Symmetry, Integrability and Geometry: Methods and Applications Spectral Curves of Operators with Elliptic Coefficients ⋆

A computer-algebra aided method is carried out, for determining geometric objects associated to differential operators that satisfy the elliptic ansatz. This results in examples of Lame curves with double reduction and in the explicit reduction of the theta function of a Halphen curve.


Introduction
The classical theory of reduction, initiated by Weierstrass, has found modern applications to the fields of Integrable Systems and Number Theory, to name but two. In this short note we only address specific cases, providing minimal historical references, listing the steps that we devised, and exhibiting some new explicit formulas. A full-length discussion of original motivation, theoretical advancements and modern applications would take more than one book to present fairly, and again, we choose to provide one (two-part) reference only, which is recent and captures our point of view [3,4].
Our point of departure is rooted in the classical theory of Ordinary Differential Equations (ODEs). At a time when activity in the study of elliptic functions was most intense, Halphen, Hermite and Lamé (among many others) obtained deep results in the spectral theory of linear differential operators with elliptic coefficients. Using differential algebra, Burchnall and Chaundy described the spectrum (by which we mean the joint spectrum of the commuting operators) of those operators that are now called algebro-geometric, and some non-linear Partial Differential Equations (PDEs) satisfied by their coefficients under isospectral deformations along 'time' flows, t 1 , . . . , t g , where g is the genus of the spectral ring. We seek algorithms that, starting with an Ordinary Differential Operator (ODO) with elliptic coefficients, produce an memories and testimonials are collected in other parts of this issue; here we only say that it was our great privilege to know him.

Elliptic covers
Curves that admit a finite-to-one map to an elliptic curve ('Elliptic Covers') are special, and are a topic of extensive study in algebraic geometry and number theory. They have come to the fore in the theory of integrable equations, in particular their algebro-geometric solutions, for the reason that a yet more special class of such curves provides solutions that are doubly-periodic in one variable ('Elliptic Solitons').
Our point of view on the latter class of curves is based on the Burchnall-Chaundy theory, via which, we connect the problem to the theory of ODOs with elliptic coefficients, much studied in the nineteenth century (Lamé, Hermite, Halphen, e.g.). It is only when such an operator has a large commutator that this theory connects with that of integrable equations. While the nineteenth-century point of view could not have phrased the special property in this way, the related question that was addressed was, when does the series expansion of solutions, in some suitable local coordinate on the curve (cf. Remark 1 in Subsection 3.1) terminate? The early work turned up important, but somehow ad hoc, properties that characterize algebro-geometric operators.

Challenge I
Detect (larger) classes of algebro-geometric operators.
We do not address this issue here. A CAS-based project would be to test whether the operator has centralizer larger than C[L]: we ask the question since Halphen [14] noticed that the equation L = 0 has solutions that can be expressed in terms of elliptic functions. A more far-reaching project would construct elliptic-coefficient, finite-gap operators starting with curves that were shown to be elliptic covers, choosing a local coordinate whose suitable power is a function pulled back from the elliptic curve, to give rise to the commutative ring of ODOs, according to the Burchnall-Chaundy-Krichever inverse spectral theory. Curves of genus 3 that are known to be elliptic 3 : 1 and 2 : 1 covers, respectively, are Klein's and Kowalevski's quartic curves: in projective coordinates, with f 2 a form of degree 2 in x and y (such that the quartic is nonsingular). Notably, Klein's curve (the only curve of genus 3 that has maximum number of automorphisms, 168) has Jacobian which is isomorphic to the product of three (isomorphic) elliptic curves [2,23,25]; Kowalevski, as part of her thesis, classified the (non-hyperelliptic) curves of genus 3 that cover 2 : 1 an elliptic curve (Klein's curve being a special case).

Challenge II
Find an equation for the spectral curve. This can be found once an algebro-geometric ODO is given. While the Burchnall-Chaundy theory allows one in principle to write the equation of the curve as a differential resultant [24], this would be unwieldy for all but the simplest cases: first, one would have to solve the ODEs of commutation [L, B] = 0 for an unknown operator B of order co-prime with L (the simplest case of a two-generator centralizer); then, once B is known, calculate the determinant of the (m + n) × (m + n) matrix that detects the common eigenfunctions of L − λ and B − µ [8].
Instead, it is much more efficient to adapt Hermite's (or Halphen's) ansatz; in the case of Lamé's equation, the case of solutions 'written in finite form' is treated in [36]; more generally, assume that an eigenfunction can be written in terms of: solving a given spectral problem: LΨ(x) = zΨ(x), where: and α is a complex number viewed as a point of the elliptic curve ν 2 = 4µ 3 − g 2 µ − g 3 . By expanding ℘(x) and φ(x, α) near x = 0 and comparing coefficients, the a j can be written in terms of (k, z) and taking the resultant of the compatibility conditions gives an equation of the spectral curve in the (k, z)-plane [7].

Challenge III
Express the eigenfunctions in terms of the theta function of the curve.

Challenge IV
Turn on the KP (isospectral-)time deformations t i and express the time-dependence of the coefficients of the operators (in particular, one such coefficient is an exact solution of the KP equation). The expression of a KP solution in terms of elliptic functions is part of the programme sometimes referred to as 'effectivization'.
Our approach to such 'challenges' is two-fold: on one hand we use the explicit expression of the higher-genus sigma functions that were classically proposed by Klein and Baker, but revived, substantively generalized, and brought into usable form at first by one of the authors with collaborators (cf. e.g. [5]). On the other hand, we design computer algebra routines which allow us to read, for example, coefficients of multi-variable Taylor expansions far enough to obtain the equations for the curves and the KP solutions for small genus.
In this paper we present classes of examples, originating with classical ansatzes of Lamé, Hermite and Halphen (specifically applied to elliptic solitons by Krichever [17]) as well as some detail of our general strategy.

Hyperelliptic case
The Lamé equation has been vastly applied and vastly studied: we refer to [18] for general information, and to the classical treatise [36] for calculations that we shall need (for richer sources please consult [11], and [12] for different perspectives). Our point of view is that the potentials n(n + 1)℘(ξ) are finite-gap when n is an integer (an adaptation of Ince's theorem to the complex-valued case) where the equation is written in Weierstrass form: (for the rational form see [18] or [36]).

The spectral curve
The Lamé curves are hyperelliptic since one of the commuting differential operators has order 2, and will have genus n for the operator in (1). There is no 'closed form' for general n, but the equations have been found by several authors and methods, cf. specifically [3,4] and references therein (work by Eilbeck et al., by Enol'skii and N.A. Kostov, by A. Treibich and by J.-L. Verdier is there referenced), [13,18,21,34]. In particular, [18] adopts a method of inserting ansatzes in the equation and comparing expansions similar to ours, and tabulates the equations for 1 ≤ g ≤ 8 (in principle, a recursive calculation will produce them for any genus), so we do not present our table (1 ≤ g ≤ 10), but use g = 3 to exemplify our treatment of reduction in Subsection 3.2 below.

Remark 1.
To motivate the project that Vadim Kuznetsov had wanted to design, as recalled in the introduction, we mention in sketch the steps of our strategy to find the spectral curves. The key ansatz is the finite expansion of the eigenfunctions; we used a formula from [36] in the case of odd n and the (equivalent) table in [1] for even n just for bookkeeping purposes; respectively, It is then a matter of finding the coefficients b r , A r . For this, we insert w in (1) and expand in powers of ℘(ξ) or (℘(ξ) − e i ) 1/2 , depending on the various cases; we also substitute for the e i in terms of the modular functions g 2 and g 3 . By comparing coefficients in the expansions we obtain a set of equations for the b r or A r , and we solve the compatibility condition of these equations. The depending on g 2 and g 3 , and three f i (z) which depend on one of the e i and g 2 and g 3 , which we found in turn. At the end, to be sure, for a given root z of one of these factors, we can determine the b r , resp. A r up to a common factor, and write the eigenfunction as a polynomial in ℘(ξ). The finiteness condition we used in expanding the eigenfunctions (a sort of Halphen's ansatz, cf. Section 1) characterizes finite-gap operators, in Burchnall-Chaundy terms, the ones that have large centralizers. It is worth quoting a speculation [28] p. 34 which regards finiteness in the local parameter z −1 : 'We do not know an altogether satisfying description of the desired class C (n) ; roughly speaking, it consists of the operators whose formal Baker functions converge for large z.'

Cover equations and reduction
Even though the Lamé curves are elliptic covers a priori, it is not easy to find the elliptic reduction, which will correspond to a holomorphic differential of smallest order at the point ∞ in (w, z)-coordinates 2 ; by running through a linear-algebra elimination for a basis of holomorphic differentials, expressed in terms of rational functions of (w, z) (here again expansions provide the initial guess, lest the programs exhaust computational power), we exhibited the elliptic differential for all our curves. Note that this gives the degree of the cover, not known a priori.
It is worth pointing out that more than n covers might be found (for genus n): indeed, according to a classical result (Bolza, Poincaré, cf. [15] where a modern proof is given), if an abelian surface contains more than one elliptic subgroup, then it contains infinitely many. Note, however, that it can only contain a finite number for any given degree (the intersection number with a principal polarization) because the Néron-Severi group is finitely generated [15]. This degree translates into the degree of the elliptic cover. We are not claiming to have found all of them, or that those which we found be minimal. For example, for genus n = 2 (where the reduction is known since the nineteenth century), finding f s = z 2 − 3g 2 , f i = z + 3e i , in expanded form the curve is: The holomorphic differential is and a second reduction (predicted by a classical theorem of Picard that splits any abelian subvariety up to isogeny) is given by the algebraic map to: This map induces the following reduction of the holomorphic differential to an elliptic differential: Notice then that in the equianharmonic case g 2 = 0 the curve is singular, w 2 = −(4z 3 −27g 3 )z 2 and we have the following birational map to the elliptic curve: and differential d℘/℘ ′ = zdz/2w. This motivated us to investigate the equianharmonic case further.
What we believe to be new is the following observation. We found it surprising that there always be at least a second reduction in the case the elliptic curve is equianharmonic (the unique one that has an automorphism of order three), but see no theoretical reason to expect that. If this were to hold for all n, one could further ask whether these curves support KdV solutions elliptic both in the first and second time variables (a private communication by A. Veselov gives such an indication, to the best of our understanding). The issue of periodicity in the second KdV variable ('time') was put forth in [30], and recently reprised in [10], but the double periodicity does not seem to have been addressed so far. We exhibit three covers in the g = 3 case (N.B. The third one is not, in general, over the equianharmonic curve); as well, we found two covers in the equianharmonic, n = 4, 5 cases, but we only provide a summary table which suggests the general pattern for two covers in the equianharmonic case.
For n = 3, f s = z, f i = z 2 − 6ze i + 45e 2 i − 15g 2 , the curve is the first cover is given by the algebraic map This map induces the following reduction of the holomorphic differential to an elliptic differential: In the equianharmonic case the above reduces to the following cover of the curve ℘ ′ 2 = 4℘ 3 − g 3 : This map induces the following reduction of the holomorphic differential to an elliptic differential: Continuing in the equianharmonic case, the second cover is given by the following algebraic map to: This map induces the following reduction of the holomorphic differential to an elliptic differential: The third cover is given by the algebraic map to the (non-equianharmonic) curve This map induces the following reduction of the holomorphic differential to an elliptic differential: For the curves: we summarize the equianharmonic covers in the table below (note: there is some small numerical discrepancy, e.g., for n = 2 our choice of normalization corresponds to the −2w of this table, but we find it more useful to have the normalization below for recognizing a pattern). The pairs appearing in the second column are the orders of the commuting differential operators.
In factorized form the coefficients are The hyperelliptic curves are singular for g = 2, 5, 8; this is important because it makes it possible to explicitly calculate the motion of the poles of the KdV solutions (Calogero-Moser-Krichever system). When all the KdV or KP differentials (of increasing order of pole) are reducible, of which case we have examples in genus 3, both hyperelliptic as we have seen, and non (Section 4), the Calogero-Moser-Krichever system is periodic in all time directions; we believe this situation had not been previously detected.

The theta function
The next challenge we discuss is the calculation of the coefficients of the ODOs in terms of theta functions; while we do not present a result that could not have been derived by classical methods, we streamlined a two-step procedure: firstly, we use Martens' thorough calculation of the action of the symplectic group on a reducible period matrix [19], to write theta as a sum, then we use transformation rules for the action under the symplectic group. The step which is not obvious however precedes all this and is the calculation of the explicit period matrix based on a reduced basis of holomorphic differentials. We give the result for one specific curve as an example. For the curve of genus 2: where we assume 0 < ξ 1 < ξ 2 < ξ 3 , which is easily related to the form given by Jacobi for the case of reduction: our calculation shows that the associated theta function can be put in the form where the θ i are the standard g = 1 Jacobi theta functions, τ 1 is the τ associated with the elliptic curve and τ 2 is the τ associated with the elliptic curve These two τ 's can be expressed explicitly in terms of elliptic integrals of the first kind K. We adopt the homology basis shown in Fig. 1. We find that where the ω i are calculated from the the homology basis shown in Figs. 2 and 3,

Proof
Using the change of variable z 2 = x, 2zdz = dx, we have The ω can be calculated by standard integrals so the period matrices we obtain can be written in the form , Following Martens, if we define where T is symplectic, T · J · T t = J, and shows that the Hopf number is 2.
The transformed matrixτ is calculated by first calculating the 2 × 4 matrix For this matrix, we use Martens' transformation formula, generalized to non-zero characteristics: in Jacobi notation, concluding the proof.
The cover π 1 is given by with holomorphic differentials given by The cover π 2 is given by with holomorphic differentials given by The cover π 3 is given by with holomorphic differentials given by The general curve C 3 is a covering of the equianharmonic elliptic curve C ∞ given by the equation The cover π is given by with holomorphic differentials given by 2zdz 3w 2 = dµ ν .

Reduction
In this section we follow a similar approach to that of Matsumoto [20] in a genus 4 problem, who developed earlier results of Shiga [27] and Picard [22]. Write the equation of C in the form We fix the following lexicographical ordering of independent canonical holomorphic differentials of C 3 , du 1 = dz/w, du 2 = dz/w 2 , du 3 = zdz/w 2 , and will define the period matrix based on the branch cuts given in Figs. 4 and 5.

Riemann period matrix
We introduce the following vector notation: where H = diag(1, 1, −1). The Riemann bilinear relation says The Riemann period matrix τ = AB −1 belongs to the Siegel upper half-space H 3 . By using the symmetries of the problem we can express all the x period integrals in terms of the two integrals (see Fig. 6) We have where ρ = exp{2iπ/3}. In the case when λ 2 2 /λ 2 1 = 5/27, the integrals simplify further, since I = −J(1 + 2ρ)/3. In this case (which we assume in all that follows) we have So we have where ρ 1 = ρ + 1. We define We find after much simplification, using the properties of ρ and the Riemann relations, that This matrix [(Ω ′ ) T Ω T ] satisfies the reduction criteria as defined by Martens [19], since if we define Expanding a theta function defined with thisτ , again following Martens, we will get a sum of 5 products of g = 1 theta functions with g = 2 theta functions. The g = 1 theta functions will have tau value τ = (1/5)(1 + ρ) and the g = 2 theta functions will have a tau of  Expanding each of the transformed genus 2 theta functions with this theta will give a product of genus 1 theta functions with τ = (1/20)(11 + ρ) and genus 1 theta functions with τ = 16(3/2 + (1/4) ρ). So finally we have 5 · 4 = 20 terms, each containing a product of three genus 1 theta functions (with fractional characteristics).

Conclusions
We contributed to the theory of spectral curves of ODOs with elliptic coefficients routine algorithms to calculate: • The algebraic equation of the curve (always, in principle); • The period matrix (only if the periods can be chosen suitably and there is an explicit solution to the action equations); • A reduction method for the theta function.
What remains to be calculated (Challenge IV) is the dependence of the coefficients on the time parameters. This is more difficult because it involves expanding an entire basis of differentials of the first kind. In [9], we were able to find the time dependence by implementing Jacobi inversion, thanks to the Hamiltonian-system theory which describes the evolution of the poles of the KP solution [17].