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University of California at San Diego

arXiv:hep-th/9302105v1 22 Feb 1993Super Elliptic CurvesJeffrey M. RabinDepartment of MathematicsUniversity of California at San DiegoLa Jolla, CA 92093jrabin@ucsd.eduJanuary 1993AbstractA detailed study is made of super elliptic curves, namely super Riemann surfaces of genusone considered as algebraic varieties, particularly their relation with their Picard groups. Thisis the simplest setting in which to study the geometric consequences of the fact that certaincohomology groups of super Riemann surfaces are not freely generated modules.

The divisortheory of Rosly, Schwarz, and Voronov gives a map from a supertorus to its Picard group,but this map is a projection, not an isomorphism as it is for ordinary tori. The geometricrealization of the addition law on Pic via intersections of the supertorus with superlines inprojective space is described.

The isomorphisms of Pic with the Jacobian and the divisor classgroup are verified. All possible isogenies, or surjective holomorphic maps between supertori, aredetermined and shown to induce homomorphisms of the Picard groups.

Finally, the solutionsto the new super Kadomtsev–Petviashvili (super KP) hierarchy of Mulase–Rabin which arisefrom super elliptic curves via the Krichever construction are exhibited.1IntroductionThe theory of elliptic curves [1, 2] is not only a rich and fascinating subject in its own right,but a confluence of several major branches of mathematics and a source of simple and explicitlycomputable examples in each. These include Riemann surfaces, algebraic groups, Abelian varieties,divisor theory, Diophantine equations, mapping class groups, and automorphic functions.Thesimple modular properties of the torus are of particular importance in conformal field theory owingto the sewing axioms, by virtue of which modular invariance on the torus guarantees this invarianceat higher genus, and in the related theory of elliptic genera.The study of super elliptic curves, meaning super Riemann surfaces of genus one considered asalgebraic varieties, was initiated in [3, 4] with the use of superelliptic functions and super thetafunctions to embed supertori in projective superspace as the sets of zeros of explicit polynomialequations, generalizing the Weierstrass equation for an elliptic curve.

Missing from this work wasany discussion of the group law on a superelliptic curve. Associated to any Riemann surface isits Picard group or Jacobian, the group of line bundles of degree zero on the surface under tensor1

product. A torus is itself a group because it is isomorphic to its Jacobian via the classical Abel map.The situation for supertori is more complicated because the Abel map turns out to be a projectionrather than an isomorphism.

The proof of this fact and extensive discussion of its consequences forthe theory of superelliptic curves are the subjects of this paper.We study specifically the supertorus with odd spin structure given informally (a more precisedefinition follows) as the quotient M = C1,1/G of the complex superplane with coordinates (z, θ)by the supertranslation group G generated by the transformationsT :z →z + 1,θ →θ,S :z →z + τ + θδ,θ →θ + δ. (1)The odd spin structure is of interest precisely because of the presence of the odd modular parameter δin addition to the usual even one τ (with the modulus τ and the theta functions, we will tolerate someexceptions to the standard convention that Greek letters denote odd quantities while Roman lettersdenote even ones).

Meromorphic functions on the supertorus are just meromorphic functions F(z, θ)on C1,1 which are G-invariant, or superelliptic. In particular, the cohomology group H0(M, O)consisting of global holomorphic functions is easily shown to be the set of functions a + αθ withconstant coefficients a, α such that αδ = 0.

For even functions, a should be even and α odd. Owingto the constraint on α, this is not simply the vector superspace C1,1 with basis {1, θ}; it is indeeda module over the Grassmann algebra Λ containing all our odd parameters, but this module isnot freely generated.

This situation occurs generically for super Riemann surfaces with odd spinstructure [5, 6] and its implications are not well understood in general. The primary motivationfor this work was to study them in this simplest case, in which complete, explicit calculations arepossible and illuminating.One is so accustomed to the fact that a sheaf cohomology group Hi(M, F) carries the structureof a finite-dimensional vector space that one forgets that the proof is nontrivial [7].

Certainly theexistence of this structure is so central to geometric applications of cohomology that one wouldhardly know where to begin without it: the Riemann-Roch theorem is only the simplest of the toolsdesigned to compute the dimensions of these vector spaces. Such tools only generalize in the supercase for generic even spin structures, the “normal case” considered in [8].

The lack of a super vectorspace structure causes difficulties in the theory and applications of super Riemann surfaces whenevera basis for a cohomology space of functions, differentials, or deformations would be desirable. (Oneshould note that because Λ is a Grassmann algebra over C, the cohomology groups do have vectorspace structures over C. However, because one may wish to vary the Grassmann algebra, it isthe module structure over Λ that is of interest.

)Certainly the super Riemann-Roch theoremholds only in the normal case or under additional assumptions. In the application to superstrings,bases for spaces of holomorphic differentials of various weights are normally used to express thesuperdeterminants appearing in the path integral measure and in the expressions for amplitudesused in finiteness and unitarity proofs.

These analyses are considerably more complicated whensuch free bases do not exist [9]. The geometry associated to the super KP hierarchies [10, 11] wouldnormally be described in terms of a super Grassmannian of vector subspaces of, say, functions on thesupercircle [12] with the Krichever map sending a supercurve and additional geometric data to thevector subspace given by a suitable cohomology group.

When the cohomology lacks a vector spacestructure, this construction fails.Presumably the correct super Grassmannian contains certainΛ-submodules as well as free subspaces, but the specific class of submodules and the geometry of2

the resulting Grassmannian have not been elucidated. The operator formalism for fermionic stringmakes use of the same Grassmannian structures [13], and the modifications which might be requiredhere in the non-normal case deserve investigation.

Sometimes the problem is finessed by consideringsplit supercurves (no supermoduli) and transporting the results to the rest of supermoduli space byusing the fermionic stress tensor as a connection. Fully justifying this procedure would require anunderstanding of how the stress tensor encodes the structure of the submodules during transportthrough the Grassmannian.This paper concentrates on how the non-free character of the cohomology affects the geometryof a superelliptic curve, particularly its relation to its Jacobian.

Section 2 develops the basics offunction theory. We exhibit the building blocks for the explicit construction of functions, the superanalogues of Weierstrass ℘functions and theta functions, as well as deriving the general constraintson the divisor of a superelliptic function.

Because the canonical bundle of a superelliptic curveis trivial, this analysis applies to meromorphic differentials of all weights as well as to functions.In Section 3 we explicitly compute the Picard group (group of line bundles), the Jacobian (spaceof linear functionals modulo periods), and the divisor class group (divisors modulo divisors offunctions) of a superelliptic curve, verifying that they are all isomorphic. This isomorphism hasbeen proven for all super Riemann surfaces in the normal case [8], but not more generally thusfar.

The Abel map from the curve to its Jacobian is obtained and observed to be a projection π:it takes the quotient of the curve by the relation (z, θ) ∼(z + αδ, θ) for all α. The origin of thisextra identification is traced to the necessity of abelianizing the nonabelian group G in order forthe quotient to admit a group structure.

Section 4 shows that, modulo this identification and anambiguity in the choice of identity element, the group operation on the Jacobian can be performedgeometrically on the curve by intersecting it with special planes in the standard superprojectiveembedding.Section 5 determines all the isogenies of superelliptic curves.These are surjectiveholomorphic mappings between supertori. For elliptic curves one proves that they are necessarilyhomomorphisms in the group structure.

Here, since a superelliptic curve does not carry the groupstructure of its Jacobian, the best one can do is to show that an isogeny induces a homomorphism ofthe Jacobians via the projection π. We also study isogenies of a superelliptic curve to itself and showthat a nonsplit curve admits only trivial endomorphisms.

Section 6 contains a major applicationof these results to the new super KP system discovered by Mulase and the author [10, 11]. Thissystem of nonlinear PDEs for the coefficients of a pseudosuperdifferential operator describes, viathe Krichever construction, the deformation of a line bundle L over an algebraic supercurve bycertain commuting flows in the Jacobian.

The pseudodifferential operator is closely related to aspecial section of L called the Baker-Akhiezer function. The algebraic supercurves involved aregenerally not super Riemann surfaces except in the special case of genus one.

In this exceptionalcase we can construct explicit solutions to the super KP system describing flows in the Jacobianof a superelliptic curve, in terms of Weierstrass elliptic functions. The result can be presented asan isomorphism between a ring of meromorphic functions on the superelliptic curve and a ring ofsupercommuting differential operators [10, 14].

It generalizes the classical result that the operatorsQ=d2dx2 −2℘(x + a),(2)P=Q3/2+= d3dx3 −3℘(x + a) ddx −32℘′(x + a)(3)arising from an elliptic curve generate a commutative ring. The parameter a should be viewed as3

a coordinate on the Jacobian and varies linearly with the flow parameters. A new feature of thesuper case is that the supercommutativity of the ring depends upon the fact that the theta functionsatisfies the heat equation.

Section 7 contains conclusions and directions for further research. AnAppendix briefly considers the problem of finding rational points on superelliptic curves.

Here thenilpotent elements of Λ linearize the problem to locating rational points on the (co)tangent lineto an elliptic curve at a rational point, so nothing of number-theoretic interest has been added.Throughout this paper, computations which employ standard methods are nevertheless given inconsiderable detail, so as to remove any mystery from the supermodulus δ and display clearly therole it plays in modifying the classical results.Before proceeding, let us return to the precise definition of the superelliptic curves we study. Wefix a finite-dimensional complex Grassmann (exterior) algebra Λ in which δ is an odd element and τan even one with Im τrd > 0.

(Throughout this paper the subscript “rd” on a Grassmann variable,supermanifold, supergroup, etc. denotes the reduction of this object by modding out the ideal ofnilpotents in Λ or in the structure sheaf.) We adopt the standard sheaf-theoretic treatment ofsupermanifolds [15] within which we are really dealing with families of superelliptic curves over theparameter superspace B = (pt, Λ).

Our covering space, informally denoted C1,1, is really the trivialfamily C1,1 × B, meaning the complex plane C equipped with the structure sheaf OC ⊗Λ[θ], whereΛ[θ] is the larger Grassmann algebra whose generators are θ and the generators of Λ. The family ofsuperelliptic curves M over B is the quotient of this family by the group G, meaning the following.The reduced space of M is the standard torus Mrd with modular parameter τrd.

The structuresheaf of M assigns to any open set U of Mrd the following ring OU. U is covered by a collection ofconnected open sets Ui of C. To each element g in G there corresponds a transformation grd in thereduced group Grd generated byTrd :z →z + 1,Srd :z →z + τrd,(4)which maps each Ui to some (possibly the same) Uj.

For OU we take all collections of functions{Fi(z, θ) ∈OUi} which are G-invariant in the sense that Fj(z, θ) = gFi(z, θ) whenever Uj = grdUi,g ∈G. Here g acts on functions via Taylor expansion in nilpotents as usual: F(z + τ + θδ, θ + δ)means F(z+τ, θ)+θδ∂zF(z+τ, θ)+δ∂θF(z+τ, θ).

If τ has a nilpotent part then the last expressionis defined by further Taylor expansion in this nilpotent part. The statement that ρ : M →B is afamily means that there is a pullback map of the functions Λ on B to functions on M; the elementsof Λ play the role of global constant functions on M and as such all the cohomology groups of Mare modules over Λ (or its even part if the sheaf is purely even or odd).For those readers less comfortable with sheaf-theoretic language, which often includes the author,we can consider the set of Λ-valued points of M rather than M itself.

This is the set of (even)maps B →Mρ→B for which the composed map B →B is the identity. For each point of Mthis is an evaluation of functions at that point by assigning even and odd values from Λ to thecoordinates z and θ respectively.

That is, it is just an abstract description of the Λ-supermanifoldsof [8], or the supermanifolds of DeWitt [16] or Rogers [17], which are genuine sets of points withGrassmann-valued coordinates. The Picard and Jacobian groups as defined here naturally appearas such sets of Λ-valued points and will be discussed as such; our constructions can be translatedinto pure sheaf-theoretic terms by those readers with the sophistication to prefer this viewpoint.The choice of Grassmann algebra will usually be left open, but two cases are worth distinguishing.4

One is the case in which δ is one of the generators of Λ. The most important example is the two-dimensional algebra having δ as its only generator (plus unity); if we let τ run through the upperhalf-plane this gives the universal Teichm¨uller family of supertori (apart from the identification of±δ).

The other is the general case in which δ is an element of Λ but not necessarily a generator.Such a family is a pullback of the universal family by a map of the base spaces, which indeed pullsback δ to some element of Λ, e.g. δ = β1β2β3 in terms of generators βi.

The most importantdistinction between these cases is that when δ is a generator it annihilates only multiples of itself,while in general it may annihilate other elements as well, e.g. multiples of β1 in the above example.2Basic Function TheoryIn order to construct explicit functions and sections of bundles on the supertorus M, in particular theBaker-Akhiezer function appearing in super KP theory, we need the building blocks correspondingto the Weierstrass elliptic function ℘(z; τ) and the theta function Θ(z; τ) (the capital letter is usedfor theta functions in this paper to avoid confusion with the coordinate θ) introduced in [4].The super Weierstrass function isR(z, θ; τ, δ) = ℘(z; τ + θδ) = ℘(z; τ) + θδ ˙℘(z; τ),(5)where by convention a dot denotes ∂τ while a prime will mean ∂z.

It is superelliptic, as are itssupercovariant derivatives DnR, where D = ∂θ + θ∂z commutes with the generators of G andsatisfies D2 = ∂z. These functions provide the standard embedding of M in projective superspacewhich we will recall in Section 4.Similarly, our super theta function will beH(z, θ; τ, δ) = Θ(z; τ + θδ).

(6)The ordinary theta function appearing here is the one often denoted Θ" 1212#(z; τ), which corre-sponds to the odd spin structure. It has a simple zero at z = 0 and the other lattice points, andsatisfiesΘ(z + 1; τ)=−Θ(z; τ) = Θ(−z; τ),Θ(z + τ; τ)=−e−πiτ−2πizΘ(z; τ).

(7)As a result, the super theta function satisfiesH(z + 1, θ) = −H(z, θ) = H(−z, θ),H(z + τ + θδ, θ + δ) = −e−πiτ−πiθδ−2πizH(z, θ),(8)where the moduli dependence of H has been suppressed. The relation between Θ and ℘is [18]d2dz2 log Θ(z; τ) = −℘(z; τ) + q,q = Θ′′′(0; τ)3Θ′(0; τ).

(9)5

The first derivative ∂z log Θ is nearly elliptic, being invariant under z →z + 1 and changing by anadditive constant under z →z + τ. Since this is also the behavior of θ according to (1), we canform the superelliptic combination [19]σ(z, θ; τ, δ) = θ + δ2πiddz log Θ(z; τ)(10)which reduces to θ in the split case where δ = 0.

This function will be of particular importance inview of the fact that it is holomorphic in the split case (when cohomology is freely generated) butonly meromorphic otherwise.To describe the meromorphic functions on M and construct them from the building blocksabove, we turn to the study of divisor theory. In the usual Cartier divisor theory, a divisor wouldbe a subvariety of codimension (1, 0), hence dimension (0, 1), given locally by an even equationF(z, θ) = 0.

The fact that such divisors are not points breaks the strong analogy between ellipticand superelliptic curves. It was the great insight of Rosly, Schwarz, and Voronov [8] (see also [20])to make use of the covariant derivative D (the superconformal structure) which exists locally on anysuper Riemann surface to define divisors of codimension (1, 1) — points — via the simultaneoussolutions ofF(z, θ) = 0,DF(z, θ) = 0.

(11)For any even function F for which the reduced function Frd(z) is not identically zero, a point (Λ-valued!) (z0, θ0) satisfying these equations is called a principal zero of F. If we write F(z, θ) =f(z)+θφ(z) and assume that (z0)rd is a simple zero of frd (in this case we are discussing a principalsimple zero of F), this amounts to the statementsf(z0) = 0,θ0 = −φ(z0)/f ′(z0).

(12)A principal pole of F is a principal zero of 1/F. A formal sum of points P niPi is a divisor of Fprovided that in a chart containing Pi = (zi, θi) we can writeF(z, θ) = E(z, θ)Yi(z −zi −θθi)ni,(13)where the product is over the Pi contained in the chart and E is holomorphic with Erd ̸= 0 inthis chart [it may not be possible to separate all the points Pi because the corresponding reducedpoints (zi)rd may coincide].

A subtlety is that a single function may have more than one divisor ifits zeros and poles are not simple. For example, on C1,1, F = (z + a)2 = z(z + 2a) with nilpotenteven constant a satisfying a2 = 0 has the two distinct divisors of zeros 2(−a, 0) and (0, 0)+(−2a, 0)as well as others.

On the supertorus, R(z, θ) has a principal double pole at (0, 0) and two simplezeros. The super theta function H(z, θ), actually a section of a bundle rather than a function, hasa principal simple zero at (0, 0).We now derive the necessary and sufficient condition for a divisorP niPi to be a divisor of somemeromorphic function F on M: the sum of the Pi with multiplicity must differ from a lattice pointby (αδ, 0) for some constant α, namelyXiniθi=nδ,Xinizi=m + nτ + αδ,(14)6

pTpSTpSp1234Figure 1: The period parallelogram, an integration contour for the proof of sum rules for the divisorof a superelliptic function. Except for orientation, sides 1 and 3 are related by the supertranslationS, sides 2 and 4 by T.for integers m, n. Of course, the total degree Pi ni must also vanish because it vanishes for thedivisor of the reduced function on the torus Mrd.The proof of the necessity follows the classical and elementary proof for elliptic curves [1] byintegrating DF/F = D log F around a period parallelogram as shown in Fig.

1, chosen to avoidthe points of the divisor. An easy computation shows that near a principal pole or zero where Fbehaves as (z −zi −θθi)ni, we haveDFF∼ni(θ −θi)z −zi −θθi= ni(θ −θi)z −zi,(15)plus holomorphic terms.

Then we evaluate the following two contour integrals (For details on thedefinition of super contour integration, see [21, 22, 23]. For closed contours it is simply Berezinintegration over θ followed by ordinary contour integration.

For an open contour lying in a simplyconnected region in which F is holomorphic, it is the change in an antiderivative Φ, with DΦ = F,between the endpoints. ):IθDFF dz=XiI −niθθiz −zidz dθ=2πiXiniθi,(16)and similarlyIzDFF dz=XiInizθz −zidz dθ=XiniI 1 +ziz −zidz=2πiXinizi.

(17)Next we evaluate the integrals over each side of the parallelogram and use the fact that F isthe same on opposite sides by superellipticity. For the first integral we note that θ is the same onsides 2 and 4, which have opposite orientations, so those contributions cancel, while sides 1 and 3are related by θ →θ + δ.

The Jacobian factors relating these integrals are unity, which is also clear7

from the antiderivative definition and the fact that D commutes with the generators of G. Hencethese contributions sum toIθDFF dz=Z1 −δDFF dz = −δZ1 D log Fdz=2πinδ,(18)the point being that only the reduced part of log F is multivalued, the nilpotent part involvingderivatives of log via the Taylor expansion. Comparing with the previous evaluation of the integralgives the sum rule for θi.

For the z integral things are slightly more complicated. Sides 1 and 3 arerelated by z →z + τ + θδ, sides 2 and 4 by z →z + 1.

Making these substitutions givesIzDFF dz=Z1(−τ −θδ)DFF dz +Z2DFF dz=2πi(m + nτ) + δZ1 θDFF dz,(19)where the last integral can have any odd value. Calling it −2πiα, we obtain the sum rule for zi.To show the sufficiency, we construct a function having any given divisor satisfying the sumrules in terms of the super theta function.

First we note the effect of a supertranslation on thedivisor of a function: if F(z, θ) has the behavior (z −zi −θθi)ni corresponding to a principal zeroor pole at (zi, θi), thenF(z −a −θǫ, θ −ǫ) ∼[z −(zi + a + θiǫ) −θ(θi + ǫ)]ni,(20)shifting the zero or pole to (zi+a+θiǫ, θi+ǫ). The odd coordinates of the divisor are shifted uniformlyby ǫ, the even coordinates uniformly by a but also nonuniformly by a term proportional to the oddcoordinates.

This changes the sum of the zi by a multiple of the sum of the θi, which is a multipleof δ, consistent with the sum rule for zi. In particular, the theta function H(z −zi −θθi, θ −θi) isholomorphic with a principal simple zero at (zi, θi).Unfortunately, this theta function is not convenient for our purposes since it does not transformby a mere phase under the group G. As a consequence of the commutation relations of supertrans-lations, the generator S sends it to a phase times H(z−zi −θθi−2δθi, θ−θi).

However, the functionH(z −zi −θθi, θ + θi) also has a principal simple zero at (zi, θi) and transforms asSH(z −zi −θθi, θ + θi) = −e−πi[τ+(θ+θi)δ+2(z−zi−θθi)]H(z −zi −θθi, θ + θi). (21)This remedy of changing the relative sign in θ −θi amounts to the usual replacement of a SUSYgenerator by a SUSY covariant derivative.Let us suppose first that Pi niPi is a degree-zero divisor for which the Pi sum exactly to a latticepoint, with no remainder αδ.

By adding the fictitious points (0, 0) −(m + nτ, nδ) we can assumethat the Pi sum to zero without changing the divisor on M. Then a superelliptic function with thisdivisor isF(z, θ) =Yi[H(z −zi −θθi, θ + θi)]ni. (22)Its invariance under the generators of the group G is easily checked using the relation (21) and thesum rules (14).8

The simplest example of a degree-zero divisor satisfying the sum rules with a nontrivial remainderαδ is ∆= (αδ, 0) −(0, 0). A meromorphic function with this divisor is easily constructed from thefunction σ introduced in Eq.

(10), namelyF∆(z, θ) = 1 −2πiασ(z, θ) = (1 −2πiαθ)[1 −αδ ddz log Θ(z; τ)],(23)where the second form shows the behavior 1 −αδz near z = 0 dictated by the divisor. Now, givenan arbitrary divisor satisfying the sum rules, subtracting the divisor ∆produces one which sumsexactly to a lattice point.

Hence a function with the original divisor is F∆times a product of supertheta functions as in Eq. (22).

This completes the construction.3The Picard and Jacobian GroupsIn this section we compute explicitly the Picard, Jacobian, and divisor class groups of the superelliptic curve M. These objects were defined and discussed in [8], where they were all shown to beisomorphic in the normal case. Some but not all of the arguments used there apply more generally;nevertheless the isomorphisms will be verified here by direct calculation.

We also exhibit the Abelmap from M to its Jacobian, which is a projection rather than an isomorphism as for classicalelliptic curves.We consider the set of line bundles over the superelliptic curve M. A line bundle is specified bytransition functions which are elements of O∗ev [5, 8, 24], the invertible, even functions, on overlapsof charts. That is, the Picard group of line bundles under tensor product is Pic(M) = H1(M, O∗ev)as usual.

The standard exponential exact sheaf sequence,0 →Z →Oev →O∗ev →1,(24)and the resulting cohomology sequence,H1(M, Z) →H1(M, Oev) →H1(M, O∗ev) →H2(M, Z),(25)imply as usual that the group of line bundles of degree zero isPic0(M) = H1(M, Oev)/H1(M, Z). (26)We can also describe a line bundle by the set of divisors of all its meromorphic sections.

Sincethe ratio of two sections is a function, this gives an isomorphism between Pic0(M) and Div0(M), thegroup of degree-zero divisors modulo divisors of meromorphic functions [5, 8]. We will compute bothgroups explicitly, verifying this isomorphism and obtaining the projection map π : M →Pic0(M).The divisor class group can be computed immediately from the results of the previous section.We first claim that every divisor ∆of degree zero is equivalent to one of the form P −P0 withP0 a fixed basepoint on M, for example (0, 0).

This is because P can always be chosen so that∆−P + P0 satisfies the sum rules (14) and is therefore the divisor of a function. What changesfrom the classical elliptic curve results is that the choice of P is not unique: evidently we are freeto add multiples of δ to the even coordinate of P without changing the equivalence class of thedivisor P −P0.

This establishes the central result of this section: the Abel map π : M →Div0(M)9

which sends a point P to the divisor class [P −P0] is a projection onto Div0(M) ∼= M/ ∼, wherethe identification is (z, θ) ∼(z + αδ, θ). In the split case δ = 0 we recover the naive isomorphism ofM with Div0(M) which might have been expected.Before we confirm this result by direct computation of the Picard group, let us pause to explain inthe context of the group structure why M cannot be isomorphic to its Picard group in general.

Theset of line bundles obviously carries the Abelian group structure given by tensor product. However,M carries no such group structure.

Recall that M is the quotient of C1,1 by the nonabelian groupG. Now, C1,1 itself can be identified with the nonabelian supertranslation group,(z, θ) · (w, χ) = (z + w + θχ, θ + χ).

(27)G is the discrete subgroup generated by (1, 0) and (τ, δ) acting by right multiplication. In view ofthe fact that [25](z, θ) · (τ, δ) · (z, θ)−1 = (τ + 2θδ, δ),(28)G is not a normal subgroup and the quotient M does not inherit the group structure.

However,C1,1 also admits an Abelian group structure via(z, θ) + (w, χ) = (z + w, θ + χ). (29)Of course, M does not inherit this group structure either, because G is not a subgroup at all.But let us take the quotient C1,1/ ∼.

On this quotient space G does act as a subgroup of theAbelian group structure, hence a normal subgroup, and the further quotient by G is the Picardgroup of M. [Something is being swept under the rug here: ∼mods out by all αδ with α in theGrassmann algebra Λ. This does not seem to include modding out by θδ as required to identify Gas a subgroup.

One must remember that the group laws are really defined on the set of Λ-valuedpoints to resolve the apparent paradox.] The moral is that the unexpected identification ∼reallyprovides the minimal modification of M which will admit an Abelian group structure as Pic0(M)must.We now turn to the direct computation of Pic0(M) from (26).

It seems cleanest to computeH1(M, Oev) as the group cohomology H1(G, Oev) with values in the functions on C1,1, followingsimilar calculations of Hodgkin [6, 26]. For an explanation of the equivalence between the sheafcohomology of M and the group cohomology of G, see [27]; the techniques of group cohomologywe use are fairly intuitive and can be found in [2, Appendix B].

In particular, there is the exactsequence0 →H1[(S), OTev] →H1(G, Oev) →H1[(T), Oev],(30)where (T), (S) are the cyclic subgroups generated by the two generators of G, and OTev are the T-invariant functions. The last cohomology group in this sequence is trivial, so we get the isomorphismH1(G, Oev) ∼= H1[(S), OTev],(31)which we use for our computation.

In geometric language this says that a torus is made from theplane by first making the cylinder with fundamental group (T), whose sheaf cohomology is trivialbecause it is noncompact. The cohomology of the torus is then computed directly from functionsOTev on the cylinder by identifying its ends with S.10

A cocycle for H1[(S), OTev] is determined by assigning to the generator S a T-invariant functionF = f(z)+θφ(z); it is trivial (exact) if F = ˜F −S ˜F for some T-invariant function ˜F = g(z)+θγ(z).This requiresf(z) + θφ(z) = g(z) + θγ(z) −g(z + τ + θδ) −(θ + δ)γ(z + τ + θδ),(32)which amounts tof(z)=g(z) −g(z + τ) −δγ(z + τ),φ(z)=γ(z) −γ(z + τ) −δg′(z + τ). (33)Because every function appearing here is T-invariant, which is to say periodic, they have Fourierseries expansions of the form,f(z) =∞Xn=−∞fne2πinz,(34)and similarly for the other functions.

Then the triviality of the cocycle becomes the conditions onthe Fourier coefficients,fn=gn(1 −e2πinτ) −δγne2πinτ,φn=γn(1 −e2πinτ) −2πinδgne2πinτ. (35)Given fn and φn, these equations can always be solved for gn and γn, except in the case n = 0 whenthe conditions for triviality aref0 = −δγ0,φ0 = 0.

(36)That is, the nontrivial cocycles are precisely the odd constants and the even constants modulomultiples of δ: H1(M, Oev) = C1,1/∼.To complete the calculation, we must compute H1(G, Z). Of course this is a lattice Z ⊕Z,but we need to know where this lattice sits inside H1(G, Oev).

An element of H1(G, Z) assignsintegers −n, m to the generators T, S respectively. In the calculation above, however, we used thetriviality of H1[(T), Oev] to represent each class in H1(G, Oev) by a cocycle which assigned zero tothe generator T. To find such a representative of our element of H1(G, Z), we pick a function g(z)such that −n = g(z) −g(z + 1), for example g(z) = nz, and subtract the trivial cocycle whichassignsT7→g(z) −g(z + 1) = −n,S7→g(z) −g(z + τ + θδ) = −nτ −nθδ,(37)obtaining the new representativeT 7→0,S 7→m + nτ + nθδ.

(38)In terms of our identification H1(M, Oev) = C1,1/ ∼, the elements of H1(M, Z) are thus preciselythe lattice points m(1, 0) + n(τ, δ) in C1,1. This explicitly shows thatPic0(M) = H1(M, Oev)/H1(M, Z) = M/∼= Div0(M).

(39)11

Next we wish to similarly calculate the Jacobian of M, defined [8] as the set of odd (Λ-)linearfunctionals on the holomorphic differentials of weight 1/2, modulo those functionals which are theperiods of the differentials around cycles. A 1/2-differential on a super Riemann surface is a sectionof the canonical bundle, the bundle whose transition functions are the Berezinian determinants ofthose of M. Since supertranslations (1) have unit determinant, this bundle is trivial for superellipticcurves, and the 1/2-differentials can be identified with functions.

The periods of such a functionare obtained by integrating it over all homology cycles. Equivalently, we can lift a function F tothe covering space C1,1 and find an antiderivative Φ with F = DΦ; the periods are the changes inΦ under the covering transformations generated by T and S. The Jacobian is then the set of oddlinear functionals on H0(M, O) = {a + θα : αδ = 0} modulo periods.

Note that we consider allglobal functions, not merely even ones, so as to obtain a Λ-module rather than a Λev-module.The periods of the function a + θα are easily found. An antiderivative is Φ = αz + θa.

Underthe translation T this changes by α, while under the other generator S it changes by ατ + δa. Theodd linear functionals which send a + θα to integral linear combinations of these two constants willbe equivalent to zero in the Jacobian.To understand the structure of the linear functionals on the functions a + θα let us begin withthe simpler case in which δ is one of the generators of the Grassmann algebra Λ.

Then the set ofα which annihilate δ is just the set of multiples of δ, and a function a + θα is a linear combinationof the functions 1 and θδ. Then an odd linear functional is determined by sending 1 to some oddconstant η, and sending θδ to some odd constant κ.

By linearity, δκ = 0, so κ = δk for an evenconstant k defined modulo δ. Hence we have found that the odd linear functionals correspondprecisely to points (k, η) in C1,1/∼.

They can be viewed as mapping 1 7→η and θ 7→k, just as if 1and θ formed a basis for the functions, except that k is only defined modulo δ. Since the periods arejust the familiar lattice points generated by (1, 0) and (τ, δ), we have explicit agreement betweenthe Jacobian and the Picard group computed eariler.

One can easily verify that the isomorphismbetween them is the one described in [8]: given a line bundle in Pic0, represent it by a divisor in theform P −P0 = (k, η) −(0, 0) and associate to it the linear functional which integrates a functionfrom P0 to P, which will also be (k, η) with our conventions.What changes in the general case in which δ is not a generator of Λ? A linear functional is stilldetermined by its effect on the functions of the forms a and θα separately.

A functional on {a} isstill determined by the odd constant η which is the image of 1, but the functionals on {θα} are notso clear. We are asking for the Λ-linear functionals on the ideal I = {α : αδ = 0}, the annihilatorof δ.

Because Λ is an example of a quasi-Frobenius, or self-injective ring [28], any such functionalis multiplication by an even constant k [29] which is determined up to constants annihilating I.Again because Λ is self-injective, these are the multiples of δ [30]. Hence the isomorphism of thePicard and Jacobian groups holds in general.

To see that Λ is indeed self-injective one can apply asimple test from [30]: the annihilator of the annihilator of any minimal ideal of Λ must be the idealitself. The unique minimal ideal in the Grassmann algebra with generators β1, β2, .

. .

, βN is the setof multiples of β1β2 · · · βN; its annihilator is the ideal of all nilpotents, whose annihilator is indeedthe minimal ideal again.12

4The Group Law in a Projective EmbeddingAs shown in [4], the superelliptic curve M can be embedded in the projective superspace P 3,2 withthe help of the super Weierstrass function R(z, θ). Indeed, the map(z, θ) 7→(R, R′, R′′, 1; DR, D3R) = (x, y, u, v; φ, ψ)(40)in the affine chart v = 1, with the extension to the points at infinity,(0, θ) 7→(0, 0, 1, 0; 0, θ),(41)embeds M as the locus of points satisfying the following homogeneous polynomial equations:y2v −4x3 + g2xv2 + g3v3 −2φψv=0,2yψv + (g2v2 −12x2)φ + δ ˙g2xv2 + δ ˙g3v3=0,2yuv + (g2v2 −12x2)y −δ ˙g2φv2=0,2(g2v2 −12x2)uv + (g2v2 −12x2)2 + 2δ ˙g2ψv3=0,(42)where g2(τ) and g3(τ) are the usual modular functions.

The last equation is redundant except wheny = 0; M is not a complete intersection.Now although M is a variety, it does not carry a group structure; its Jacobian, which does, isnot a variety since varieties cannot have the kind of singularities produced by the identification ∼:the reduced space is a smooth manifold but not every f(z) is a function on M/∼even locally [31].What then becomes of the standard geometric implementation of the group law by intersecting anelliptic curve with lines?We attempt to follow the usual construction by taking a meromorphic function F on M givenbyF = aR + R′ + αDR + βD3R + b. (43)This is the restriction to M of a linear function on P 3,2 (in the chart v = 1),F = ax + y + αφ + βψ + bv.

(44)The conditions for F to have a principal zero at some point on M, F = DF = 0, translate into thelinear equations of a plane,ax + y + αφ + βψ + bv=0,aφ + ψ −αy −βu=0,(45)to be solved simultaneously with the equations of M. Note that this is hardly a generic plane,but rather a very special one encoding the notion of a principal zero. It is given by simple linearequations only because the embedding of M was constructed using the covariant derivative D whichalso encodes the superconformal structure.

We can adjust the four parameters a, b, α, β so that Fhas principal simple zeros at any two given points Pi = (zi, θi), i = 1, 2 on M. The naive expectationwould be that F has a principal triple pole at (0, 0) and, as a consequence of our function theory,there is a third point of intersection with M at P3 such that P1 + P2 + P3 = 0 mod ∼. This turns13

out to be wrong on two counts. First, using the fact that the singular part of R(z, θ) is 1/z2, wefind for the singular part of FF∼az−2 −2z−3 −2αθz−3 + 6βθz−4=(az −2 −3aθβ + 2θα)(z −θβ)−3,(46)so that the triple pole is actually located at (0, β).

This is a consequence of the fact that the mostsingular term in F is the nilpotent βD3R term. We could not have avoided this by including anequally singular even term R′′ in F, since then the condition DF = 0 for a principal zero wouldinvolve D5R, which is not one of the projective coordinates in our embedding.

Next, there willindeed be a third point of intersection, another simple zero of F at P3, but there is also a fourthintersection at the location of the triple pole itself: (0, β) embeds in P 3,2 as (0, 0, 1, 0; 0, β), whichis easily seen to satisfy the homogeneous equations (45). Thus the group law is realized in the formP1 + P2 + P3 −3(0, β) = 0 mod ∼.

(47)This is a translate of the standard group law, with the identity shifted to the point (0, 3β) in thefiber of M at infinity. Note that the point which plays the role of the identity varies with thechoice of points P1, P2 to be added, since β depends on this choice, but it can always be locatedgeometrically as the fourth intersection of the curve with the plane.

The existence of this fourthintersection could have been expected from the fact the the reduction of this embedding of M is notthe usual degree 3 embedding of an elliptic curve in P 2, but the degree 4 embedding in P 3 using℘, ℘′, and ℘′′, in which there is indeed an extra intersection at infinity [32].5IsogeniesAn isogeny of elliptic curves is a holomorphic map f from one to the other with the translationsymmetry normalized out by requiring f(0) = 0. One proves that an isogeny is either constant oronto, and that it is always a homomorphism of the group structures.

Since a super elliptic curvedoes not have a group structure, the super generalization will be that an isogeny F induces a grouphomomorphism via the projection maps to Pic0:Pic0(M1)π−11→M1F→M2π2→Pic0(M2). (48)The homomorphism is independent of the inverse chosen for π1.

We will also discuss isogenies froma super elliptic curve to itself and show that only a split curve can admit nontrivial endomorphisms.This is due to a conflict between the linear nature of an isogeny and the quadratic constraint whichis implicit in the superconformal structure of M.Given two superelliptic curves Mi = C1,1/Gi over Λ, with Gi generated by supertranslationsof the form (1) with parameters τi, δi, an isogeny will be a holomorphic map F : M1 →M2 withF(0, 0) = (0, 0). (We will eventually require the map to be surjective as well.) Its lift to the coveringspace C1,1 takes the form,(z, θ) 7→F(z, θ) = [F(z, θ), Ψ(z, θ)] = [f(z) + θφ(z), ψ(z) + θg(z)],(49)with f(0) = ψ(0) = 0.

Note that an isogeny is not assumed to be superconformal, but merelyholomorphic, even though the groups Gi act superconformally.14

In order that the map (49) descend to the quotient spaces Mi, it is necessary and sufficient thatacting on (z, θ) with a generator of G1 must change F(z, θ) by the action of some element of G2,which must be independent of z by continuity and the discreteness of the group. Therefore, we haveF(z + 1, θ) = F(z, θ) + k + lτ2 + lΨ(z, θ)δ2,(50)Ψ(z + 1, θ) = Ψ(z, θ) + lδ2,(51)F(z + τ1 + θδ1, θ + δ1) = F(z, θ) + m + nτ2 + nΨ(z, θ)δ2,(52)Ψ(z + τ1 + θδ1, θ + δ1) = Ψ(z, θ) + nδ2,(53)with integers k, l, m, n. If we use (49) to write these conditions in terms of f, φ, ψ, g, we obtainf(z + 1) −f(z) = k + lτ2 + lψ(z)δ2,(54)φ(z + 1) −φ(z) = lg(z)δ2,(55)ψ(z + 1) −ψ(z) = lδ2,(56)g(z + 1) −g(z) = 0,(57)f(z + τ1) −f(z) = m + nτ2 + nψ(z)δ2 −δ1φ(z + τ1),(58)φ(z + τ1) −φ(z) = ng(z)δ2 −δ1f ′(z + τ1),(59)ψ(z + τ1) −ψ(z) = nδ2 −δ1g(z + τ1),(60)g(z + τ1) −g(z) = −δ1ψ′(z + τ1).

(61)The analysis of these equations is somewhat tedious, but straightforward. Eqs.

(57) and (61)imply that δ1g(z) is an elliptic function, and entire, hence a constant. (A simple argument usingthe filtration of Λ shows that this is true even though τ1 may have a nilpotent part.) Given this,Eqs.

(56) and (60) say that ψ′(z) is elliptic, hence constant. Calling the constant γ and using thenormalization ψ(0) = 0, we have ψ(z) = γz.

According to (56), γ = lδ2. From (60),δ1g(z) = nδ2 −γτ1 = (n −lτ1)δ2,(62)so that δ2 must be a multiple of δ1 [and vice versa if we assume g(z) is invertible].

Consequently,multiplying any equation by δ1 will kill terms containing either δi, and terms involving ψ(z)δi arealready zero.With this information, Eqs. (55) and (59) say that δ1φ(z) is elliptic, so constant.

Then (54)and (58) say that f ′(z) is elliptic, which together with the normalization f(0) = 0 gives f(z) = azwhere the constant a = k +lτ2. Eqs.

(57) and (61) give that g(z) is elliptic; so g(z) = c, a constant,and (55) and (59) make φ′(z) a constant, so φ(z) = αz + β with α = lcδ2 according to (55). Havingexpressed all the unknown functions in terms of a few constants, all eight equations are satisfiedprovided the constants satisfy a few relations.

Eq. (58) requires δ1β = m+nτ2 −aτ1; Eq.

(59) givesδ1a = ncδ2 −ατ1 = (n −lτ1)cδ2; and Eq. (60) implies δ1c = nδ2 −γτ1 = (n −lτ1)δ2.

Collecting allthese results, the general form of an isogeny is given byf(z) = az,φ(z) = αz + β,ψ(z) = γz,g(z) = c;(z, θ) 7→[az + θ(αz + β), γz + θc],(63)wherea = k + lτ2,γ = lδ2,α = cγ,δ1a = (n −lτ1)cδ2,δ1c = (n −lτ1)δ2,δ1β = m + nτ2 −(k + lτ2)τ1. (64)15

Having obtained this general form, we can use it to answer several questions about isogenies ofsuper elliptic curves. First let us ask whether an isogeny, which is only holomorphic by definition,is in fact a superconformal map.

A map F(z, θ) = [F(z, θ), Ψ(z, θ)] is superconformal provided thatDF = ΨDΨ; in our case this says thatαz + β + θa = γcz + θc2. (65)This requires α = γc, which is one of the conditions (64); a = c2, which need only hold modulo theannihilator of δ1 according to (64); and β = 0, which is a completely new restriction.

We concludethat not every isogeny is superconformal; the superconformal ones take the special form,(z, θ) 7→(c2z + θγcz, γz + θc). (66)Next, we see that while isogenies of ordinary elliptic curves are either constant or onto, this isnot true for super elliptic curves.

If the parameter a is nilpotent, for example, a nonconstant isogenymay have a constant reduction, so that it is not surjective. This is simply because the presence ofnilpotents can lead to a wider range of singularities for maps in general.

We prefer not to considersuch singularities, so we assume from now on that all our isogenies are surjective, which requiresthat the reduced parameters ard and crd be nonzero. The important consequence of this is that δ1is a multiple of δ2 as well as vice-versa.We now prove that a surjective isogeny of super elliptic curves induces a well-defined homomor-phism of their Picard groups via the diagram (48),Pic0(M1)π−11→M1F→M2π2→Pic0(M2).

(67)A point (z, θ) of Pic0(M1) is the image under π1 of any point (z + ǫδ1, θ) of M1 for any ǫ. Theisogeny F sends this point to(z + ǫδ1, θ) 7→[az + aǫδ1 + θ(αz + β) + θαǫδ1, γz + γǫδ1 + θc](68)in M2. Then π2 removes any multiple of δ2 from the first coordinate.

The result is indeed in-dependent of ǫ, showing that the composite map is well-defined, because the surjectivity makesδ1 a multiple of δ2. This also eliminates the term γǫδ1 from the second coordinate, because theconditions (64) include γ = lδ2.Now, at the level of the Picard groups, we can drop α, which is a multiple of δ2, from (63) andwrite an isogeny as(z, θ) 7→(az + θβ, γz + θc).

(69)But this is a linear map, and the group law is simply addition in these coordinates, so the map isa group homomorphism as claimed.Next we examine isogenies of a super elliptic curve M onto itself (endomorphisms). Settingτ1 = τ2 = τ, δ1 = δ2 = δ in the general formulas, we obtain in this case(z, θ) 7→[az + θ(αz + β), γz + θc],(70)withδc = δ(n −lτ),δa = δc2,(71)a = k + lτ,γ = lδ,α = clδ,δβ = m + nτ −(k + lτ)τ.

(72)16

In the special case when M is split, δ = 0, we lose the conditions (71) and obtain the simple form,(z, θ) 7→(az + θβ, θc),(73)a = k + lτ,0 = m + nτ −(k + lτ)τ. (74)In particular, c is now arbitrary; there is no relation like a = c2 in this case.We see that in the split case, multiplication by an integer k, (z, θ) 7→(kz, kθ), is an endomor-phism, which was to be expected since M and its Picard group coincide in this case.

But this isnot true more generally, since this map violates the condition δa = δc2 which is a vestige of thesuperconformal action of the group G. In fact, for δ ̸= 0, this implies ard = c2rd, which gives thequadratic constraint,l2τ 2rd −(2n + 1)lτrd + (n2 −k) = 0. (75)This must hold in addition to the usual quadratic constraint appearing in the theory of complexmultiplication, which here arises from reducing the condition on δβ in (72),lτ 2rd + (k −n)τrd −m = 0.

(76)When l ̸= 0 we are indeed describing complex multiplication, meaning an endomorphism witha complex.By eliminating the quadratic term between these equations, we conclude that τrdis rational, not complex, a contradiction which shows that a nonsplit M cannot admit complexmultiplication. However, even in the case l = 0 when a is an integer, the constraints give k = n2in addition to the usual k = n and m = 0, so that M admits only the trivial endomorphismsk = n = 0, 1.6Supercommuting Differential Operators from SuperElliptic CurvesThe beautiful Krichever theory which produces solutions to the Kadomtsev–Petviashvili (KP) hi-erarchy of nonlinear PDEs from geometric data consisting of a line bundle over an algebraic curvetogether with some coordinate choices is by now well-known [33, 34].

The simplest explicit exampleuses a line bundle L of degree zero over an elliptic curve M to construct the commuting pair ofordinary differential operators,Q=d2dx2 −2℘(x + a),(77)P=Q3/2+= d3dx3 −3℘(x + a) ddx −32℘′(x + a),(78)where Q3/2+is the differential operator part of Q3/2 computed in the larger algebra of formal pseudo-differential operators. The correspondence which associates Q and P to the meromorphic functions℘(z) and −℘′(z)/2 on M respectively sets up an isomorphism between the commutative ring ofdifferential operators generated by Q, P and the ring of meromorphic functions on M with polesonly at z = 0, which is generated by ℘(z) and −℘′(z)/2.

As L varies through the Picard groupPic0(M), the parameter a changes and the ring of operators is isospectrally deformed. In fact, there17

is an infinite set of linear coordinates tn for Pic0(M) on which a depends linearly, with Q satisfyingthe KP equations,∂Q∂tn= [Qn/2+ , Q]. (79)The corresponding construction of solutions to the supersymmetric KP hierarchies was workedout recently [10, 11, 14].

One surprise was that the geometric data involve a line bundle over aspecific type of algebraic supercurve, which cannot be a super Riemann surface except in the specialcase of genus one. Another was the fact that linear flow in the Picard group of a fixed supercurveis described by a new super KP hierarchy discovered by Mulase and myself, and not by eitherof the previously known hierarchies due to Manin–Radul or to Kac–van de Leur.

It follows thatexplicit solutions to this new super KP hierarchy can be constructed using the information about thePicard group of a super elliptic curve developed in the previous sections. In this section we exhibitand discuss these solutions.

We change our notation slightly to conform to the conventions of theliterature on KP theory: the standard coordinates on the covering space C1,1 of the supertorus Mwill now be denoted by (w, φ), so that (z, θ) can be reserved for a different set of local coordinateson M to be introduced below.We begin with an overview of the construction to be carried out. In a small disk U around thepoint P0 : (w, φ) = (0, 0) we introduce new coordinates (z, θ) such that z−2 and θz−3 extend toglobal holomorphic functions on M −P0.

We fix a nontrivial line bundle L of degree zero on M andnote that it is holomorphically trivial on each of the Stein patches U and M −U, hence completelydescribed by a transition function across the overlap, a small annular neighborhood of ∂U, whichwe can take to be the circle z = 1. We embed L in a family of bundles L(x, ξ) by multiplying itstransition function by an extra factor exp(xz−1 +ξθ).

Although these bundles have no holomorphicsections, they have one which has the form (z−1+ holomorphic) near P0 (note that this is differentfrom having a principal simple pole there); the expression of this section in the coordinates (z, θ) inthe chart M −U is the Baker-Akhiezer function B(z, θ, x, ξ) [although we will express it in termsof the covering space coordinates (w, φ) instead]. It is the basic object in the theory and we willconstruct it explicitly in terms of super theta functions.

We observe that successive derivatives ofB with respect to x and ξ produce sections having poles of higher orders at P0 and constitute abasis for the space of meromorphic sections on M with poles only at P0. This allows us to set up anisomorphism between the ring of functions having poles only at P0 and a ring of super differentialoperators as follows.

Given such a meromorphic function F, FB is a section with poles at P0 only,so it must be a linear combination of derivatives of B. But this is to say that it arises from B by theaction of a certain differential operator OF, so we associate this operator to F. It can be computedfor an explicit F by matching the singular and constant terms in the Laurent expansions of FBand OFB about P0.

We will exhibit a set of generators for this ring analogous to Q, P above, anddiscuss how they flow under the deformations of L described by the super KP equations.We start with the specification of the new coordinates (z, θ). In order that z−2 extend to aholomorphic function away from P0, we choosez−2 = R(w, φ) + 23c = ℘(w; τ + φδ) + 23c,(80)where c is a constant and R is the super Weierstrass function introduced earlier.

Similarly, in orderthat θz−3 extend holomorphically we use a function with behavior φw−3 near P0, setting−2θz−3 = DR(w, φ) −2γ = δ ˙℘(w; τ) + φ℘′(w; τ) −2γ,(81)18

where γ is another constant, and we recall that a dot means ∂τ while a prime denotes ∂w. Usingthe Laurent expansion of ℘(w) [18] we obtain the relation between the two sets of coordinates,z−1 = w−1[1 + 13cw2 + ( g240 −c218 + ˙g240φδ)w4 + · · ·],(82)θ = φ −cφw2 + γw3 + (c22 −g28 )φw4 −(cγ + ˙g240δ)w5 + · · · .

(83)Following the construction of the Baker function in the non-super theory [33], we express it asa ratio of super theta functions times a prefactor which is the exponential of a function with thebehavior xz−1 + ξθ = xw−1 + ξφ + holomorphic terms. Such a prefactor isexp[x∂w log H(w, φ) + ξφ].

(84)It has the correct singular part because of∂w log H(w, φ) = w−1 + (q + φδ ˙q)w + · · · ,(85)where q is still the ratio of theta constants introduced in (9). It is invariant under the coveringtransformation T : w →w + 1, φ →φ, while under the other generator S it acquires a phaseexp(−2πix + ξδ) = exp −2πi(x −ξδ2πi).

(86)As our “pre-Baker function” ˆB we take the product of this with a ratio of theta functions trans-forming by the opposite phase, namelyˆB = exp[x∂w log H(w, φ) + ξφ]H(w −a −φα −x + ξδ2πi, φ + α)H(w −a −φα, φ + α),(87)as is easily verified using (21).The parameters a and α describe the given line bundle L: its divisor is (a, α) −(0, 0). It hasa section given by 1 outside the disk U, and z−1H(w −a −φα, φ + α) inside.

Equivalently, itstransition function across ∂U (inside to outside) is z/H(w −a −φα, φ + α). Then the transitionfunction of the deformed bundle L(x, ξ) isz exp(xz−1 + ξθ)H(w −a −φα, φ + α).

(88)Now ˆB is to be viewed as a section of this bundle in the outside chart M −U; dividing by thetransition function gives the same section in the inside chart U as a nonvanishing holomorphicfunction (the mismatch between the exponential factors) times z−1H(w −a −φα −x + ξδ2πi, φ + α),from which we see that the deformed bundle has divisor (a + x −ξδ2πi, α) −(0, 0). (We assume thatall constants and parameters are small enough that the supports of these divisors are inside U.) Inparticular, x shifts the even coordinate of Pic0(M) linearly and could be viewed as such a coordinateitself, but ξ does not shift the odd coordinate α.

In fact, because of the identification ∼, ξ inducesno flow on the Picard group at all but only changes the trivialization of the bundle.19

The pre-Baker function can be normalized so that, apart from the exponential prefactor, itsTaylor series in powers of w and φ begins with constant term unity.We will need this seriesthrough the quadratic terms in order to match singular parts of Laurent series later:ˆBn=Θ(a; τ + αδ)Θ(a + x −ξδ2πi; τ + αδ)ˆB=exp[x∂w log H(w, φ) + ξφ]{1 + φαL′ + φδ ˙L −wL′−φαw(L′′ + L′2) −φδw( ˙L′ + L′ ˙L) + 12w2(L′′ + L′2)+12φαw2(L′′′ + 3L′L′′ + L′3) + 12φδw2( ˙L′′ + 2L′ ˙L′ + L′′ ˙L + ˙LL′2) + · · ·}/N,(89)where we have introduced the abbreviationsL = L(x, ξ, τ, δ) = log Θ(a + x −ξδ2πi; τ + αδ),L′ = ∂xL,˙L = ∂τL,(90)and the normalization constant N is the series in braces with x and ξ set to zero. Although theseries has constant term unity, leading to the behavior 1/z for this section near P0, we see that thereare also terms proportional to φ, leading to additional singularities like φ/z.

To obtain the trueBaker function, we must subtract these off. Because of the exponential prefactor, derivatives of ˆBnwith respect to x or ξ produce new sections1 containing additional factors z−1 and φ respectively,so ∂ξ ˆBn has a φ/z singularity.

Subtracting the appropriate multiple of this yields the true Bakerfunction,B=e[···]N−1{1 −wL′ + αφwL′′ + δφw ˙L′ + αδ2πiwL′L′′ + 12w2(L′′ + L′2)+12φαw2(L′′′ + 2L′L′′) + 12φδw2( ˙L′′ + 2L′ ˙L′) −αδ4πiw2(L′L′′′ + 2L′′L′2) + · · ·}. (91)It is now tedious but straightforward to work out the explicit correspondence between meromor-phic functions F on M holomorphic away from P0 and differential operators OF in x, ξ by matchingthe singular terms in the series for FB = OFB.

For example, the operator corresponding to thesuper Weierstrass function R(w, φ), with a double pole at P0, has the form Q = d2 + ω∂+ u, withω=2[α℘′(a + x; τ) + αδξ2πi ℘′′ + δ ˙℘],u=2{−℘+ ξδ2πi℘′ −αδ ˙℘+ αδ ˙q + αδ2πi[℘′∂x log Θ −(℘−q)2]},(92)where all the functions have the same arguments as ℘′(a + x; τ), all odd parameters having beenexplicitly expanded out, and d = ∂x, ∂= ∂ξ. It follows from the general theory, and can be verifiedexplicitly, that the function −R′(w, φ)/2 having a triple pole must correspond to P = Q3/2+ .

Forany such second-order operator Q, one findsP = Q3/2+= d3 + 32ω∂d + 32ud + 34ω′∂+ 34u′. (93)1 It may not be clear that derivatives of ˆB are still sections of L(x, ξ).

The point is that ˆB is a global function onM −U for all x, ξ, so its derivatives are too. They must extend into U as meromorphic sections since no essentialsingularity has been introduced.20

A set of generators for the ring of functions holomorphic offP0 must contain an odd functionin addition to R, −R′/2; this is conveniently taken to be σ(w, φ) of Eq. (10), which corresponds tothe simple first-order operatorΣ = ∂+ δ2πid.

(94)The supercommutativity of the generators Q, P, Σ of the isomorphic ring of operators can be verifiedexplicitly. Although it is not manifest from the form of (92), both Q and P depend on x, ξ onlythrough the combination x −ξδ2πi [see (86),(87) for the origin of this], and this is precisely thestatement that they commute with Σ.

We also have Σ2 = 0. The vanishing of [Q, P] leads to a pairof third-order differential equations for ω, u, namelyωxxx + 3ωωxξ + 6ωux + 6uωx + 3ωxωξ=0,(95)uxxx + 3ωuxξ + 3ωxuξ + 6uux=0.

(96)One finds that, exactly as in the non-super case, the first equation is satisfied identically in virtueof the identity℘′′′ = 12℘℘′(97)satisfied by the Weierstrass function. However, the second equation requires, in addition to thisidentity, the relationg2 = 12(q2 −2πi ˙q)(98)between the modular function g2 and the theta constant q. I have found similar relations in theliterature on elliptic functions, though not in just this form; however, it is a simple consequence ofthe fact that the theta function satisfies the heat equation [18],4πi ˙Θ(w; τ) = Θ′′(w; τ).

(99)As a consequence, its logarithm f = log Θ satisfies4πi ˙f = f ′′ + f ′2. (100)From the relation (9) between Θ and ℘we get the Laurent expansionf ′ = w−1 + qw −g260w3 + · · · ,(101)and the desired relation (98) follows by using this in (100) and equating the coefficients of w2 on bothsides.

This illustrates that the super KP system contains information about the modular dependenceof the theta functions, through the coupling between τ and θ in the superelliptic functions, whichdoes not appear in the solutions to ordinary KP (although changes in moduli do figure in theadditional symmetries of the KP hierarchy).Finally, we describe the flows on the Picard group (further deformations of L) which lead to thesuper KP equations for Q. These depend on an infinite set of parameters tn which are (Grassmann)even or odd for even or odd n respectively.

They multiply the transition function of L(x, ξ) by anadditional factor exp t2nz−n or exp t2n+1θz−n respectively. At this point the properties of the newcoordinates (z, θ) become important.

Because z−2 extends to a holomorphic function on M −U,all the flows t4, t8, . .

. are trivial since they can be undone by a change of bundle trivialization on21

M −U. Because θz−3 extends holomorphically, the same is true for t7, t11, .

. ..

The parameter t2can be identified with x, since they produce the same deformation. The first nontrivial even flowis by exp t6z−3, and we need to understand the Baker function for the new bundle this produces.It should have an exponential prefactor having this singular behavior.

For this purpose we employthe function −R′(w, φ)/2 with singular part w−3 = z−3 −cz−1 + · · ·. Thus we need only multiplyour previous Baker function by exp −t6R′/2 and replace x by x + ct6 to obtain the new one.

Theeffect on the resulting differential operators is the replacement a →a + ct6 showing explicitly theflow on the Jacobian where a is the even coordinate. The flow would be trivial if we had chosenc = 0; the motivation for introducing this constant is precisely to get a nontrivial t6 flow.Similarly, for the first nontrivial odd flow by exp t3θz−1 an exponential prefactor with thisbehavior isexp t3∂η log H(w −φη, φ + η) = exp t3[φΘ′(w; τ)Θ(w; τ) + δ˙Θ(w; τ)Θ(w; τ)].

(102)This function is invariant under the generator T, but acquires a phaseexp −t3(πiδ + 2πiφ)(103)under S. To obtain a well-defined pre-Baker function we compensate this phase by shifting theparameter α in the numerator factorH(w −a −φα −x + ξδ2πi, φ + α)(104)in (87) by α →α −t3, which is the flow on the Jacobian in this case. The next odd flow t5θz−2 isactually trivial because there is a global function with this behavior, namely−t5D∂w log H(w, φ) = t5[θz−2 + θ(c3 −q) + · · ·].

(105)The mechanics of this triviality is rather interesting: if this function is used to form an exponentialprefactor for the pre-Baker function, a shift of the parameter ξ will be required due to the termproportional to θ. We know that ξ only changes the trivialization of a bundle, and indeed thechange in ˆB resulting from this shift is subtracted offalong with the φ/z poles in forming B, sothat the differential operators are unchanged.The higher flows can all be computed in the same manner.

Because there are global functionswith leading singularities z−n and θz−n for all n ≥2, we can use them as prefactors for the Bakerfunction (that is, to change the bundle trivialization in M −U) until any flow is reduced to alinear combination of those for n = 1. (In other words, any deformation can be reduced to a linearcombination of the single even and odd generators for Pic0.) Then its effect can be read offas alinear shift in the Jacobian coordinates a and α.

It is not always true, however, that the even flowsonly shift a while the odd flows only shift α. In general each flow can shift both in the nonsplitsituation.

The flow parametrized by t10, for example, acts bya →a + (g28 + 56c2)t10,α →α −˙g28 δt10,(106)showing how the supermodulus δ permits a flow in both even and odd coordinates.22

The differential operator Q + 23c corresponding to the function z−2, with its parameters shiftedin this manner, gives a solution to the new super KP hierarchy of [10, 11]. Unfortunately, unlike thestandard KP theory, this hierarchy has no simple formulation in terms of Q itself, but is written inthe Sato form in terms of the wave pseudodifferential operator K which conjugates Q into a simpleform:Kd2K−1=Q + 23c,(107)∂K∂t2n=−(KdnK−1)−K,(108)∂K∂t2n+1=−(K∂dnK−1)−K.

(109)I have not tried to obtain an explicit expression for K.7ConclusionsIn this paper we have developed the theory of super elliptic curves with an emphasis on the role of thesupermodulus δ and the non-freely generated character of the cohomology modules. We discussedthe building blocks for superelliptic functions, the super Weierstrass and theta functions, and provedthe necessary and sufficient conditions for a divisor to be the divisor of a superelliptic function.

Wecomputed the Picard, Jacobian, and divisor class groups of a superelliptic curve, explicitly verifyingthe isomorphisms between them, and found that the Abel map π : M →Pic0(M) was a projectionin the nonsplit case. The agreement between the different methods of calculation — cohomology forthe Picard group, duality of modules for the Jacobian, function theory for the divisor class group— is very satisfying.

We showed that the group law can be implemented in a projective embeddingby intersecting M with planes chosen to encode the notion of principal zero, modulo the kernel ofπ and an ambiguity in the group identity element. We determined the general form of an isogenyof superelliptic curves, proving that it always induces a homomorphism of their Picard groups, andthat a nonsplit curve admits trivial endomorphisms only.

Finally, we applied this machinery tothe explicit calculation of the supercommutative rings of differential operators which constitute thesolution to the new super KP hierarchy corresponding to flow in the Jacobian of a superellipticcurve. The Baker function was expressed in terms of super theta functions and used to work outthe differential operators corresponding to simple superelliptic functions, generalizing the classicalQ, P pair of ordinary KP theory.It would be natural to seek extensions of this theory in two directions: higher-genus superRiemann surfaces, and supercurves of genus one which are not super Riemann surfaces.

For superRiemann surfaces of higher genus the primary motivation is again to understand the consequences ofthe non-freely generated cohomology. One should again construct the Picard, Jacobian, and divisorclass groups as explicitly as possible and check their isomorphism in the general nonsplit case.

AnAbel map from the surface to its Jacobian should be constructed and investigated. Function theoryon the surface should be studied in terms of the pullback of theta functions from the Jacobian.

Ahigher-genus analogue of the simple substitution τ →τ +θδ which converts ordinary theta functionsto super ones should be found. As here, the duality properties of modules which determine thestructure of the Jacobian should be understandable on the basis of Λ being self-injective, and this23

should be used to develop Serre duality for cohomology groups as Λ-modules rather than as C-modules as in [35]. One should study the map from super Riemann surfaces with local coordinatesto states in the operator formalism, and the geometry of the Grassmannian of such states when thering of functions with poles at a single point is not freely generated.The motivation for studying genus-one supercurves which are not super Riemann surfaces, orAbelian supergroups on two generators whose action on C1,1 need not be superconformal, is toconstruct more general solutions to super KP hierarchies.

(One should also find nontrivial endo-morphisms of such curves with the relaxing of the superconformal constraint.) We know from [10]that the Manin–Radul and Kac–van de Leur super KP hierarchies describe simultaneous deforma-tions of the supercurve M and the line bundle L over it, specifically by changing the patching of thecoordinate θ along with that of the line bundle across ∂U.

Even if M is initially a super Riemannsurface, this property will not be preserved by the flow. Hence one needs to repeat enough of theanalysis of this paper for general genus-one curves to construct the Baker functions for families ofline bundles over such curves.

One may learn something about where the locus of super Riemannsurfaces sits inside the larger moduli space of genus-one curves by studying the corresponding su-per KP solutions, e.g. what is special about the rings of differential operators when M admits asuperconformal structure?

Without the covariant derivative D one will have to settle for Cartierdivisors which are not sums of points. On the other hand one may be able to exploit the remarkablecorrespondence [36] between general supercurves and untwisted N = 2 super Riemann surfaces, andthe resulting involution in the moduli space under which N = 1 super Riemann surfaces are fixedpoints.

Perhaps this involution plays a role in the super KP theory.AppendixA natural question is whether anything of number-theoretic interest results from seeking rationalpoints on super elliptic curves. By analyzing a simple example we will see that this essentiallyamounts to finding rational points on the (co)tangent plane — more generally, the jets — of anordinary elliptic curve at a rational point.

This answers our question in the negative, since rationalpoints on planes are abundant and easy to find.When we consider super elliptic curves over Q, the generators of the lattice cannot always bereduced to the form (1). Instead we must consider the more general form,T :z →z + ω1 + θδ1,θ →θ + δ1,S :z →z + ω2 + θδ2,θ →θ + δ2,(110)with δ1δ2 = 0.

As in [4], we find that the affine part of the super elliptic curve is embedded in C2,2by the map,(z, θ) 7→(R, R′; DR, D3R) = (x, y; φ, ψ),(111)where R(z, θ) = ℘(z; ω1 + θδ1, ω2 + θδ2), as the set of solutions of the polynomial equations,y2 −4x3 + g2x + g3 −2φψ = 0,2yψ −(12x2 −g2)φ +2Xi=1δi(∂ωig2x + ∂ωig3) = 0. (112)24

We fix Λ to be the Grassmann algebra on just two generators β1, β2, and consider the affinesupertorus in C2,2 given by the equations,y2 −4x3 + g2x + g3 −2φψ = 0,2yψ −(12x2 −g2)φ + aβ1x + bβ2 = 0,(113)where g2, g3, a, b are rational.Now Λ is a four-dimensional vector space, and using the basis{1, β1, β2, β1β2} we can writex = xrd + x12β1β2,y = yrd + y12β1β2,φ = φ1β1 + φ2β2,ψ = ψ1β1 + ψ2β2. (114)By a rational point we understand one whose components in this basis are rational.

Inserting theseexpressions into the polynomial equations (113), we obtainy2rd −4x2rd + g2xrd + g3 = 0,(115)2yrdψ1 −(12x2rd −g2)φ1 = −axrd −b,(116)2yrdψ2 −(12x2rd −g2)φ2 = 0,(117)2yrdy12 −(12x2rd −g2)x12 = 2(φ1ψ2 −φ2ψ1). (118)The first equation says that (xrd, yrd) must be a rational point on the reduced elliptic curve.

Thethree remaining equations share a common structure which can be understood by recalling theinvariant differential of the reduced curve,2yrddyrd −(12x2rd −g2)dxrd = 0. (119)This can be viewed as defining a linear map from rational values of dxrd to rational values of dyrd, orvice versa, at the chosen rational point of Mrd; the derivative map defined over the rationals.

Sim-ilarly here we get a map from rational (φ1, φ2, x12), playing the role of dxrd, to rational (ψ1, ψ2, y12)analogous to dyrd, which is a deformation of the derivative map and is computed by solving linearequations only. A Grassmann algebra with more generators will bring higher derivatives into playthrough later terms in the Taylor expansions of the equations (113).References[1] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York,1984.

[2] J.H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, New York 1986.

[3] A.M. Levin, “Supersymmetric elliptic curves,” Funct. Analysis and Appl.

21, 243–244 (1987);“Supersymmetric elliptic and modular functions,” ibid. 22, 60–61 (1988).

[4] J.M. Rabin and P.G.O.

Freund, “Supertori are algebraic curves,” Commun. Math.

Phys. 114,131–145 (1988).25

[5] S.B.Giddings and P. Nelson, “Line bundles on super Riemann surfaces,” Commun. Math.

Phys.118, 289–302 (1988). [6] L. Hodgkin, “Problems of fields on super Riemann surfaces,” J. Geom.

and Phys. 6, 333–348(1989).

[7] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York 1977. [8] A.A. Rosly, A.S. Schwarz and A.A. Voronov, ”Geometry of superconformal manifolds,” Com-mun.

Math. Phys.

119, 129–152 (1988). [9] E. D’Hoker and D.H. Phong, “Conformal scalar fields and chiral splitting on super Riemannsurfaces,” Commun.

Math. Phys.

125, 469–513 (1989); K. Aoki, E. D’Hoker, and D.H. Phong,“Unitarity of closed superstring perturbation theory,” Nucl. Phys.

B342, 149–230 (1990). [10] J.M.

Rabin, “The geometry of the super KP flows,” Commun. Math.

Phys. 137, 533–552(1991).

[11] M. Mulase, “A new super KP system and a characterization of the Jacobians of arbitraryalgebraic supercurves,” J. Diff. Geom.

34, 651–680 (1991). [12] A.S. Schwarz, “Fermionic string and universal moduli space,” Nucl.

Phys. B317, 323–343(1989).

[13] L. Alvarez-Gaum´e, C. Gomez, P. Nelson, G. Sierra, and C. Vafa, “Fermionic strings in theoperator formalism,” Nucl. Phys.

B311, 333–400 (1988). [14] M. Mulase, “Geometric classification of Z2-commutative algebras of super differential opera-tors,” MSRI preprint 06929-91, August 1991.

[15] Yu. I. Manin, Topics in Noncommutative Geometry, Princeton University Press, Princeton1991.

[16] B. DeWitt, Supermanifolds, Cambridge University Press, Cambridge 1984. [17] A. Rogers, “A global theory of supermanifolds,” J.

Math. Phys.

21, 1352–1365 (1980). [18] K. Chandrasekharan, Elliptic Functions, Springer-Verlag, Berlin-Heidelberg-New York 1985.

[19] L. Mezincescu, R.I. Nepomechie, and C.K. Zachos, “(Super)conformal algebra on the (su-per)torus,” Nucl.

Phys. B315, 43–78 (1988).

[20] Yu. I. Manin, “Neveu-Schwarz sheaves and differential equations for Mumford superforms,” J.Geom.

and Phys. 5, 161–181 (1988).

[21] D. Friedan, “Notes on string theory and two-dimensional conformal field theory,” in Proceedingsof the Workshop on Unified String Theories, ed. D. Gross and M. Green, World Press, Singapore1986.

[22] I.N. McArthur, “Line integrals on super-Riemann surfaces,” Phys.

Lett. B206, 221–226 (1988).26

[23] A. Rogers, “Contour integration on super Riemann surfaces,” Phys. Lett.

B213, 37–40 (1988). [24] J.M.

Rabin and P. Topiwala, “Super Riemann surfaces are algebraic curves,” U.C.S.D. preprint(1988).

[25] H. Kanno, K. Nishimura, and A. Tamekiyo, “Two-point functions on supertorus and supermodular invariance,” Phys. Lett.

B202, 525–529 (1988). [26] L. Hodgkin, “A direct calculation of super Teichm¨uller space,” Lett.

Math. Phys.

14, 47–53(1987). [27] D. Mumford, Abelian Varieties, pp.

22–24, Oxford University Press, London 1970. [28] E. Kirkman, J. Kuzmanovich, and L. Small, “Finitistic dimensions of Noetherian rings,” J.Alg.

147, 350–364 (1992). [29] H. Cartan and S. Eilenberg, Homological Algebra, p. 8, Princeton University Press, Princeton1956.

[30] J. Dieudonn´e, “Remarks on quasi-Frobenius rings,” Ill. J. Math.

2, 346–354 (1958). [31] J.M.

Rabin and L. Crane, “How different are the supermanifolds of Rogers and DeWitt?,”Commun. Math.

Phys. 102 123–137 (1985).

[32] A. Weil, Number Theory: An Approach Through History from Hammurapi to Legendre, pp.135–139, Birkhauser, Boston 1984. [33] G. Segal and G. Wilson, “Loop groups and equations of KdV type,” Publ.

Math. IHES 61,5–65 (1985).

[34] M. Mulase, “Lectures on algebraic geometry of the KP system,” ITD preprint 90/91–13, U.C.Davis. [35] C. Haske and R.O.

Wells, Jr., “Serre duality on complex supermanifolds,” Duke Math. J.

54,493–500 (1987). [36] S.N.

Dolgikh, A.A. Rosly, and A.S. Schwarz, “Supermoduli spaces,” Commun. Math.

Phys.135, 91–100 (1990).27

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