Equations for arithmetic pointed tori

193 Pages·2010·2.26 MB·English

Equations for
arithmetic pointed tori
Thesiscommittee:
Prof.dr.P.Bayer,UniversitatdeBarcelona
Prof.dr.G.L.M.Cornelissen,UniversiteitUtrecht
Prof.dr.S.J.Edixhoven,UniversiteitLeiden
Prof.dr.M.vanderPut,RijksuniversiteitGroningen
Dr.J.Voight,UniversityofVermont
Cover: Thewhitepatternontheforegroundofthecoverconsistsoftheedgesof
partofthequadrilateraltilingoftheupperhalf-planeconstructedinSection1.2
forthearithmetic(1;e)-groupe2d1D6i(seetheAppendixforthisnotation).
Thebluebackgroundcontainstheoctagonaltilingassociatedwiththegroup
e3d1D6ii,asconstructedinSection1.3.
ISBN:978-90-393-5373-8.
PrintedbyWo¨hrmannPrintService,Zutphen.
Copyright(cid:13)c 2010byJ.R.Sijsling. Allrightsreserved.
Equations for
arithmetic pointed tori
Vergelijkingen voor
arithmetische gepunte tori
(meteensamenvattinginhetNederlands)
Proefschrift
terverkrijgingvandegraadvandoctoraandeUniversiteitUtrecht
opgezagvanderectormagnificus,prof.dr.J.C.Stoof,
ingevolgehetbesluitvanhetcollegevoorpromoties
inhetopenbaarteverdedigenopmaandag30augustus2010
desmiddagste4.15uur
door
Jan Roelof Sijsling
geborenop23december1983
teArnhem
Promotor: Prof.dr.F.Beukers
OpgedragenaanmijngrootvaderJanSijsling
Forwhowouldlose,
thoughfullofpain,thisintellectualbeing,
thosethoughtsthatwanderthrougheternity...
—JohnMilton,ParadiseLost

Contents
Introduction ix
1 Fundamentaldomains 1
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Dirichletdomains . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Explicithomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Quaternions 21
2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Eichlerorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Moreorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Classandtypenumbers . . . . . . . . . . . . . . . . . . . . . . . 36
3 Curves 39
3.1 Shimuracurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 FromK+ toK1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
B B
3.3 Arithmetic(1;e)-curves . . . . . . . . . . . . . . . . . . . . . . . 50
4 Traces 53
4.1 Quaternioniccuspforms . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 ComputingHeckeoperators . . . . . . . . . . . . . . . . . . . . . 60
4.3 Shimuracongruence . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Uniformizations 69
5.1 Dualgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 SearchingY (p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
0
6 Covers 85
6.1 Bely˘ımaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 (1;e)-covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
viii Contents
7 Equations 115
7.1 Themodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Provingcorrectness . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A Tables 151
Bibliography 159
Indexofnotation 163
Samenvatting(SummaryinDutch) 165
CurriculumVitae 173
Dankwoord(Acknowledgements) 175
Introduction
LetGL+(R)begroupofmatricesin M (R)whosedeterminantispositive,and
2 2
letPGL+(R)bethequotientofGL+(R)byitscenterR×. Denotingthegroup
2 2
ofholomorphicautomorphismsofthecomplexupperhalf-planeHbyAut(H),
onehasthewell-knownisomorphism
PGL+(R) −∼→Aut(H)
2
(cid:20)(cid:18) a b (cid:19)(cid:21) (cid:18) az+b(cid:19)
7−→ z 7→ .
c d cz+d
Let Γ be a Fuchsian group of the first kind, that is, a discrete subgroup of
PGL+(R)offinitecovolume. ThenthegroupΓactsonH,andonecanconsider
2
thequotient
Y(Γ) = Γ\H.
Thisthesisisdevotedtostudyingthesequotientsandtheirarithmeticnature
foraspecialfinitelistofPGL+(R)-conjugacyclassesofΓcutoutbyarather
2
restrictivesetofconditions,asin[Tak83]. Thegreaterpartofthisintroduction
isdedicatedtodescribingtheseconditionsandtogivingabroaderoverviewby
relatingthemtoseveralcongruentareasofmathematics. Afterthis,wedescribe
ourstudyoftheresultinglist.
Lame´ equations
ThefirstconditionsthatweimposeonΓareofageometricnature: wedemand
that
• ThequotientY(Γ)iscompactandofgenus1;and
• Theramificationlocusofthecanonicalprojection
H −π→Y(Γ)
consistsofasinglepointPofY(Γ),whereπramifieswithindexe ≥2.
Intheterminologyof[Bea95],ΓisaFuchsiangroupofsignature(1;e). Being
acompactRiemannsurface,Y(Γ)hasanalgebraicstructure: infact,because
ofthecompactnesscondition,itisaprojectivealgebraiccurveoverC. Ithas
x Introduction
adistinguishedpoint, namely P. WearethereforejustifiedincallingY(Γ) a
pointedcomplextorus,althoughintherestofthisthesis,wewillusuallyreferto
itasa(1;e)-curve.
OurmotivationforconsideringsuchΓandY(Γ)isthefollowing. Consider
themultivaluedinversemap
Y(Γ)9π9−K1 H.
ThisinversemapisintimatelyrelatedtoadifferentialequationonY(Γ). Indeed,
chooseaWeierstrassequation
y2 = p(x)
forY(Γ)suchthatPisthepointatinfinity. Thenπ−1 canbedescribedasthe
quotientoftwosolutionsoftheLame´differentialequation
(cid:20) (cid:21)
d
(y )2−(n(n+1)x+A) u =0 (0.1)
dx
onY(Γ). Herewedenote
1 1
n = − <0,
2e 2
and A ∈Cisanaccessoryparameter. Thisdifferentialequationhasexactlyone
singularpoint,givenbyP,whichisregularsingular.
Lame´ equationscanbethoughtofasbeingthesimplesthighergenusanalogue
ofthehypergeometricdifferentialequations
(cid:20) d2 d (cid:21)
x(1−x) +(c−(a+b+1)x) −ab u =0 (0.2)
dx2 dx
(with a,b,c ∈ C) on P1. These hypergeometric equations describe similar
multivaluedinversemaps
Y(∆)99KH,
namely those arising from triangle groups ∆, which are the Fuchsian groups
whosesignatureequals(0;p,q,r)forsome p,q,r ∈Z≥2.
Thepresenceoftheextraparameter AintheLame´ equation(0.1)isanew
feature that does not occur in the hypergeometric case: like the parameter
n in (0.1), the parameters a, b and c in (0.2) are determined by the signature
(0;p,q,r)oftheassociatedtrianglegroup∆.
We now turn to the second demand on Γ that relates these groups to a
classicalconstructionduetoShimura.
Arithmeticity
Thegeometricdemandsabovearesatisfiedbyacontinuumofdiscretegroups
Γ ⊂PGL+(R),evenuptoconjugacy. Indeed,letEbeanellipticcurve,andlet
2
e ∈Z≥2beaninteger. Considerthemaximalcover
Eee −→ E (0.3)