Higher genus Abelian functions associated with algebraic - ROS

307 Pages·2009·2.52 MB·English

Higher genus Abelian functions
associated with algebraic curves
Matthew England
Submitted for the qualification of
Doctor of Philisophy
Heriot Watt University
School of Mathematics and Computer Science
Department of Mathematics
December 2009
Thecopyrightinthisthesisisownedbytheauthor. Anyquotationfromthethesis
oruseofanyoftheinformationcontainedinitmustacknowledgethisthesisas
thesourceofthequotationorinformation.
Abstract
We investigate the theory of Abelian functions with periodicity properties defined from
an associated algebraic curve. A thorough summary of the background material is given,
including a synopsis of elliptic function theory, generalisations of the Weierstrass σ and
℘-functionsandaliteraturereview.
The theory of Abelian functions associated with a tetragonal curve of genus six is con-
sidered in detail. Differential equations and addition formula satisfied by the functions are
derived and a solution to the Jacobi Inversion Problem is presented. New methods which
centre on a series expansion of the σ-function are used and discussions on the large com-
putations involved are included. We construct a solution to the KP equation using these
functions and outline how a general class of solutions can be generated from a wider class
ofcurves.
We proceed to present new approaches used to complete results for the lower genus
trigonalcurves. Wealsogivesomedetailsonthethetheoryofhighergenustrigonalcurves
before finishing with an application of the theory to the Benney moment equations. A
reduction is constructed corresponding to Schwartz-Christoffel maps associated with the
tetragonal curve. The mapping function is evaluated explicitly using derivatives of the σ-
function.
i
Acknowledgments
ThanksfirsttomysupervisorProfessorChrisEilbeckforhisguidancethroughoutmystud-
iesandhisproofreadingofthisdocument. ProfessorEilbeckismycoauthorin[39],upon
which Chapter 3 of this document is based. Many thanks also to Dr. John Gibbons for his
patientexplanationsofsomecomplicatedideas. Dr. Gibbonsismycoauthorin[40],upon
whichChapter6ofthisdocumentisbased.
NextIwouldliketoacknowledgeDr. Y.OˆnishiforhisassistancewithLemma3.5.2,Dr
A.Nakayashikiforhissuggestionthatledtotheconstructionofthebasisinequation(3.55)
and Dr. V. Enolski for useful conversations. Acknowledgments should also be given to the
anonymous referees of my papers [39] and [40], whose comments have in turn improved
thisdocument.
Finally, I would like to thank Sue England, Cliff England and Pinar Ozdemir for their
continuedsupport.
ii
Contents
Abstract i
Acknowledgments ii
Contents iii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Whatiscontributedinthisthesis . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Guidetothisdocument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 BackgroundMaterial 10
2.1 Ellipticfunctiontheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Meromorphicfunctions . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Ellipticfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.3 TheWeierstrass℘-function . . . . . . . . . . . . . . . . . . . . . . 17
2.1.4 Weierstrass’quasi-periodicfunctions . . . . . . . . . . . . . . . . 24
2.1.5 Theadditionformulae . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Abelianfunctiontheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.1 Curves,surfacesanddifferentials . . . . . . . . . . . . . . . . . . 32
2.2.2 Abelianfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.3 TheKleinianσ-function . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.4 TheKleinian℘-functions . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 Literaturereview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.1 Thegenustwogeneralisation . . . . . . . . . . . . . . . . . . . . . 51
2.3.2 Developingthegeneraldefinitions . . . . . . . . . . . . . . . . . . 53
2.3.3 Abelianfunctionsassociatedtotrigonalcurves . . . . . . . . . . . 55
2.3.4 Recentandadditionalcontributions . . . . . . . . . . . . . . . . . 56
3 Abelianfunctionsassociatedwithacyclictetragonalcurveofgenussix 57
3.1 The(4,5)-curveandassociatedfunctions . . . . . . . . . . . . . . . . . . . 59
3.1.1 Constructingthefundamentaldifferential . . . . . . . . . . . . . . 60
iii
3.1.2 Abelianfunctionsassociatedwiththe(4,5)-curve . . . . . . . . . . 64
3.1.3 TheQ-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 ExpandingtheKleinianformula . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.1 Generatingrelationsbetweenthe℘-functions . . . . . . . . . . . . 73
3.2.2 SolvingtheJacobiInversionProblem . . . . . . . . . . . . . . . . 76
3.3 TheSatoweights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4 Theσ-functionexpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4.1 Propertiesoftheσ-function . . . . . . . . . . . . . . . . . . . . . 84
3.4.2 Constructingtheexpansion . . . . . . . . . . . . . . . . . . . . . . 86
3.5 RelationsbetweentheAbelianfunctions . . . . . . . . . . . . . . . . . . . 94
3.5.1 BasisforthefundamentalAbelianfunctions . . . . . . . . . . . . . 95
3.5.2 DifferentialequationsintheAbelianfunctions . . . . . . . . . . . 99
3.6 Additionformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.7 ApplicationsintheKPhierarchy . . . . . . . . . . . . . . . . . . . . . . . 109
4 Highergenustrigonalcurves 112
4.1 Thecyclictrigonalcurveofgenussix . . . . . . . . . . . . . . . . . . . . 113
4.1.1 Differentialsandfunctions . . . . . . . . . . . . . . . . . . . . . . 113
4.1.2 ExpandingtheKleinianformula . . . . . . . . . . . . . . . . . . . 116
4.1.3 Theσ-functionexpansion . . . . . . . . . . . . . . . . . . . . . . 119
4.1.4 RelationsbetweentheAbelianfunctions . . . . . . . . . . . . . . . 122
4.1.5 Additionformula . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.2 Thecyclictrigonalcurveofgenusseven . . . . . . . . . . . . . . . . . . . 130
4.2.1 Differentialsandfunctions . . . . . . . . . . . . . . . . . . . . . . 130
4.2.2 ExpandingtheKleinianformula . . . . . . . . . . . . . . . . . . . 133
4.2.3 Theσ-functionexpansion . . . . . . . . . . . . . . . . . . . . . . 134
4.2.4 RelationsbetweentheAbelianfunctions . . . . . . . . . . . . . . . 135
5 NewapproachesforAbelianfunctionsassociatedtotrigonalcurves 137
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 BilinearrelationsandB-functions . . . . . . . . . . . . . . . . . . . . . . 142
5.2.1 DefiningtheB-functions . . . . . . . . . . . . . . . . . . . . . . . 142
5.2.2 Derivingbilinearrelations . . . . . . . . . . . . . . . . . . . . . . 145
5.2.3 Thecyclic(3,4)-case . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.2.4 Thecyclic(3,5)-case . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.2.5 Highergenuscurves . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.3 Quadraticrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.3.1 Derivingquadraticrelations . . . . . . . . . . . . . . . . . . . . . 154
5.3.2 Thecyclic(3,4)-case . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.3.3 Thecyclic(3,5)-curve . . . . . . . . . . . . . . . . . . . . . . . . 165
iv
5.3.4 Highergenuscurves . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.4 Calculatingthesecondbasisinthe(3,5)-case . . . . . . . . . . . . . . . . 170
5.4.1 Possiblefunctionsforinclusioninthebasis . . . . . . . . . . . . . 170
5.4.2 Derivingbasisentries . . . . . . . . . . . . . . . . . . . . . . . . . 172
6 ReductionsoftheBenneyequations 175
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.2 Benney’sequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2.1 Reductionsofthemomentequations . . . . . . . . . . . . . . . . . 178
6.2.2 Schwartz-Christoffelreductions . . . . . . . . . . . . . . . . . . . 179
6.3 Atetragonalreduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.4 Relationsbetweentheσ-derivatives . . . . . . . . . . . . . . . . . . . . . 185
6.4.1 Thedefiningstratarelations . . . . . . . . . . . . . . . . . . . . . 185
6.4.2 Furtherrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.5 Evaluatingtheintegrand . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.5.1 Theexpansionofϕ (u)atthepoles . . . . . . . . . . . . . . . . . 193
2
6.5.2 FindingasuitablefunctionΨ(u) . . . . . . . . . . . . . . . . . . . 197
6.5.3 EvaluatingthevectorB . . . . . . . . . . . . . . . . . . . . . . . 200
6.6 Anexplicitformulaforthemapping . . . . . . . . . . . . . . . . . . . . . 201
Appendices 203
A BackgroundMathematics 204
A.1 Propertiesofellipticfunctions . . . . . . . . . . . . . . . . . . . . . . . . 204
A.2 Jacobiθ-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
A.3 Multivariateθ-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
A.4 TheWeierstrassGapSequence . . . . . . . . . . . . . . . . . . . . . . . . 213
A.5 TheSchur-Weierstrasspolynomial . . . . . . . . . . . . . . . . . . . . . . 217
A.5.1 WeierstrassPartitions . . . . . . . . . . . . . . . . . . . . . . . . . 217
A.5.2 Symmetricpolynomials . . . . . . . . . . . . . . . . . . . . . . . 221
A.5.3 Schur-WeierstrassPolynomials . . . . . . . . . . . . . . . . . . . . 223
A.6 Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
B Independencefromtheconstantc 230
C Resultsforthecyclictetragonalcurveofgenussix 236
C.1 Expansionsinthelocalparameter . . . . . . . . . . . . . . . . . . . . . . 236
C.2 Theσ-functionexpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
C.3 The4-index℘-functionrelations . . . . . . . . . . . . . . . . . . . . . . . 239
v
D Newresultsforthecyclictrigonalcurveofgenusfour 248
D.1 ExpressionsfortheB-functions . . . . . . . . . . . . . . . . . . . . . . . 248
D.2 Quadratic3-indexrelations . . . . . . . . . . . . . . . . . . . . . . . . . . 254
E Stratarelationsforthecyclictetragonalcurveofgenussix 289
E.1 Relationsforu ∈ Θ[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
E.2 Relationsforu = u . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
0,N
Bibliography 296
vi
Chapter 1
Introduction
Recent times have seen a revival of interest in the theory of Abelian functions associated
with algebraic curves. An Abelian function may be defined as one that has multiple in-
dependent periods, in this case derived from the periodicity property of an underlying al-
gebraic curve. The topic can be dated back to the Weierstrass theory of elliptic functions
whichweuseasamodel.
The ellipticfunctions haveoften beendescribed as oneof thejewels ofnineteenth cen-
tury mathematics and have been the subject of study from mathematicians including Abel,
Gauss, Jacobi, Legendre, Riemann and Weierstrass. They have been of great importance
since their original definition and have been applied in a variety of mathematical areas.
Over the last three decades their relevance in physics and applied mathematics has also
been greatly developed. This has in turn inspired renewed interest in the theory of Abelian
functions in which solutions to a number of the challenging problems of mathematical
physicsoccurnaturally.
In this chapter we will start in Section 1.1 by describing the motivation for our work.
We discuss the theory of the Weierstrass ℘-function and introduce the generalisation that
we work with. Note that formal definitions and proofs will be presented in Chapter 2 with
the material in this chapter just for introductory purposes. We will highlight the important
areasofthetheorywhicharestudiedindetailduringtheremainderofthedocument.
In Section 1.2 we will proceed to identify the key contributions that are made to the
topicbythisdocument. Thisincludesnewclassesofproblemsstudied,newtechniquesand
methods used to solve the problems and new interpretations of the generalisation. Finally,
inSection1.3wepresentaguidetothisdocument.
1
Chapter1: Introduction Section1.1: Motivation
1.1 Motivation
Let℘(u)betheWeierstrass℘-function. WedefinethisformallyinChapter2andshowthat
ithastwocomplexperiodsω ,ω :
1 2
℘(u+ω ) = ℘(u+ω ) = ℘(u), forall u ∈ C. (1.1)
1 2
Functionsthataredoublyperiodicareknownasellipticfunctionsandhavebeenthesubject
ofmuchstudysincetheirdiscoveryinthe1800s. The℘-functionisaparticularlyimportant
elliptic function which has the simplest possible pole structure for an elliptic function. It
haspolesofordertwowhichoccuronlyonthosepointsthatareasumofintegermultiples
oftheperiods.
The℘-functionsatisfiesanumberofinterestingproperties. Forexample,itcanbeused
toparametriseanellipticcurve,
y2 = 4x3 −g x−g , (1.2)
2 3
where g and g are constants. It also satisfies the following well-known differential equa-
2 3
tions,
(cid:0)℘(cid:48)(u)(cid:1)2 = 4℘(u)3 −g ℘(u)−g , (1.3)
2 3
℘(cid:48)(cid:48)(u) = 6℘(u)2 − 1g . (1.4)
2 2
Weierstrassintroducedanauxiliaryfunction,σ(u),inhistheorywhichsatisfied,
d2
(cid:2) (cid:3)
℘(u) = − log σ(u) . (1.5)
du2
This σ-function plays a crucial role in the generalisation and in applications of the theory.
Aparticularlyinterestingresultitsatisfiesisthefollowingtwotermadditionformula.
σ(u+v)σ(u−v)
− = ℘(u)−℘(v). (1.6)
σ(u)2σ(v)2
TakinglogarithmicderivativesofthiswillgivethestandardadditionformulafortheWeier-
strass℘-function. Inthisdocumentwepresentgeneralisationsofequations(1.3)−(1.6)for
newclassesoffunctions.
Klein developed an approach to generalise the Weierstrass ℘-function to a function
with two variables and four periods. Such functions were defined as hyperelliptic and the
approachisdescribedinBaker’sclassictexts[7]and[10]from1897and1907respectively.
2
Chapter1: Introduction Section1.1: Motivation
Itiscenteredaroundageneralisationoftheσ-functionwhichisthenusedtodefinenew
℘-functions by equation (1.5). This approach has motivated the general definition of what
wenowcallKleinian℘-functions.
Recallthathyperellipticcurvesarealgebraiccurveswithequations,
y2 = f(x),
where f(x) is a polynomial of degree greater than four. The original functions of Klein
andBakerwereassociatedwiththesimplesthyperellipticcurve,(withf(x)ofdegreefive).
Baker later constructed examples using another hyperelliptic curve, but a theory for the
functionsassociatedtoanarbitraryhyperellipticcurvedidnotfollowuntilthe1990swhen
Buchstaber,EnolskiiandLeykinpublished[19].
This was followed by a general definition for Abelian functions, structured by the un-
derlying algebraic curves to which the functions are associated. In this document we work
withcyclic(n,s)-curvesdefinedby,
yn = xs +λ xs−1 +...λ x+λ ,
s−1 1 0
whereλ ,...,λ areconstantsand(n,s)apairofcoprimeintegers. Notethattheoriginal
0 s−1
elliptic curve in equation (1.2) was an (n,s)-curve with (n,s) = (2,3) while the hyperel-
lipticcurvesarethose(n,s)-curveswithn = 2ands > 4.
Ineachcasethegenusofthecurveisauniqueintegerassociatedtothecurvethatplays
akeyrole. Inparticular,thefunctionsbecomemultivariatewithg variables,
σ = σ(u) = σ(u ,u ,...u ).
1 2 g
The elliptic curve has genus one and hence the Weierstrass ℘-function has one complex
variable. Thegeneralisationshowever,willhavegenusgreaterthanone.
We will define the Kleinian ℘-functions as logarithmic derivatives of σ(u) in analogy
with equation (1.5). However, we introduce a new subscript notation to make clear which
variablesareusedinthedifferentiation.
∂2
(cid:2) (cid:3)
℘ (u) = − log σ(u) .
ij
∂u ∂u
i j
Further derivatives are described by adding more subscripts and we may refer to the func-
tionsasn-index℘-functions.
∂ ∂ ∂
(cid:2) (cid:3)
℘ (u) = − ... log σ(u) , i ≤ ··· ≤ i ∈ {1,...,g}. (1.7)
i1,i2,...,in ∂u ∂u ∂u 1 n
i1 i2 in
The generalised σ and ℘-functions have properties in analogy to the Weierstrass functions.
3