ON THE KLEINIAN CONSTRUCTION OF ABELIAN FUNCTIONS OF CANONICAL ALGEBRAIC

21 Pages·1999·0.81 MB·English

ON THE KLEINIAN CONSTRUCTION OF ABELIAN
FUNCTIONS OF CANONICAL ALGEBRAIC CURVES
J C EILBECK, V Z ENOLSKII, AND D V LEYKIN
1. INTRODUCTION
The discovery of classical and quantum completely integrable systems
ledtoanincreaseininterestinthetheoryofAbelianfunctionsintheoretical
physicsandappliedmathematics. Thisareawasconsideredtraditionallyas
afieldofpuremathematics. Thisnewtrendmakesitnecessarytoreconsider
classicalresultsintheareafromthepointofviewofmodernapplications.
In this paper we consider an arbitrary algebraic curveV of genus g and
constructthefieldofmeromorphicfunctionsonitsJacobivarietyJac V in
termsofKleinian -functions,
2
u ln u i j 1 g
ij
u u
i j
wherethevectoru Jac V and istheKleinian -function. Theeffective
construction of the -function plays the principal role in our approach. It
is defined on the universal space of Jacobians, which is the fibration with
the base given by the space of moduli, M V of the curveV of dimension
d 3g 3 and a fibre generated by Jacobi variety Jac V . The Kleinian
functionrepresentsanaturalgeneralizationoftheWeierstrassellipticfunc-
tionto the caseof anarbitraryalgebraiccurve, andhas thefollowingprop-
erties:
The -function is automorphic with respect to the action of the sym-
plecticgroupSp 2g .
The -function is an entire function on the universal space and is ex-
pandedin apowerserieswhoseentriesaremonomials
u 1 u g 1 d
1 g 1 d
whereu Jac V , M V , g, d, 0 .
TheresearchdescribedinthispublicationwassupportedinpartbygrantsfromtheCivil
ResearchDevelopmentFoundation,CRDFgrantno. UM1-325,theINTASgrantno. 96-
770 and the Engineering and Physical Sciences Research Committee, grant GR/L06119
(JCEandVZE).
1
2 JCEILBECK,VZENOLSKII,ANDDVLEYKIN
The addition formula for the function inherits the form of the addi-
tiontheoremfortheWeierstrass functionandiswrittenas
u v u v
polynomialin
u 2 v 2 i j
ThisconstructionbeganwithWeierstrass[22,23]andKlein[13]andwas
welldocumentedin[1,2];recentlyB.M.Buchstaberandtwooftheauthors
reviewedthehyperellipticcase[4,5,6].
Theprincipalresultof[4,5]isthediscoveryofa g 2 g 2 matrix
H ofarank 4whoseentriesare functionsandthemoduliofthecurveV
andwhichdictatesallthetheory: the4 4minorsgivetheembeddingofthe
Kummer variety Kum V in the projective space, 3 3 minors describe in
the same way the Jacobi variety Jac V , and the associated KdV hierarchy
is constructed in terms of 2 2 minors of the matrix H. The hyperelliptic
Kleinian functions were also developed in [16, 10] for a description of the
latticeKdVsystem.
Thepresentpaperisaimedatdevelopingtheanalogousmatrixrealization
of the Kleinianconstructionof Abelian functionsfor an arbitraryalgebraic
curve, including the case of a singular curve. The paper is based on the
recentresultsof[7],wheretheconstructionoftheKleinian functionswas
given for a non-hyperellipticcurve, and contains a set of explicit formulae
torealizetheapproachof[7].
The paper starts from an example of a hyperelliptic curve of genus two
for which we give the basic formulaeof the theory and their application to
completely integrable systems. We concentrate further on the construction
of the principle objects – Kleinian functions for a wide class of alge-
braic curves (the so called canonical curves). With this purpose we derive
thecanonicalabeliandifferentialsusingtheWeierstrassgaptheoremasthe
main working tool. We obtain as the result the Kleinian formula (3.28),
which is a generating one for deriving the complete set relations between
functions and their derivatives. The principle result of the paper is the
explicitsolution of the Jacobi inversionproblem, which is an alternativeto
that given by M.No¨ther [17]. We consider as a main example the case of a
non singular trigonal curve, for which we give the complete set of formu-
lae, analogous to those given in [1, 2] for hyperelliptic curve. The paper
is completed by a short discussion on the application of our approach to
completely integrable equations and of the further perspectives of the de-
velopmentofthetheory.
KLEINIANCONSTRUCTIONOFABELIANFUNCTIONS 3
2. KLEINIAN FUNCTIONS OF THE GENUS TWO HYPERELLIPTIC CURVE
Inthissectionweconsiderthesimplestexample-theKleinianfunctions
of an algebraic curve of genus two, and demonstrate how these functions
workin completelyintegrablesystems.
WeconsidertheRiemannsurfaceofacurveV x y ofgenusg 2,
(2.1) y2 4x5 x4 x2 x2 x
4 3 2 1 0
5
4 x a a a
k i j
k 1
equippedwithahomologybasis ; H V .
1 2 1 2 1
Introduce the canonical basis in the space of holomorphic differentials
duT du du 1 V asfollows
1 2
dx xdx
du du
1 2
y y
TheassociatedcanonicalmeromorphicdifferentialsofthesecondkinddrT
dr dr havetheform
1 2
x 2 x2 12x3 x2
3 4
(2.2) dr dx dr dx
1 2
4y y
The2 2matricesoftheirperiods,
2 du 2 du
l l
k kl 12 k kl 12
2 dr 2 dr
l l
k kl 12 k kl 12
satisfytheequations,
ı
T T 0 T T 1 T T 0
2
2
which generalizes the Legendre relations between complete elliptic inte-
gralsto thecaseg 2.
Thefundamental functioninthiscaseisanaturalgeneralizationofthe
Weierstrasselliptic functionandisdefinedasfollows
u
det 2 4 a a
1 i j 5 i j
exp uT 2 1u 2 1u 1
4 JCEILBECK,VZENOLSKII,ANDDVLEYKIN
where 8 1and v isthe functionwithanoddcharacteristic
1 2
,
1 2
v expı m T m 2 v T m
m 2
Wedenote
2 2
u ln u u ln u
11 u2 12 u u
1 1 2
2
u ln u
22 u2
2
The multi-index symbols etc. are defined as logarithmic derivatives
i jk
bythevariableu u u onthecorrespondingindicesi j k etc.
i j k
TheequationsoftheJacobiinversionproblem,
x x
1 2
u du du
1 1 1
a a
1 2
x x
1 2
u du du
2 2 2
a a
1 2
areequivalentto analgebraicequation
(2.3) x u x2 u x u 0
22 12
thatis,thepair x x isthepairofrootsof(2.3). So wehave
1 2
(2.4) u x x u x x
22 1 2 12 1 2
Thecorrespondingy is expressedas
i
(2.5) y u x u i 1 2
i 222 i 122
Thereis the followingexpressionfor the function u in terms of x x
11 1 2
andy y :
1 2
F x x 2y y
1 2 1 2
(2.6) u
11 4 x x 2
1 2
where
2
(2.7) F x x xrxr 2 x x
1 2 1 2 2r 2r 1 1 2
r 0
All the possible pairwise productsof the functions are expressed as
ijk
followsin terms of and constants of the defining equation
22 12 11 s
KLEINIANCONSTRUCTIONOFABELIANFUNCTIONS 5
(2.1). Wegivehereonlybasisequations
2 4 4 3 4 2
222 11 3 22 22 12 22 4 22 2
1 1
2 2 2
222 122 2 1 12 11 22 2 3 12
4 2
12 22 4 12 22
2 4 2 4 2
122 0 11 12 4 12 22 12
All such expressions may be rewritten in the form of an extended cubic
relationasfollows. Forarbitraryl k 4 thefollowingformulaisvalid[2]
1 H l
(2.8) lT Tk det
4 kT 0
where T andH isthe4 4matrix:
222 221 211 111
1 2 2
0 2 1 11 12
1 4 1 2 2
H 2 1 2 11 2 3 12 22
2 1 2 4 2
11 2 3 12 4 22
2 2 2 0
12 22
The vector satisfies the equation H 0, and so the functions
22 12
and arerelatedbytheequation
11
(2.9) detH 0
Theequation(2.9)definesthequarticKummersurface in 3 [12].
The functionsareexpressedasfollows
ijkl
1
(2.10) 6 2 4
2222 22 2 3 4 22 12
(2.11) 6 2
2221 22 12 4 12 11
1
(2.12) 2 4 2
2211 22 11 12 2 3 12
1
(2.13) 6
2111 12 11 2 12 1 22 0
2
1 1
(2.14) 6 2 3
1111 11 0 22 1 12 2 11 2 0 4 8 1 3
These equations can be identified with completely integrable partial dif-
ferential equations and dynamical systems, which are solved in terms of
Abelianfunctionsofhyperellipticcurveofgenustwo.
TodemonstratethisstatementweconsiderthestationaryVeselov-Novikov
equation,
u u 3 vu 3 wu 0
xxx yyy x y
(2.15) w u v u
x y y x
6 JCEILBECK,VZENOLSKII,ANDDVLEYKIN
where 0areconstants. Thenthefollowingpropositionisvalid
Proposition2.1. The stationary flow of the Veselov-Novikov equation is
satisfiedifweset
u x y 2 x y
22
1
(2.16) v x y 2 x y
12 4
3
1
w x y 2 x y
11 2
3
Proof. A straightforward substitution of (2.16) into (2.15) and use of the
relations(2.11),(2.13)andH 0.
Because the Kleinian functions are the coordinates of the
22 12 11
Kummer surface, the stationary Veselov-Novikov equation, being associ-
atedwithahyperellipticcurveofgenustwo,describestheKummersurface.
ThelinkbetweenthestationaryVeselov-NovikovequationandtheKummer
surfacewasrecentlydiscussedbyFerapontov[11].
It can be also shown, that the equations (2.10)-(2.14) describe hierar-
chiesofKdVand”sine-Gordon”equationsassociatedwiththehyperelliptic
curveofgenustwo.
In what follows we develop the Kleinian construction of Abelian func-
tionsforacertainclassofnon-hyperellipticalgebraiccurves.
3. KLEINIAN CONSTRUCTION OF ABELIAN FUNCTIONS
3.1. Thecurve. LetV beanalgebraiccurvegivenbyanirreducibleequa-
tion
n
(3.1) f x y 0 f x y a x yn a x yn k
0 k
k 1
where a x are polynomials in x and a x and a x , k 1 n have no
k 0 k
commonfactors. The curveis called singularif a x 1and nonsingular
0
otherwise. In other words, a singular curve is the curve which has points
y x .
Definition3.1. The orderN of an arbitraryrationalfunction x y on the
curveV is the number N of common solutions x y x y of the
1 1 N N
equations
(3.2) f x y 0 x y 0
We shall call the positive integers n s orders of the curve V because,
clearlyxisa functionofordernandyisa functionoforders.
The definitions given above permit us to formulate the Weierstrass gap
theorem,whichservesasaprincipalworkingtoolinwhatfollows.
KLEINIANCONSTRUCTIONOFABELIANFUNCTIONS 7
Theorem 3.1(WeierstrassLu¨ckensatz). ForV thereexistsanumbergof
positive integers n n (“gap sequence”) such that for each number n
1 g i
theequations
(3.3) f x y 0 x y 0 ord x y n
V i
havenosolutionsand
all exceptionalintegers(“gaps”)belongtotheinterval 1 2g 1
0 n n n 2g
1 2 g
if the numbers n and n – are permitted (“non-gaps”) then the num-
i j
bers pn qn ,where p q arealsonon-gaps.
i j
Thefollowingcorollariesarevalid
Corollary3.1.1. Considerthegapsequencewiththenumberofgapsg. Let
nbethelowestfromthenon gaps. Denotebys,i 1 n 1thelowest
i
non-gapssuchthats modn i n. Thenthefollowingequalityisvalid
i
n 1 s
i
(3.4) g
n
i 1
Corollary3.1.2. Suppose that the condition of Corollary 3.1.1 are satis-
fied. Lets min s s . Thenthefollowingequalityisvalid
1 n 1
n 1 s 1
(3.5) g where 0
2
The Weierstrass gap theorem introduced two important positiveintegers
gand whicharetheprincipalgeometricalinvariantsofthecurve.
Definition3.2. The nonnegativeinteger is the numberof multiple points
ofthecurve,whichcannotexceed n 1 s 1 2. Thenumberofgapsg,
which is the difference between the maximal number of multiple points of
a curve of orders n s and number of multiple points, which it actually
has,is calledthegenus1 ofthecurve.
Todescribethenumber analyticallywecomputethediscriminant(iny)
D x ofthecurve(3.1),
y
n 1
(3.6) D x d i x r x
y i
i 1
where 2. Zerosofd x i 1 n 1,where
i
j
d x x
i ij j ij
j 1
1Inclassicalliterature–Geschlechtordeficiency
8 JCEILBECK,VZENOLSKII,ANDDVLEYKIN
are multiple points, whence zeros of the polynomial r x are the branching
pointsofthecurve. Letuswritethenumber
n 1
(3.7) 1
i i
i 1
where we denote 0, as number of multiple points. The curve is
callednon-degenerateif 0anddegenerateotherwise.
IntroduceonV the structure of one-dimensionalcompact complexman-
ifold – the Riemann surface of algebraic curve by introducing a local
parametrization of the point y x y x in the vicinity of a point
a b ,where is thelocalcoordinate:
a b if a b is anregularpoint
a b m if a b is anbranchingpoint
1 1 if a b isbranching
y x s n
pointatinfinity
a m b if a b is anmultiplepoint
ofmultiplicitym
We will employ further the same notation for the plane curve and the Rie-
mannsurface–V. Combiningnotionsintroducedso far,wecomeupwith
Definition3.3. ThealgebraiccurveV oforders n s iscallednonsingular
non-degeneratecanonicaliffa x 1, 0and n s areco-prime.
0
It is clearly that the Weierstrass gap sequence of the nonsingular non-
degenerate canonical curveV of orders n s is generated by two coprime
integersnands. Alternatively,theWeierstrassgapsequenceofanonsingu-
lardegeneratecanonicalcurveV oforders n s isgeneratedbytheintegers
nands s s ,where1 k n 1.
1 k
The canonical algebraic curve has a branching point at infinity, where
the coordinates x and y are given as x 1 y 1 2
n s
n s where is the local coordinate. The polynomial f x y defining the
canonicalcurvecanbewrittenin theform
f x y yn xs termsoflowerorder
Letusfixintheformofapropositiontheimportantpropertyofthenonsin-
gularcurve
Proposition3.2. The order of a monomial xPyQ, P Q on nonsingular
non-degeneratecurveof orders n s equals
(3.8) ord xPyQ nP sQ
V
KLEINIANCONSTRUCTIONOFABELIANFUNCTIONS 9
Example3.1. Consider all canonical curves of genus g 4 and construct
theassociatedWeierstrassgapsequences.
Nonsingularcanonicalhyperellipticcurve
0 1 2 3 4 5 6 7 8 9
Thesequenceis generatedbytheorders2and9.
Nonsingularcanonicaltrigonalcurve
0 1 2 3 4 5 6 7 8 9
Thesequenceis generatedbytheorders3and5.
Singularcanonicaltrigonalcurve
0 1 2 3 4 5 6 7 8
Thesequenceis generatedbytheorders3,7and8.
Singularcanonicalfour-sheetedcovering
0 1 2 3 4 5 6 7 8 9
Thesequenceis generatedbytheorders4,5and6.
3.2. The differentials and integrals. All the construction is based on the
explicitrealizationofthefundamentalsecondorder2–differential.
Definition3.4. The 2–differential d y x;w z on V V is called a fun-
damental second order 2-differential if it is symmetric d x y;z w
d z w;x y and has the only pole of the second order along the diagonal,
x zin thevicinityofwhichitcan begivenas
d d
(3.9) d y x;w z O 1 d d
2
where – are local coordinates of the points x and z respectively. We
shalllookforarealizationofd y x;w z intheform[1]
F y x;w z dxdz
(3.10) d y x;w z
x z 2f x y f z w
y w
whereF y x;w z isapolynomialofitsvariables.
Holomorphic differentials can be represented locally at every point
X Y V in the form du h d , where h – is the holomorphic func-
tionand –isthelocalparameterinthevicinityofthepoint X Y . Forthe
algebraic curve of genus g there exist exactly g independent holomorphic
differentials,whichcanbewrittenintheform
xPiyQidx
(3.11) du i 1 g
i
f x y
y
10 JCEILBECK,VZENOLSKII,ANDDVLEYKIN
whereord xPiyQi,i 1 gareexactlygfirstnon-gapsoftheWeierstrass
V
gapsequence. Introducetheg-vector
(3.12) T
i j 1 g
whose components xPiyQi are ordered by increasing order of the
i j
monomials:
1 x
1 00 2 10
Definition3.5. Let x be a polynomial in x of order ord x n and
x
F x polynomialsofordersk,suchthattheequality
k
x z n
zn kF x
k
x z
k 0
is valid. Then Dk z F x is the umbral derivative of order k of the
z k
polynomial z .
Definition3.6. Therationalfunction x y onthenonsingularcurveV is
called an entire rationalfunction,if x y iff x . Entire rational
functionsgeneratearing,whichwe denoteby V .
Let
x y T 1 x y x y
1 n 1
beabasisin V ,where
Dn if x y
y
(3.13) x y i 0 n 1
i
d x
i
where Dk – is the umbral derivativein y of order k, and f x y is the poly-
y
nomialdefiningthecurve.
Thenit followsfromthedefinitionoftheumbralderivativeandtheexis-
tenceofsuchfunctions
(3.14) x y T x y x y x y
0 1 n 1
that
f x Y f x y
(3.15) x y T x Y
Y y
Thefunctions have,clearly,theform
i
x y d x yn i 1 i 0 n 1
i i