduction of the Benney equations 33 where c is c = 6 λ σ λ 4/3 σ 3 [ 6A λ σ λ 5/3 λ σ 3 6A λ 8/3 σ 3 4λ 4/3 λ σ σ 3 u, u, 6A 3 λ 8/3 σ 3 λ λ σ 6A 3 λ σ 6A 4 λ σ 6A 4 λ 8/3 σ3 A 4 λ 7/3 σ σ 3 u,3 A 4 λ 7/3 σ σ 3 u,4 6A 4 λ σ λ σ 3 σ λ 4/3 σ 3 σ 6A 4 λ 8/3 σ3 A λ 7/3 σ 3 σ ] 3λ 5/3 σ σ 3 λ σ 3 σ λ 5/3 σ 3 σ 33 λ λ 3 σ3 A 3 λ 7/3 σ σ 3 Using substitutions 77-8 we find and so { lim u u 3 λ /3 [ = 3 { λ /3 ] 3 λ /3 c = 3 λ σ 3 σ u [ A 3 w λ λ [ A 3 ] u A 3 u A 4 u 4 λ λ ] u, A 3 u, A 4 u,4 } σ 3 u λ /3 σ The Taylor series for the second term in equation 94 is [ lim σ ] [ ] σ u = u u σ σ λ /3 σ 3 w σ λ /3 σ 3 } 9 σ 3 u λ /3 Ow. 96 σ σ σ λ /3 σ σ 3 λ /3 σ σ 33 λ /3 σ σ 3 λ /3 σ 3 σ 3 λ 3 σ σ 3 λ /3 σ σ 33 σ σ λ /3 σ σ 3 3λ /3 σ σ ] Ow again, using the substitutions 77-8 for the second and third order sigma derivatives, this becomes [ lim σ ] [ ] [ u = ] λ Ow. 97 u u σ 3 w 3 Since λ /3 lim [λp p] = O p we set the constant C to be [ C = 3 λ /3 A 3 λ, p λ λ u, A 3 u, A 4 u,4 ] 9 λ /3 To summarise, the mapping we require is given by the following result: σ 3 λ u.98 σ 3 λ