polyhedral groups
Shigeru MUKAI 3
May,2003
Abstract : Let A
d
b e the moduli space of p olarized ab elian surfaces of
typ e (1;d). For d =2;3 and 4,the Satake compactication of A
d
is isomor-
phic to the quotient of P 3
by an action of PSL(2;Z=d)2PSL(2;Z=d). Let
PSL(2;Z=5)=:G
5
PGL(2)b etheicosahedralgroup andPGL(2)P 3
the
natural embeddingintotheprojectivespace of222 matrices. Asmall resolu-
tion of theSatakecompactication of A
5
(at thepoit cusps)is isomorphicto
thequotientoftheblow-up
~
P 3
atthe60p oints('G
5
)byanactionofG
5 2G
5 .
LetX(d)b ethemo dulispaceofellipticcurvesEwithafullleveldstructure,
i.e.,asymplecticisomorphismb etweenthestandardsymplecticmo dule
2[Z=d)]:=
Z=d8Z=d;
0 1
01 0
andthegroupE
d
ofd-torsionp ointswiththeWeilpairing. Themodularcurve
X(d)isrationalifandonlyifd5. Inparticular,thenitegroupPSL(2;Z=d)
isaregularp olyhedralgroup G
d
PGL(2)ford=2;3;4and5. Thecompact-
ied mo dularcurveX(d)isidentiedwiththecircumscribingRiemannsphere
of the regular p olyhedron P
d
with t vertices,where t is the number ofcusps.
TheorderofG
d
isequaltodt.
d 2 3 4 5
P
d
triangle tetrahedron o ctahedron icosahedron
t 3 4 6 12
G
d S
3
A
4
S
4
A
5
Regularp olyhedralgroupsarealsocloselyrelatedwithHilb ertmo dularsur-
facesofsmalldiscriminant.
3
Supp ortedinpartbytheJSPSGrant-in-AidforExploratoryResearch? and?(Germann
side)?.
Example Let O p
5
b e thering ofintegers of thequadratic eld Q( 5) and
put
0=SL(2;O p
5 :
p
5)=
a b
c d
1
2 mo d
p
5; a;b;c;d;2O p
5
; ad0bc=1
:
Then0 actsonthepro duct H2H of two copiesof theupp erhalf planes.
Let
~
0b ethegroupgeneratedby0andtheswitchinvolutionofH2H. Thenthe
Hilb ertmo dularsurfaceY
0 :=
~
0nH2H addedwith6p ointcuspsisisomorphic
totheprojectiveplaneP 2
. ThisY
0
hasanactionoftheicosdahedralgroupG
5 .
Moreover,X
0
:=0nH2H isthedoublecoverofY
0
withbranchaG
5
-invariant
planecurveofdegree10.
Inthe3-dimensionalcasethewreathpro duct2kG
d
playstheroleofG
d . Let
P 1
2 P 1
P 3
b etheSegreemb edding. Theambientspaceistheprojectivization
of the space of 2 by 2 matrices and the quadric P 1
2P 1
parametrizes the
rank one marices (mo dulo constantmultiplication). Hence thecomplement is
naturally identiedwith PGL(2). This P 3
is an equivariant compactication
of the algebraic group PGL(2) and the p olyhedral group G
d
acts on it from
b oth sides. Let b e the involution of this P 3
interchanging
a b
c d
and
itscofactor matrix
d 0b
0c a
. Thexedlo cus istheunionoftheconstant
matrices
a 0
0 a
andthetracelessones
a b
c 0a
. interchangesthetwo
factorsofP 1
2P 1
. Sothebipolyhedralgroup2kG
d
actsonP 3
.
For ap olarized abeliansurface(X ;L)of type(1;d),a (symplectic)isomor-
phismb etweenthestandard symplecticmo dule2[Z=d]andthegroup
K(L):=fx2X jT 3
x L'Lg
withtheWeilpairingiscalledacanonicallevelstructure. Byacanonicalcolevel
structure,wemeana(symplectic)isomorphismb etween2[Z=d]andthequotient
group X
d
=K(L),whichhasa symplecticstructureasthecomplementofK(L)
inX
d
. Wedenotethemo dulispaceofp olarizedab eliansurfaceswithcanonical
leveland colevel structure by A(1;d) and A(d;1), respectively. The forgetful
morphisms
A(1;d)0!A
d
and A(d;1)0!A
d
arebothGaloiscoveringswithGaloisgroup PSL(2;Z=d). Thebreproduct
A wbl
d
:=A(1;d)2
A A(d;1)
iscalledthemo dulispaceof ab eliansurfaces witha weakbilevelstructure.
For a p olarized ab elian surface (X ;L) oftype (1;d),its dual
^
X hasa nat-
ural polarization
^
L of the same type such that
^
L
L
= d
X
. The colevel
structure of (X ;L) is equivalentto the level structure of its dual (X;L) and
vice versa. Therefore, themodulispace A w bl
d
hasanaction ofwreath pro duct
2kPSL(2;Z=d).
Remark1 Themo dulispaceA
d
isthequotientoftheSigelupp erhalf space
ofdegree2bythefullparamodulargroup
1
4 +
0
B
B
@
Z Z Z dZ
dZ Z dZ dZ
Z Z Z dZ
Z 1
d
Z Z Z
1
C
C
A
\Sp
4 (Q):
A pair (;) of isomorphisms : 2[Z=d]
! K(L) and : 2[Z=d]
! K(
^
L) is
calledacanonicalbilevelstructureof(X ;L). Themo dulispaceA bl
d
ofp olarized
ab eliansurfaces oftype(1;d)withbilevelstructure(X ;L;;)isthequotient
bythesubgroup
1
4 +
0
B
B
@
dZ dZ dZ
dZ dZ dZ d 2
Z
dZ dZ dZ
dZ 1
C
C
A
\Sp
4 (Z):
ThemodulispaceA w bl
d
isthequotientof A bl
d
bytheinvolution
(X ;L;;)7!(X ;L;;0);
whichcorresp ondstotheelement
0
B
B
@ 1
01
1
01 1
C
C
A 2Sp
4 (Q):
Theorem (1) For d = 2;3 and 4, the Satake compactication of A w bl
d is
(2kG
d
)-equivariantlyisomorphic totheprojective 3-spaceP(M
222 C).
(2)Thereexistsa(2kG
5
)-equivariantmorphism
:
~
P 3
0!A w bl
d
ontotheSatakecompacticaitonand contractsthestricttransformsofthe 72
special lines(see below) to the 72 point cusps, where
~
P 3
is the blow-up of P 3
with center G
5
. (The normal bundles of the strict transforms are isomorphic
toO(04)8O (04).) isanisomorphismelsewhere. Moreover,theexceptional
divisorsoverthe60pointsG
5
aretheHilbertmodularsurfaceY
0
inExampleand
parametrizetheComesattisurfaces,i.e.,abeliansurfaceswithrealmultiplication
byO p
5 .
(3) In both cases (1) and (2), P 2P P(M
222
C) parametrizes the
productsowtwoelliptic curves (ofdegree1 andd).
Letp
1
;:::;p
t
bethecuspsoftheellipticmo dularcurveX(d),d=2;3;4;5.
Thenbythetheorem,the2tlinesp
i 2P
1
andP 1
2p
i
,1it,onP 1
2P 1
are
1-dimensional(Satake)b oundariesofA w bl
d . A
wbl
2
andA wbl
3
arethecomplement
ofthese2tlinesinP 3
. Inordertodescrib eA w bl
4
andA w bl
5
,weneedthefollowing:
Denition A line in P 3
joining two p oints [g
1
] and[g
2 ] of G
d
PGL(2) is
specialifg
1 g
01
2 2G
d
isof orderd.
Thenumberofsp eciallinesisiqualto9,16,18and72ford=2;3;4and5.
Inthecased=2;3,thesp eciallinesparametrizethep olarized ab eiansurfaces
(X;L)whichhavesymplecticautomorphismoforderd.
Prop osition (1) The moduli space A wbl
4
is the complement of 12 lines p
i 2
P 1
; P 1
2p
i
andthe 18speciallinesinP 3
.
(2)The moduli spaceA w bl
5
isthe complementof the strict trransformof 12
linesp
i 2P
1
; P 1
2p
i
andthe 72speciallinesinthe blow-up
~
P 3
.
Remark 2 Let K
4
betheklein's subgroupoftheo ctahedral group G
4 . The
actionofK
4 2K
4 onP
3
istheprojectivizationoftheSchrodingerrepresentation
oftheHeisenberggroup. Eachofthe15involutionsinK
4 2K
4
hastheunionof
twoskewlinesasxedp ointlo cus. The18and12linesin(1)oftheprop osition
coincidewiththese 30xedlines.
Acknowledgement This note isessentially[?]. But a confusionamong the
bilevelstructureandtheweakoneiscorrected. Thiswasfoundinthediscussion
of the author with Prof. Klaus Hulek in his stay at Universitat Hannover in
2003. Heisverygrateful foritshospitality.
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ResearchInstituteforMathematicalSciences
KyotoUniversity
KitashirakawaOiwake-cho,Sakyo-ku
Kyoto606-8502,Japan
e-mail address: [email protected]
