## Abstract

Burchnall & Chaundy (Burchnall & Chaundy 1928 *Proc. R. Soc. A* **118**, 557–583) classified the (rank 1) commutative subalgebras of the algebra of ordinary differential operators. To date, there is no such result for several variables. This paper presents the problem and the current state of the knowledge, together with an interpretation in differential Galois theory. It is known that the spectral variety of a multivariable commutative ring will not be associated to a KP-type hierarchy of deformations, but examples of related integrable equations were produced and are reviewed. Moreover, such an algebro-geometric interpretation is made to fit into A.N. Parshin's newer theory of commuting rings of partial pseudodifferential operators and KP-type hierarchies which uses higher local fields.

## 1. Introduction

Thirty years after the appearance of the first papers that extended the historical symbiosis between algebraic geometry and physics (such as the origin of Riemann surfaces from potential theory) to partial differential equations, this fruitful interaction has not waned and some questions that remain open call for new links. The main problem covered by this paper is the construction of higher-dimensional spectral varieties: these should replace the spectral curve in the classification of commutative rings of (partial, replacing ordinary) differential operators (PDOs). The new links that are suggested by the emerging examples are: differential algebra (extending the theory of Burchnall and Chaundy to ideals in rings of PDOs), differential Galois theory and representation theory.

*Contents*. The paper begins with a historical note (§2), namely the spectral Prym construction for Schrödinger operators at one energy level, the recent examples solved exactly in terms of Kleinian functions and an interpretation of such constructions in terms of differential algebra. At the end, recent examples of non-separable completely integrable rings of PDOs are reviewed; these were constructed using representation theory. In §3 the algebraic theory is laid out. In §4, geometric constructions proposed by M. Sato and realized by A. Nakayashiki are related to such theory. Sato's approach follows the other striking geometric feature of integrable equations, namely the linearization of the flows over an infinite Grassmannian. It is subtle to detect where the two constructions, spectral curve and Grassmannian flows, take place simultaneously. A.N. Parshin yet more recently used the Grassmannian approach, and hierarchies of KP type were attached to his construction of commutative rings of formal pseudodifferential operators in several variables: this seems at the moment the most mysterious and promising area for finding spectral varieties. Section 6 brings together again the integration problem with the differential algebra and offers a technique for answering some algebraic questions that were otherwise open.

## 2. Schrödinger operators

A natural generalization of the one-variable theory would begin with two-variable Schrödinger operators −Δ+*q*(*x*_{1}, *x*_{2}), whose spectrum is in some sense algebro-geometric, providing eigenfunctions *ψ*(*x*_{1}, *x*_{2}, *z*_{1}, *z*_{2}) defined on an algebraic surface where *z*_{1}, *z*_{2} are local parameters.in Dubrovin *et al.* (1976), the authors worked out the asymptotic properties of the eigenfunction restricted to each energy level *X*_{λ}, assumed to be an algebraic curve, and solved the inverse spectral problem only for the case when one energy level is algebro-geometric, under suitable analyticity assumptions (cf. also Feldman *et al.* (1992)), unless the potential is essentially a sum of one-variable algebro-geometric potentials; this case is sometimes called separable

### (a) Algebro-geometric solutions: one energy level

#### (i) Solution I

By solving a direct spectral problem, Novikov & Veselov (1986) gives a survey of the original results (Veselov & Novikov 1984*a*,*b*) and were able to produce exact Schrödinger potentials, including some that are not separable, algebro-geometric for one energy level. We report their formula for the zero energy level (in this paper we do not consider reality conditions).

*If the Riemann surface* *of genus g admits an involution* *that fixes exactly* *pairs of points*, , , *then the Prym variety of the cover* , *which has dimension* , *where* , *parametrizes potentials* *such that* , . *Moreover, these potentials admit a sequence of time deformations* *such that the corresponding Schrödinger operators have the same property*. *The operator* *satisfies the hierarchy of PDEs*

*More specifically, in the case that all time evolutions satisfy the* ‘*reality condition*’ , *are suitable operators in* *of order* *and* *of order* . *The reality condition is satisfied for a curve* *carrying an antiholomorphic involution* *such that* , *The eigenfunction* *is determined by choosing the inverse of local parameters to be* *and* *at the points* , *respectively* , , , *and requiring* *to have asymptotic expansions*,*where* , *and prescribed poles* *such that the divisor* *belongs to the translate of the Prym variety of the cover* *that contains the divisor* , *where* *is the canonical divisor of the curve* ; *and* . *The potential is given as an exact expression in the theta function* *of the Prym variety*.

Here several normalizations were made to define the Prym–Abel map , the point *e* of the Prym corresponding to a (generic) divisor *D*, the differentials of the second kind and the constant . We do not reproduce all the notation which is not relevant in this paper.

As noted in Veselov & Novikov (1984*b*), once appropriate reality conditions are given, the Schrödinger potentials thus obtained by algebraic geometry make up a strictly larger class than the separable potentials , which give a subclass of codimension 1 for , e.g. and thus generically do not admit more than one energy level.

#### (ii) Solution II

More recently, several authors have revisited Klein's idea of generalizing to genus-2 curves, the interplay between algebra and geometry, achieved by the Weierstrass function. One application resulted in the explicit construction of Schrödinger potentials, is also algebro-geometric for one energy level. These are parametrized by a different variety, although Buchstaber *et al.* (2002) announce work that relates it to the one described above (theorem 2.1). We report on this second construction here not only due to intrinsic interest but also due to the only case that was worked out so far is genus 2. The programme in principle should work for hyperelliptic curves of any genus and the open question is to characterize explicitly the corresponding subvarieties of Schrödinger operators, algebro-geometric for one energy level. The number of variables is the same as the genus.

Before stating the result, we sketchily recall definitions/notations from Buchstaber *et al.* (2002). The Kleinian function can be constructed by analogy with the *g*=1 case for any (nonsingular) hyperelliptic curve,As such, we only recall that depends on a *g*-dimensional complex vector ** x**, that the function and function are defined as: , and that the Baker function is defined on ; the eigenfunction is given by times the exponential of a scalar depending on

**and**

*x***.**

*α**With the above notation and g*=2, *the following equation holds*:*for* , *when the variable* *α**is restricted to the one-dimensional Bloch variety defined by*(*here* *Ω**and* *are suitable half-periods*).

Unlike the parameter space of solution I, which is a linear subvariety of , on which linear flows can be defined, the variety *X* of solution II is a curve of genus 4. No linear flows appear to be defined for this class of solutions.

#### (iii) Differential algebra aspect

This is the focus of this paper, and the general set-up will be given in §3, but we highlight the properties of the Schrödinger operators in this respect. The question of finding (and deforming) commutative rings of partial differential operators (PDOs) from algebraic geometry, which was settled in the one-variable case by the KP flows for affine rings of algebraic curves (the Schrödinger-operator case being the one of lowest order, corresponding to hyperelliptic curves), cannot be answered by the same kind of ring-theoretic characterization for the Schrödinger PDOs which are algebro-geometric for one energy level. The situation is the following (Novikov & Veselov 1986): if we define a two-variable PDO to be ‘algebraic’ when there exist non-trivial PDOs for whichthen for given algebraic operators, there exists a polynomial such thatand the common eigenfunction is meromorphic on the Riemann surface . Rank 1, second-order algebraic operators, where the rank is taken to be the dimension of the space of common eigenfunctions for generic , are characterized in theorem 2.1. Thus, ‘commute’ on the space of common eigenfunctions, and so do the higher-order deformations: or ; the ‘stationary flows’ are such operators associated to a meromorphic function *f* on ,

### (b) Representation-theoretic solution: Variety of energy levels

If instead we insist on the question of commutative rings of PDOs (loosely speaking, the corresponding energy levels would correspond to Fermi surfaces), to the best of our knowledge the state of the art on the Schrödinger operator is provided in work by Chalykh & Veselov (1990, 1993), and specifically on the Lamé operator in work by Chalykh *et al.* (2003). To pose the question, let's recall that in the one-variable case the ‘non-reducible’ rings according to Burchnall–Chaundy terminology (Burchnall & Chaundy 1923, 1928) are those containing an operator *L* such that is not a polynomial ring. We will define the ring of differential operators in §3 and here just recall that the second order such have potential written in terms of the theta function of a (possibly singular) hyperelliptic curve; in particular, for the Lamé operators,where is the Weierstrass function associated to a lattice or the limit obtained when one or both periods of the lattice go to infinity. The appropriate generalization of this problem to several variables was posed by Chalykh & Veselov (1990):

A commutative ring of PDOs in *n* variables is called complete if it contains *n* operators with independent symbols, and supercomplete if it is complete and not contained in any commutative ring generated by *n* operators. Such a ring is called separable if it contains a non-trivial operator in fewer variables.

With this definition, Chalykh & Veselov (1990) pose the question of determining the supercomplete, non-separable rings that contain a Schrödinger operator , and formulates the

The only Schrödinger operators that are contained in a supercomplete, non-separable ring are of the formwhere is the set of positive roots for a simple complex Lie algebra of rank *n*; is some positive scalar product in , invariant under the action of the Weyl group; and for some .

In Chalykh & Veselov (1990, 1993), the authors present progress towards the sufficiency part of the conjecture, namely the fact that such an *L* belongs to a supercomplete ring, for the degenerate cases ( of rank 1 or 0). Note the examples for , the first one resembling the one-variable rank 1 case

Subsequent work, in Veselov *et al.* (1993, 1996), Berest & Loutsenko (1997) and Chalykh *et al.* (1999), was devoted to commuting families of operators, including non-Coxeter, (namely not of the form given in conjecture 2.1), belonging to large commutative rings, in fact given explicitly at least in the rational case, together with the attendant eigenfunctions. In consequence, definition 2.1 was refined into ‘algebraic integrability’, and in that sense, in Chalykh *et al.* (2003) more of the conjecture is proven, by inputting techniques of differential Galois theory, which was first done in Braverman *et al.* (1996). The main technique of proof consists in finding eigenfunctions, although most of the time inexplicitly; also not quite explicit is the construction of differential operators that commute with a given *L*. We try to give a flavour for this theory.

A generalized Lamé operator is a Schrödinger operator , , with elliptic potential *u* of the following form a finite set of affine–linear functions on , such that the resulting potential has the properties of periodicity and quasi-invariance.

Periodicity and quasi-invariance are crucial generalizations of the one-dimensional case (Calogero–Moser potentials); to simplify matters, we assume the former to mean that *u* is periodic with lattice , where is the dual of the lattice generated by (loosely speaking) the linear part of the transformations ** α** contained in . The property of quasi-invariance is an important analytic condition, corresponding to the equations defining the Korteweg-de-Vries(KdV) locus in the one-variable case; is required to be divisible by , where is one of the hyperplanes comprising the singular locus of , is assumed to be of the form for some positive integer , and is the reflection with respect to .

Definition 2.1 is refined into algebraically integrable.

An *n*-variable operator *L* is said to be completely integrable if it is a member of a commutative family of differential operators, which are algebraically independent. A Schrödinger operator is said to be strongly integrable if the commuting operators have algebraically independent homogeneous constant highest symbols for which the systemhas the unique solution . Lastly, *L* is called algebraically integrable if it belongs to a commutative algebra of rank 1 (speaking again somewhat loosely here, this means that the local space of common solutions has rank 1 for the generic point of the variety corresponding to the ring); in our situation, this implies that there are additional commuting operators, which are not algebraic functions of the .

The generalized Lamé operators are all algebraically integrable.

*Fact* (Chalykh *et al.* (2003) and references therein). For a reduced irreducible root system in , *W* the corresponding Weyl group and for each homogeneous *p* in the ring of W-invariant polynomials, which is freely generated by *n* elements , there exists a differential operator with highest symbol *p* commuting with *L*, and the family is commutative.

*Any generalized Lamé operator which is completely and strongly integrable*, *i.e. admits a commutative family of differential operators* *which have meromorphic coefficients*, *is periodic with respect to the same lattice*, *so that they are defined on the torus* ( *is an appropriate lattice of rank n*), *and have algebraically independent homogeneous constant highest symbols* *for which the system**has the unique solution* *is algebraically integrable*.

as well aswith denoting the half periods .

A non-Coxeter example was found in Veselov *et al.* (1996); for any numbers *l*, *m*,

What is most remarkable about these examples is that the double Bloch functions can be constructed explicitly, providing meromorphic sections of a line bundle on a suitable projective variety *X*, a finite quotient of the torus where is the coroot lattice. These sections make it possible to compute in principle a third PDO that commutes with and (cf. above fact), providing algebraic integrability.

A possible link with the Treibich–Verdier Elliptic Solitons (Treibich & Verdier 1990; Treibich 1999). In the one-variable case, the classification of algebraically integrable Schrödinger operators is very rich; the link with the KdV equation was first indicated by Novikov (1974), and the classification of the deformations into an algebraically completely integrable system was achieved by Krichever (1980). The naive example of an algebraically integrable Lamé operator in *n* variables is then , with algebraic Lamé ODOs; in fact the period lattices need not be the same for each *i*. Even in this case, I do not believe the spectral variety is understood in all its aspects, as I will try to illustrate in §6. Indeed Chalykh *et al.* (2003) pose the question of comparing the variety Spec , which is affine, and the ‘Hermite-Bloch’ variety that parametrizes the Bloch eigenfunctions of *L*. In one variable, the most elusive elliptic solitons are those whose spectral curve Spec is a non-exceptional tangential cover of the elliptic curve , and the first layer of these (they are attached to a numerical constraint given by four integers , one for each point of period 2 of ) were found in Treibich (1999). Their description runs as follows:

*For any lattice* *and any quadruple* *of positive integers*, *the even* -*periodic function* *is a finite-gap potential if and only if* *and* *satisfies the following equation*:(in Treibich (1999), *Treibich describes geometrically the full set of such solutions*) *The form of these potentials and that of the examples above suggest that by letting the point* *vary*, *one might find a relation between elliptic solitons in one variable* (*even though the finite-gap property holds only for a finite number of* 's) *and the Hermite-Bloch variety for two variables* , *which is in fact a finite cover of* .

## 3. Differential algebra

In this section, we reconsider the problem of spectral varieties for commutative rings of differential operators from a purely algebraic point of view. Constructions of algebraic geometry can be adapted to rings with a differential action, the most extensive reference being Kolchin's work, cf. (Kolchin 1973). While these techniques have proved very efficient in one variable, however, the field remains relatively unexplored in several variables (cf. (Ritt 1950)), mainly—in my opinion—because the kernel of a differential operator becomes infinite dimensional. In fact, a very nice correspondence (which can be interpreted as a Fourier duality) between ideals of polynomial rings and solutions of ideals of constant-coefficient differential operators is clearest for ideals whose solution space is finite dimensional, cf. [Sturmfels (2002) ch. 10]. In this paper, I would like to highlight some features of differential algebra which—again in my opinion—give finite-gap integration coherence and a wide reach.

### (a) The one-variable case

The use of differential algebra and its geometric interpretation actually precedes any formal treatment. In the light of its reinterpretation in the 1970s, it is now generally agreed that the work of Burchnall and Chaundy in the 1930s (Burchnall & Chaundy 1923, 1928) provides a classification of commutative rings of differential operators, with some constructions going further back to Floquet (1879), Wallenberg (1902) and Schur (1905) (e.g. for a very learned historic account cf. Gesztesy & Weikard 1998). The phenomenon that gives elegance to the one-variable case is that commutative rings cannot be very large, as follows from the series of facts we now list; for each fact, we illustrate the situation in several variables.

We denote by the ring of differential operators; since some facts can be stated for both cases, we will use the same symbol for one or several variables. We will also go back and forth between different conventions on the coefficients of the operators. For algebraic considerations, it would be best to use formal power series, so thatwhere we abbreviate by and in the one-variable case, by . However, it may also be convenient to consider coefficients that are (complex)analytic, at least in the neighbourhood of a given point or meromorphic.

*Fact I*. In one variable, a non-constant can be normalized to have the formby using the two generators of the automorphism group of , change of variable and conjugation by a function. In several variables, the automorphism group of the Weyl algebra does not appear to be known (Dixmier 1968; Makar-Limanov 1984). In the sequel, we tacitly normalize the *L* whose centralizer we consider, and assume .

*Fact II*. In , if two operators commute with a given (non-constant) *L* they commute with each other. Therefore, any maximal-commutative subalgebra of must be a centralizer (of any one of its non-trivial elements). In several variables, of course the same statement is not true, as and both commute with , however, under favourable conditions the following analogue holds (Braverman *et al.* 1996), cf. also (Makar-Limanov 1978): if *X* is an *n*-dimensional manifold and an irreducible *n*-dimensional affine algebraic variety, then a ‘quantum completely integrable system (QCIS)’ on *X* is defined to be an embedding of algebras . Given a QCIS , then the centralizer of in is commutative. The main reason for this is that is finite-dimensional over , a general algebraic fact.

*Fact III*. (Schur 1905) In one variable,whereis an element of the algebra of formal pseudodifferential operators.and , with *d*=order *L*, an equation that can be solved recursively. Thus, can be thought of as the inverse of a local parameter *t* on the spectral curve . What is key here is that in one variable, a -algebra of transcendence degree 1 is finitely generated (by an inductive argument in the ring of formal power series in one variable). This is not longer true in higher dimensions, and we will analyse an example below (§6).

*Fact IV*. Burchnall and Chaundy posed the question of classifying ‘non-degenerate’ operators *L*, namely those whose centralizer is not a polynomial ring. As follows from fact III, these are rings of algebraic curves. We make the assumption that such a ring contain a normalized *L*, and the additional assumption that it has rank 1, defining the rank of a subset *S* of to be the greatest common divisor of the orders of all its elements. Then, Burchnall & Chaundy (1928) provides the solution to both direct and inverse spectral problem. We sketch the construction. If a non-degenerate commutative algebra of rank 1 is given, we can associate to it by fact III a spectral curve , even if is not maximal-commutative, and a line bundle (more precisely, a torsion-free sheaf) on a compact curve , whose fibre at each finite point P is the space of common eigenfunctions of , with eigenvalues the image of in . Burchnall and Chaundy's observation was that the operation of ‘transference’, which amounts to conjugation by the greatest common divisor of the ring (at a point P) gives an isospectral algebra and translates the line bundle on by . This shows that any point on the Jacobian can be reached by at most *g* steps if *g*=genus *X*, and since the inverse spectral construction then allows one to recover explicitly the coefficients of the operators of a commutative algebra in terms of the theta functions of , the classification is complete.

*Fact V*. The next aspect is that of isospectral deformations. Since the tangent directions to a Jacobian can be organized in an independent set of *g* commuting flows, one can view the coefficients of the operators as functions of several variables. It is also remarkable that one can define a connection on a trivial bundle over the infinite-dimensional affine space , whose fibre at each points is the ring , with zero curvaturewhere is the Zakharov–Shabat connection (Mulase 1984). This way, we obtain a hierarchy of nonlinear PDEs, called the KP hierarchy, satisfied by the Jacobian theta functions. It is little known that H. F. Baker wrote the KdV equation for genus-2 theta functions (Baker 1907; Matsutani 2000). There is, however, another flat connection defined on the theta bundle over , whose Fourier–Mukai transform gives the same equations, and we will return to this in §4. These two approaches should coincide for the particular KP solutions coming from curves, although the conversion, to my knowledge, has not been worked out; a language suited for both was developed in Álvarez-Vázquez *et al.* (1998), by giving a formal rendition of the Krichever map. This is the key technique in generalizing the KP hierarchy to higher-dimensional spectral varieties.

*Fact VI*. Sato's powerful interpretation of certain modules over the ring of formal pseudodifferential operators as a ‘universal Grassmann manifold’ UGM (Sato & Sato 1983; Sato 1989) finally gave a stunning geometric model for the KP hierarchy. We need again a quick review, which will be of use in §4 to illustrate its multivariable generalization.

The order of a pseudodifferential operator , defined to be the largest

*i*such that , gives and a filtration by subspaces and of elements of order less than or equal to*i*.The vector space has an induced filtration

and a ‘standard basis’ . The universal Grassmann manifold (UGM) is defined to be the set of subspaces *W* of *V* such that is finite. The KP hierarchy can be defined as a sequence of deformations on the dense open subset , which is in one-to-one correspondence with the following set of submodules of

The KP flows are finally linear on this setand the link with the flows on (cf. fact V) is the following: a -module of the type considered is cyclic, , where *W* is of the form and the equations for *W* becomeso that .

Sato's theory has provided us with a dictionary between the commutative algebras of operators we are considering and the geometry of an infinite-dimensional Grassmannian. The function is a formal eigenfunction for the ring with eigenvalue

To the -module , we can associate the space of formal power series , and we can view the deformations as multiplication of the transition function of a line bundle on a disc around the point at infinity. Further, note that all commutative algebras have been conjugated into the ring of pseudodifferential operators with constant coefficients, so that the focus has shifted from the study of commutative algebras to the study of one element such that is not a polynomial ring. Lastly, in the Grassmannian interpretation, Sato gave Plücker coordinates for the elements of and the explicit expression of the coefficients of *W* in terms of the ‘ function’, through an infinite-dimensional version of the Borel–Weil theorem (Pressley & Segal 1986).

### (b) Differential Galois theory

To my knowledge, the use of the automorphism group of a suitable differential field in order to study algebras of commuting differential operators has not been developed. I will outline four threads that I have found, and which seem to give four as yet unrelated strategies: (i) using a field generated by solutions of a given differential operator with finite Galois group, (ii) using a field generated by solutions to a system of nonlinear evolution equations, which turns out to be the function field of an Abelian variety where the system linearizes, (iii) using the field generated by the coefficients of a given differential operator, and (iv) using the field generated by the operator themselves.

Given any finite group

*G*, there exist both a Riemann surface*X*whose automorphism group is isomorphic to*G*(Madan & Rosen 1992), and a differential equation whose Picard–Vessiot group is*G*(van der Put & Ulmer 2000), although the proofs are non-constructive and it seems rather rare that an explicit differential equation with Picard–Vessiot group and coefficients that are meromorphic functions on*X*have been exhibited, such as the one in Hurwitz (1893) for the Klein quartic. On the other hand, differential equations with elliptic coefficients have been studied in great depth, but there does not seem to be much work available on their Picard–Vessiot extension. In Braverman*et al.*(1996) and Chalykh*et al.*(2003), however, the authors introduce differential Galois theory in the study of quantum integrable systems (roughly speaking, rings of commuting differential operators) and prove the following theorem.A quantum completely integrable system is algebraically integrable if and only if its differential Galois group is commutative.This criterion allows the authors to check a conjecture of Chalykh and Veselov for many classes of generalized Lamé operators (Chalykh*et al.*2003). In the coarsest possible terms, a quantum Hamiltonian system, namely a (maximal) commutative algebra of partial differential operators, is completely integrable if it contains ‘many’ operators and algebraically integrable if it has ‘few’ simultaneous eigenfunctions; this is the reason for the condition stated above on the Galois group, which acts on the solutions of a differential system viewed as functions of . In particular, the work in Chalykh*et al.*(2003) entails looking closely at some beautiful geometric properties of PDOs with (generalized) elliptic coefficients. We now switch to the second point of view.Differential algebra, or ‘Differential equations from an algebraic standpoint’ in the original title of Ritt (1950), a field pioneered by J. F. Ritt and developed mostly by E. R. Kolchin, consists in viewing differential fields as function fields of ‘differential manifolds’ much in the same way as algebraic geometry associates a field of rational functions to an algebraic variety. The scope of differential algebra cannot be conveyed in a few paragraphs. Rather, leaving aside a description of its nature, links with solution of differential equations, and achievements, I will use an example given in Buium (1986), to offer a view of integrable systems that has not been pursued, and to advocate this method to be extended to the case of partial differential algebra, in which barely anything seems to be known (Ritt (1950) seems the only place where the general theory is laid out). Of course, there have been extensive developments in the theory of -modules, but it seems to me that this simpler approach may be better suited for certain questions. Buium's point of view is that differential function fields with no moveable singularity ‘lead to systems which can be linearized by means of Abelian functions’, and since the algebraically completely integrable Hamiltonian systems have this property, they should be made to fit into the differential theory. Our example in hindsight is just a, possibly singular, elliptic curve

*F*. Think of its functions as a differential field as follows: if where are indeterminates and the derivation is defined bywhere*a*is a non-zero number and are generic (singular varieties may arise if they are not), then and (the invariants of Euler's equations for a rigid body with fixed centre of gravity) are such that . Let*L*be the algebraic closure of , and*F*be the subvariety of given by equationsThen*F*is an elliptic curve, which gives a non-singular projective model of since it is birationally isomorphic to the image of the embedding given by:Moreover, has no movable singularity, unlike whose divisor of movable singularity equals the plane , because (if are homogeneous coordinates for which ). Indeed, it can be checked that , by constructing a global one-form on*F*for which , because this guarantees that is a generator of the*L*-vector space of global vector fields over*F*. Letwhere for (the equations for*F*become ). In local coordinates, on the open set defined by , for . This setting gives not only a beautiful dictionary between differential equations and geometry, but also information about the solutions which can be expressed in terms of theta functions for the elliptic curve.Here, one decides to use the independent variable as a function on a variety, and we will just look at two examples.

The key idea that Hurwitz (1893) used to find the Fuchsian differential equation:whose differential Galois group is the same as the automorphism group of the Klein curve (Singer & Ulmer 1993),was to use as independent variable

*J*a basic invariant, preserved by the 168 projective linear transformations that fix the quartic. In a similar vein, linear differential operators with elliptic coefficients, which were extensively studied in the nineteenth century, may belong to an ‘algebraic class’.For the Lamé operators(where is the Weierstrass -function for a given lattice), the existence of a differential operator

*B*of odd order that commutes with*L*is equivalent to the condition , for an(y) integer*n*. The object that encodes the data of solution: does not so much correspond to the field generated by the solutions, as to the points of the common spectrum , an algebraic curve*X*. The operator has eigenfunctions that can be expressed algebraically in terms of elliptic functions (Whittaker & Watson 1996*A*) for those eigenvalues that are branchpoints of the curve*X*. Both the independent variable*x*and the common eigenfunction, a normalized section of a line bundle over*X*, can then be viewed as depending on a point of , or more precisely on , since Abelian varieties are self-dual, via a geometric Fourier transform. Possibly, then, these different problems, to each of which there corresponds a Picard–Vessiot group, one acting on the differential field and the other on can be viewed as ‘embedded’ into one another. I do not think this has been done nor that there was given a way to relate one concept of ‘algebraicness’ for a differential operator to the other.As was explained in facts I–IV above, commutative algebras correspond to spectral curves. In this context, the linear disjointness theorem for differential rings allows us to revisit and complete an elementary observation, proposed (but not pursued) by Wilson (1985); this proof grew out of discussions with Phyllis J. Cassidy, during a visit within the special programme ‘Model Theory of Fields (MSRI, spring 1998)’ (MSRI's hospitality under NSF Grant DMS-9701755 is gratefully acknowledged) and is a joint result.

*If L is a normalized element of**with order L*>0,*and**is such that gcd*(*orderB*,*orderL*)=1*and the differential resultant**is independent of x*,*then*.The differential resultant (cf. e.g. Previato 1991) of two differential operators, analogous to the algebraic resultant, is the determinant of a matrix whose entries are differential polynomials in the coefficients of the operators, which vanishes if and only if the operators have a common eigenfunction. We will define and study a generalization to several variables in section 6 which includes the one-variable as a special case.In one direction, the result was known to Burchnall & Chaundy (1923), but we revisit the proof which otherwise relies on

*ad hoc*calculations. Assume that*L*and*B*commute. Then the coefficients of are independent of*x*by ‘linear disjointness’. Indeed, on the common eigenvalues of*L*and of*B*gives a dependence relation on monomials in whose coefficients*a priori*depend on*x*. Inside the differential algebra , the subalgebras with viewed as functions on the plane curve given by the equation and the subalgebra of operators whose coefficients are differential polynomials in those of*L*and*B*are clearly linearly disjoint over , so that monomials in which are linearly dependent over must be linearly dependent over . Conversely, we assume that the determinant of the resultant matrix be independent of*x*, where*m*=order*L*and*n*=order*B*. If , are fundamental systems of solutions for , , where is a point of the curve ), then by multiplying the matrix by the Wronskian matrix of*f*'s,*g*'s and their derivatives, we see that there exist two operators*R, T*of orders strictly less than*n, m*, respectively, such that (since this operator has independent solutions). This shows that*L, B*have a common eigenfunction for all eigenvalues and this would be an infinite dimensional family, yielding . If this is the case, then Burchnall and Chaundy showed that*L, B*commute (Burchnall & Chaundy 1923). ▪It may be possible to give another proof by differential algebra, as follows: for a given pair satisfying the assumptions, the differential ideals generated by the conditions, and have constant coefficients, are the same. We checked this for , by introducing a grading for the differential indeterminates

*b, c*(for fixed*a*in a differential field), and using differential Gröbner bases and Ritt's definition of characteristic set for a differential ideal.

## 4. Sato's ring

Sato's striking interpretation of the KP hierarchy as the Plücker equations for an infinite dimensional Grassmannian (Sato & Sato 1983; Sato 1989) takes place in the ring as we saw in 3.1, fact VI. It is in this context that he proposed a multivariable generalization (Sato 1989) again by deformations of -modules. Nakayashiki (1991, 1994) fulfilled that project: he produced rings of matrices whose entries are differential operators in *g* variables, isomorphic to the ring where *A* is a *g*-dimensional principally polarized Abelian variety and a smooth theta divisor; note that cannot be a Jacobian as soon as (Andreotti & Mayer 1967). In Nakayashiki (1991), a sequence of time derivatives are introduced in the data, so as to let them flow (linearly on Pic) and derive the analogue of the KP equations for the function. I suspect that these can be translated into the equations that Barsotti introduced in Barsotti (1983), and which characterize theta functions, so that the characterization of Jacobian theta functions by KP equations (‘Novikov's Conjecture’; cf. Taĭmanov 1997) extends to theta functions of general Abelian varieties by these generalized KP. Rothstein revisited this work (Rothstein 1996*a*,*b*) and defined some generalized flows attached to extra choices of trivializations (Rothstein 2002), and Mironov worked out the coefficients for a Nakayashiki ring in two variables in terms of theta functions (Mironov 2000). In this section, we briefly reproduce Nakayashiki's construction because it would seem worthwhile to pursue an explicit expression for the rings, the equations and the solutions, as well as a relationship with the generalized KP hierarchy to be given in §5. Our goal is to sketch the idea and connect it with Sato's set-up from §3, but we omit detailed formulae that can be found in Nakayashiki (1991).

Let *X* be a smooth projective variety, , an ample divisor, the Picard variety of *X* and the Poincaré line bundle. By choosing a standard basis of the torsion-free part of and normalized holomorphic differentials on *X*, the ring of differential operators can be identified with , where are coordinates on the universal cover of .

Let be the Fourier–Mukai transform (Mukai 1981) of the sheaf , where are the two projections from , and let be a normalized basis of differentials of the second kind with poles only on

*D*. Then can be given the structure of a -module by the connections viewed as operators on the sections (, a base point, and*z*, are points of the universal cover of*X*). with this structure is called a Baker–Akhiezer module. If is the stalk at , where is the projection of a point*c*of the universal cover of , then the elements of are called Baker–Akhiezer functions. When*X*is a curve of genus*g*, these coincide with the Baker–Akhiezer functions defined by Krichever, if*D*is a point .When

*X*=*A*is a principally polarized Abelian variety of dimension*g*, and a theta divisor, then can be identified with*A*by , where is the translation map. Therefore, if is the bundle over*A*defined by the cocycle ), where is the period matrix and , and , then the theta functions with characteristicsgive a basis of , andwhere , and the normalized line bundle satisfies .If is smooth, then is generated over by for any . Moreover, if , is a free -module of rank , and the filtration is such that the action of first-order differential operators satisfies for a sufficiently large

*i*. Then the mapdefined by , where the vector gives a basis of , gives an isomorphism of the ring of functions on*A*with poles at most on (of any multiplicity) with a (commutative) ring of matrices of differential operators on*A*. Note that if we identify with the ring of convergent power series , the entries of the matrices are elements of the ring of differential operators we considered in §3.Analogue of the KP flows. Sato in Sato (1989), remarked that there is no natural multivariable generalization of the KP hierarchy, because for the analogous rings , filtered by order of , there is no natural choice of a free left submodule of , for deforming -submodules of . He then advocated the above-described algebro-geometric example, where the ‘codirection’ is naturally given by an equation for the theta divisor, under the identification with local parameters near a point of the spectral variety: Indeed, the following time deformations are defined in Nakayashiki (1991):where we set , and denote by the vector if ; denotes . can be embedded in as a -submodule, , in such a way that for and the -submodule of , satisfies where is a suitable collection of indices from , and . Then , for in the complementary index set, and are suitable -generators of of the form + [an operator whose terms have multi-indices belonging to ].

As observed in Mironov (2002

*a*), the functions inare independent of the time variables, so there is not really a deformation beyond the*g*-dimensional variety , which indeed is . In Mironov (2002*b*),*k*time deformations independent of the ‘spectral variables’ are constructed, by using as spectral variety the transverse intersection of*k*translates of the theta divisor (loosely speaking), at the cost of increasing the size of the matrix representation. In Rothstein (2002), similar methods are applied to define (in principle) time flows on extensions of , for a specific*X*of dimension 2 (the Fano surface of lines on a smooth cubic threefold) whose has dimension 5.

## 5. Parshin's ring

In Parshin (1999), Parshin proposed a different construction, based on the theory of higher local fields, in which the commuting partial differential operators are scalar. In (Lee 2002*a*,*b*), Lee was able to give a Sato–Grassmannian interpretation of Parshin's rings; see also (Plaza-Martín 2000). We observe that Parshin's ring contains Sato's ring properly; the two constructions have not yet been compared, but this should be possible at least in the geometric sense of (Osipov 2001). An *n*-dimensional local field *K* (with ‘last’ residue field ) is the field of iterated Laurent series , with structure of a complete discrete valuation ring with residue field an -dimensional local field. Note that the order of the variables matters, in the sense that does not contain the same elements of , e.g. the former contains elements of unbounded positive degree in , although they are isomorphic. These are suited to give local coordinates on an *n*-dimensional manifold, since the inverse of a polynomial in , say, can be written as the inverse of the highest-order monomial times something entire, so as a Laurent series it is bounded in both variables. Whereas, the symbols cannot be given a ring structure unless we want to define sums of infinitely many complex numbers, because involves infinitely many indices unless we bound *j* (or *i*) from above. With this definition, Parshin constructs a -dimensional skew-field , infinite-dimensional over its centre, namely the (formal) pseudodifferential operators,

The order of the variables is also singled out in the definition of the grading.

If with , we say that the operator *L* has order *m* and write . If , then is a decreasing filtration of by subspaces and , where and consists of operators involving only nonnegative powers of . The highest term (h.t.) of an operator *L* is defined by induction on *n*. If and order *L*=*m*, then . If with , then we let . We consider also the subring of , and .

Note that *E* is much larger than Sato's ring when *n*>1.

(P)

*An operator**is invertible in E if and only if the coefficient f in the highest-order term of L is invertible in the ring*.*If f in**is an m*-*th power in*(*resp*.*for*)*then there exists*,*unique up to multiplication by m-th root of unity, an operator*(*resp*. )*such that*.*Thus*,*is a discrete valuation ring in**with residue field*.*Let*.*Then**for all i, j if and only if there exists an operator**such that*,*for all i*.*For**as in*(ii),*the flows**commute*,*and if**satisfies**then**evolves according to them*.The Baker function associated to an operator is defined to be where , for an auxiliary set of variables . Sincethe Baker function has the form where is a formal power series in and for an

*L*that flows as defined above.For a vector , the operator on functions is defined by again with the multi-index notation (here ). A -function for the operator

*S*is a formal solutions of the equation . It can be shown that such a function exists by solving recursively for its coefficients.

It should be interesting to work out an analytic Grassmannian suited to the Parshin flows, using these formal solutions to interpret suitable subspaces as -modules. This is the intent of the framework in Dupré

*et al.*(2004); in Lee (2003), a suitable formal Grassmannian is defined, basically modelled on the power series in the last variable.

## 6. Toy models

The question of classification of commutative subalgebras of the Weyl algebra seems quite difficult for ; in fact the classification is not explicit even when *n*=1, unless the algebras have rank 1, cf. §3. This would be a subcase of operators with more general coefficients; polynomial/rational coefficients have proved very robust under the bispectral involution ( in the notation above; Horozov 2002) and this is one more reason why some calculations are possible. In Kasman & Previato (2001) we posed two questions, and in §6 we briefly recall the answers we reached; the method should be capable of application to a geometric theory of the many examples (§§3 and 4) of commutative rings.

*Question 1*. Does a maximal-commutative subalgebra of in *n*>1 variables have to be finitely generated, as in *n*=1? The answer is no. Note that when it is, can be viewed as an affine variety, and it turns out to have dimension (Braverman *et al.* 1996).

*Question 2*. When we have a finitely generated subalgebra of , is there a way to identify explicitly the variety , for example by algebraic equations satisfied by the generators, given by a differential resultant, as was the case for dimension 1 (cf. §3)? Here we did not find an answer, other than provide a (very inefficient) definition of a differential resultant which is an algebraic equation, and remark that it will be identically zero in most cases! We conclude this paper with some suggestions that can be explicitly tested and would provide theoretical information.

*The answer to question 1*. We use Darboux transformations to construct commutative rings of operators with non-constant coefficients from constant-coefficients ones; we use algebraic arguments (induction on the degree; calculations on eigenfunctions of constant-coefficients operators) to show that one of these algebras is maximal-commutative, and not finitely generated. One of the simplest examples is: let (for any parameter ), and let . Then is the centralizer ofis maximal-commutative in , and is not finitely generated as a ring over .

*A method for question 2*. We used Macaulay's definition of algebraic resultant for polynomials in *n* variables to calculate a polynomial in variables that a set of commuting operators in *n* variables satisfies identically: . We note that for the example given above the polynomial *p* is identically zero, so it does not give any relation of algebraic dependence, and there is no ‘geometric’ reason, unlike for other examples where the variety has no relevant points in affine space but does have points at infinity.

*Question 3*. We were unable to show that, in analogy to the case of one variable, the coefficients of the resultant are independent of . This is true in the examples we computed and should hold for theoretical reasons: translation in should give an isospectral ring. In several variables, it is not at all clear how to construct common eigenfunctions of commuting operators, cf. Chalykh *et al.* 2003, but at least for the cases when these are known, such as (separable) Schrödinger potentials (§§2, 3) those associated by Nakayashiki to Abelian varieties (§4), and those constructed from some (polynomial) convergent -functions (§5), this result ought to be proved.

## Acknowledgments

Partial support under grant NSF-DMS-0205643 as well as support by the Weyl Fund at IAS (2002–2003) are very gratefully acknowledged. Also, much gratitude is extended to the anonymous referee for a very detailed reading, which resulted in important updates of results and citations.

## Footnotes

One contribution of 15 to a Theme Issue ‘30 years of finite-gap integration’.

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