The Moduli Space of Real Abelian Varieties with Level Structure

27 Pages·2003·0.24 MB·English

Compositio Mathematica139: 1–27, 2003. 1
#2003KluwerAcademic Publishers. Printed inthe Netherlands.
The Moduli Space of Real Abelian Varieties
with Level Structure
?
MARK GORESKY1 and YUNG SHENG TAI2
1School ofMathematics, Institute forAdvancedStudy,Princeton N.J., U.S.A.
email: [email protected].
2Department ofMathematics, Haverford College, Haverford PA, U.S.A.
(Received: 18October2001; accepted in finalform3 June 2002)
Abstract. ThemodulispaceofprincipallypolarizedAbelianvarietieswithrealstructureand
withlevelN¼4mstructure(withm51)isshowntocoincidewiththesetofrealpointsofa
quasi-projectivealgebraicvarietydefinedoverQ,andtoconsistoffinitelymanycopiesofthe
quotientofthespaceGLðn;RÞ=OðNÞ(ofpositivedefinitesymmetricmatrices)bytheprincipal
congruence subgroupoflevel N in GLðn;ZÞ.
MathematicsSubjectClassification(2000). Primary:14K10,14P05,53C35;Secondary:11G10,
14L35,20G20.
Key words. Abelian variety, real algebraic, symmetric space, arithmetic group, anti-
holomorphicinvolution, modulispace,Comessatti lemma, Baily-Borel compactification.
1. Introduction
1.1. Let h ¼Spð2n;RÞ=UðnÞ be the Siegel upperhalf space of rank n. The quotient
n
space Spð2n;ZÞnh has three remarkable properties: (a) it is the moduli space of
n
principally polarized Abelian varieties, (b) it has the structure of a quasi-projective
complex algebraic variety which is defined over the rational numbers Q, and (c) it
has a natural compactification (the Baily–Borel Satake compactification) which is
defined over the rational numbers.
NowletC ¼GLðn;RÞ=OðnÞbethesymmetricconeofpositivedefinitesymmetric
n
matrices and let Z¼GLðn;ZÞnC . One might ask whether similar statements hold
n
for Z. Is Z in some sense a moduli space for principally polarized real Abelian
varieties?Doesitadmitthestructureofarealalgebraicvariety,possiblydefinedover
Q? If so, does it admit a compactification which is also defined over the rational
numbers? The answer to all these questions is ‘no’. In fact, Silhol [Si] constructs
the moduli space of real principally polarized Abelian varieties and he shows that
itisa(topological)ramifiedcoveringofZ.MoreoverSilholconstructsacompactifi-
cationofthismodulispace,analogoustotheBaily–Borelcompactification.However,
neither the moduli space nor this compactification has an algebraic structure.
?
ResearchpartiallysupportedbyNSFgrants#DMS9900324andDMS0139986.
2 MARKGORESKYANDYUNGSHENGTAI
In this paper we show that all three statements (a), (b), and (c) above may be
recovered if we consider real Abelian varieties with an appropriate level structure.
Tobeprecise,letN¼4mforsomem51,letG ðNÞbetheprincipalcongruencesub-
‘
group of GLðn;ZÞ oflevel N,andlet ZðNÞ¼G ðNÞnC .Then (a) the moduli space
‘ n
of real principally polarized Abelian varieties with level N structure consists of
finitelymanycopiesofZðNÞ.Thesevariouscopiesareindexedbyacertain(nonAbelian
cohomology) set H1ðC=R;GðNÞÞ. Morever, (b) this moduli space may be naturally
identified with the real points X of a quasi-projective algebraic variety X which is
R
definedovertherationalnumbers.SeeTheorems4.2and6.6.Finally,(c)thevariety
X has a compactification which is defined over Q.
C
1.2. InfactthevarietyX isjustthemodulispaceGðNÞnh ofAbelianvarietieswith
C n
principalpolarizationandlevelN¼4mstructure.(Here,GðNÞdenotestheprincipal
congruencesubgroupofSpð2n;ZÞoflevelN.)Thefirstobservationisthatthereisan
involutiontonSpð2n;RÞwhosefixedpointsetisGLðn;RÞ,whichpassestoananti-
holomorphicinvolutiononX .ThisimpliesthatthesetofrealpointsofX consists
C C
of finitely many arithmetic quotients of the symmetric cone C . What is not so
n
obvious, however, is that these quotients are copies of a single arithmetic quotient
G ðNÞnC or that this set of real points may be realized as the moduli space of real
‘ n
principally polarized Abelian varieites with level N structure. Indeed, these state-
ments are likely false for N6(cid:6)0 ðmod4Þ. The key technical tool (which is needed
for both these facts) is the lemma of Comessatti and Silhol ([C, Si, Si2]) describing
certain conditions on the period matrix of a principally polarized Abelian variety
A which are necessary in order for A to admit a real structure.
1.3. Let V¼V denote the Baily–Borel Satake compactification of X . It is
C C
an algebraic variety defined over Q. Complex conjugation t:V !V is an anti-
C C
holomorphic involution whose fixed point set is the set of real points V . This set
R
is compact and it contains X as an open set. One might ask whether V is a
R R
compactification of X , that is, whether V ¼X(cid:2) , where X(cid:2) denotes the closure
R R R R
of X in V . We have been able to prove this (Section 10) in the special case
R C
N¼2k (that is, for the principal congruence subgroup Gð2kÞ of level 2k) for k52,
and we suspect it is otherwise false unless n¼1; (cf. Section 5). However, we
havebeenabletoshow(forgeneralGð4mÞ)thatthecomplementV (cid:8)X(cid:2) hasahigh
R R
codimension. The variety V has a stratification
C
V ¼V0 [V1 [(cid:10)(cid:10)(cid:10)[Vn
C C C C
suchthateachVr isaunionofarithmeticquotientsGnh offinitelymanycopiesof
C n(cid:8)r
the Siegel space of rank n(cid:8)r. The largest stratum V0 is just X . Denote by Vr the
C C R
t-fixed points in Vr. In Proposition 9.2 we show that
C
V0 [V1 (cid:11)X(cid:2) (cid:11)V :
R R R R
Although the moduli space X consists of finitely many disjoint copies of the
R
locallysymmetricspaceZðNÞ,thecompactificationX(cid:2) isnotadisjointunion:some
R
of these copies of ZðNÞ may become glued together along the boundary.
THEMODULISPACEOFREALABELIANVARIETIESWITHLEVELSTRUCTURE 3
1.4. The questions in the first paragraph, and others like them, were posed by
G. Shimura, J. Millson and M. Kuga [J] in the early 1970’s, in the context of
attempts to associate automorphic forms on groups of Hermitian type to auto-
morphic forms on groups of non-Hermitian type. One of many missing ingredients
inthetheoryofautomorphicformsforgroupsofnon-Hermitiantypearisesfromthe
factthattheassociatedlocallysymmetricspacedoesnotappeartohaveanalgebraic
structure, or to be associated with elliptic curves or Abelian varieties. On the other
hand, suppose X is a moduli space for Abelian varieties with certain polarization,
C
endomorphism,andlevelstructures,andsupposethatXhasamodeldefinedoverR.
In [Sh1], Shimura showed that the set of real points X does not necessarily give a
R
moduli space for real Abelian varieties (with the corresponding PEL structures).
ThesequestionswerepartiallyinvestigatedbyA.Adler[A],H.Jaffee[J],S.Kudla
[K],M.Kuga,J.Millson,K.-Y.Shih[Shh],andG.Shimura[Sh1,Sh3,Sh4].Kudla,
forexample,enumeratedthepossiblerealformsanddeterminedthenumberofcon-
nected components of the set of real points of certain compact arithmetic quotients
Gnhn of a product of upper half planes. Related results concerning products hn of
r
Siegel upper half-spaces appear in [Sh3]. Since 1975 however, this line of investiga-
tionappearstohavebeenabandoned.Indeed,itisunlikelythatthisworkcouldhave
progressed much further without (some analog of) the lemma of Comessatti and
Silhol. We believe Theorems 4.2 and 6.6, and the related results in [GT], are but
two examples of a much more general phenomenon involving arithmetic quotients
of symmetric cones associated to (formally real) Jordan algebras.
2. Non-Abelian Cohomology
2.1. Let G be a reductive algebraic group defined over Q, let K be a maximal
compact subgroup of G¼GðRÞ, and let G(cid:11)GðQÞ be an arithmetic subgroup.
Set D¼G=K and X¼GnG=K with projection p:D!X. Suppose t:G!G is an
involution which preserves K and G. Then t passes to an involution on X and we
denote by Xt the fixed point set. Let hti be the group f1;tg and let H1ðhti;GÞ be
the first non-Abelian cohomology set. For any g2G let f :hti!G be the mapping
g
f ð1Þ¼1andf ðtÞ¼g.Thenf isa1-cocycleiffgtðgÞ¼1,inwhichcaseitscohomo-
g g g
logy class is denoted ½ f (cid:18). A cocycle f is a coboundary iff there exists h2G so that
g g
g¼tðhÞh(cid:8)1. To every such 1-cocycle f we associate the ‘g-twisted’ involutions
g
tg:D!D by x7!tðgxÞ and tg:G!G by g07!tðgg0g(cid:8)1Þ. Let Dtg be the fixed point
setinDoftheinvolutiontgandletGtg bethefixedgroupinGoftheinvolutiontg.
To describe Xt it is necessary to understand the various Dtg : if Z2D, then
pðZÞ2Xt iff there exists g2G such that Z2Dtg. For completeness, we recall the
proof of the following theorem ([Ro1, Ro2, RoS]) of Rohlfs.
THEOREM 2.2. Suppose G is torsion free. Then the association
f 7!XðtgÞ¼pðDtgÞ¼GtgnDtg
g
4 MARKGORESKYANDYUNGSHENGTAI
determinesaonetoonecorrespondencebetweenthecohomologysetH1ðhti;GÞandthe
connected components of the fixed point set Xt.
Proof2:3: Thetwistedinvolutiontg:D!Dactsbyisometriesso([H]ChapterI
Section 13.5) the fixed point set Dtg is nonempty. If x;x0 2Dtg then the unique
geodesicconnectingthemisalsofixedbytg,soDtg isconnected.ItsimageinXisa
connectedcomponentXðtgÞofXtwhichdependsonlyonthecohomologyclassoff .
g
It is easy to check that f and f are cohomologous iff XðtgÞ\Xðtg0Þ6¼f. &
g g0
2.4. In general, the cohomology set H1ðhti;GÞ may be difficult to compute, the
connected component XðtgÞ may be difficult to describe, and distinct components
may fail to be isomorphic. We will introduce additional hypotheses which will
allow us to address these three issues. Let G (cid:11)GðQÞ be a t stable arithmetic sub-
0
group which contains G. Let y:G!G be the Cartan involution corresponding to
K. Consider the following possible hypotheses (cf. Remark 2.7, Theorem 4.2 and
Proposition 4.6).
(1) G is Zariski connected, t:G!G is defined over R, its fixed point set
Gt ¼GtðRÞ is Zariski connected, and y preserves Gt,
(2) H1ðhti;KÞ is trivial,
(3) H1ðhti;GÞ!H1ðhti;G Þ is trivial,
0
(4) G contains G as a normal subgroup,
0
(5) t acts trivially on GnG , and
0
(6) G is torsion free.
WeremarkthatifG satisfiesconditions(3)and(4)thenthereisanarithmeticgroup
0
G with G(cid:11)G(cid:11)G which satisfies conditions (3), (4), and (5), namely
0
G¼GðG;G Þ¼fa2G j tðaÞa(cid:8)1 2Gg: ð2:4:1Þ
0 0
LEMMA 2.5. Suppose G(cid:11)G (cid:11)GðQÞ are arithmetic subgroups. Then the following
0
statements hold.
ðaÞ Under hypothesis ð1Þ above, Gt is reductive, y restricts to a Cartan involution of
Gt, and Kt is a maximal compact subgroup of Gt.
ðbÞ Under hypothesis ð2Þ above, Dt ¼Gt=Kt.
ðcÞ Under hypothesis ð3Þ above, for each cohomology class ½ f (cid:18)2H1ðhti;GÞ there
g
exists h2G such that g¼tðhÞh(cid:8)1, in which case,
0
Dtg ¼hDt and Gtg ¼hGt ¼hGth(cid:8)1:
ðdÞ Under hypotheses ð3Þ, ð4Þ, and ð5Þ this association f 7!h determines a one to one
g
correspondence
H1ðhti;GÞffiGnG =Gt: ð2:5:1Þ
0 0
ðeÞ Underhypothesesð3Þ,ð4Þ,ð5Þ,andð6ÞthefixedpointsetXt isthedisjointunionof
THEMODULISPACEOFREALABELIANVARIETIESWITHLEVELSTRUCTURE 5
isomorphic copies
a
Xt ¼ hGtnhDt
h2GnG0=G0t
of the quotient GtnDt.
Proof 2:6: Part (a) is proven in [Sa] Ch. I Theorem 4.2 and Corollary 4.5 (pages
15 and 17). Now consider part (b). Clearly, Gt=Kt (cid:11)Dt so it suffices to show that
Gt acts transitively on Dt. Let x¼gK2Dt. Then tðgÞK¼gK so the element
k¼g(cid:8)1tðgÞ lies in K. Moreover, f is a cocycle, so by (2) there exists u2K with
k
k¼utðuÞ(cid:8)1 ¼g(cid:8)1tðgÞ. Then gu¼tðguÞ2Gt and x¼guK. Part (c) is straight-
forward. Part (d) follows from the long exact sequence
associatedtothesequence1!G!G !GnG !1.Part(e)followsfromRohlfs’
0 0
theorem and parts (c) and (d) however it is also easy to verify directly. First check
that Xt ¼pðS hDtÞ. If h2G then tðhÞh(cid:8)1 2G by (5), which implies that
h2G0 0
pðhDtÞ(cid:11)Xt.Ontheotherhand,ifx2DandpðxÞ2Xt thenthereexistsg2Gsuch
that tðxÞ¼gx. By (6), f is a cocycle, so by (3) there exists h2G such that
g 0
g¼tðhÞh(cid:8)1, hence x2hDt. To see that the union (2.5.1) is disjoint, suppose
pðhDtÞ\pðh0DtÞ6¼f. Then there exists x;x0 2Dt and g2G such that ghx¼h0x0,
hence tðghÞx¼tðh0Þx0. This gives
x¼tðh(cid:8)1g(cid:8)1h0Þx0 ¼tðh(cid:8)1g(cid:8)1h0Þðh(cid:8)1g(cid:8)1h0Þ(cid:8)1x:
Foranyu2G ,(5)impliesthattðuÞu(cid:8)1 2G,whichistorsionfree.Takeu¼h(cid:8)1g(cid:8)1h0.
0
Then tðuÞ¼u so u2Gt and h0 ¼ghu2GhGt. &
0 0
Remark2:7: IfGissemisimple,connectedandsimplyconnectedandift:G!G
is defined over R then, according to a theorem of Rasˇevskii˜ ([Ra1, Ra2]) and
Steinberg [St], the complex group GðCÞt ¼GtðCÞ is connected, so its group of real
points Gt is Zariski connected.
3. Symplectic Group
3.1. The symplectic group G¼Spð2n;RÞ may be realized as the group of 2n by 2n
real matrices
((cid:2)A B(cid:3)(cid:4)(cid:4)At D(cid:8)tCB¼I )
Spð2n;RÞ ¼ (cid:4)
C D (cid:4)At C;tBD symmetric
((cid:2)A B(cid:3)(cid:4)(cid:4) AtD(cid:8)BtC¼I )
¼ (cid:4)
C D (cid:4) AtB;CtD symmetric
6 MARKGORESKYANDYUNGSHENGTAI
IfJ¼(cid:2) 0 In(cid:3)isthestandardsymplecticformonR2ntheng2Spð2n;RÞifftgJg¼J.
(cid:8)In 0
The inverse of such a symplectic matrix is
(cid:2)A B(cid:3)(cid:8)1 (cid:2) tD (cid:8)tB(cid:3)
¼ ð3:1:1Þ
C D (cid:8)tC At
Identify GLðn;RÞ,!Spð2n;RÞ with its image under the embedding
(cid:2) (cid:3)
A 0
A7!
0 At (cid:8)1
A Cartan involution y of Spð2n;RÞ is given by yg¼JgJ(cid:8)1, that is,
(cid:2) (cid:3) (cid:2) (cid:3)
A B D (cid:8)C
y ¼
C D (cid:8)B A
ItsfixedpointsetistheunitarygroupK¼UðnÞwhichisembeddedinthesymplectic
group by
(cid:2) (cid:3)
A B
AþiB7! :
(cid:8)B A
The symplectic group acts transitively on the Siegel upper halfspace
h ¼fZ¼XþiY2M ðCÞ j tZ¼Z and Y>0g
n n(cid:25)n
by fractional linear transformations: if g¼(cid:2)AB(cid:3) then
CD
gZ¼ðAZþBÞðCZþDÞ(cid:8)1:
Then Stab ðiIÞ¼UðnÞ, and h ffiG=K.
G n
3.2 Let I ¼(cid:2)(cid:8)I0(cid:3)2GLð2n;RÞ. Consider the involution t: Spð2n;RÞ!Spð2n;RÞ
(cid:8) 0 I
which is defined by tðgÞ¼I gI , that is,
(cid:8) (cid:8)
(cid:2) (cid:3) (cid:2) (cid:3)
A B A (cid:8)B
t ¼ ð3:2:1Þ
C D (cid:8)C D
The following properties of t are easily verified by direct calculation:
(1) t is an involution of G, i.e. tðgh(cid:8)1Þ¼tðgÞðtðhÞÞ(cid:8)1.
(2) tðgÞ¼g,g2GLðn;RÞ
(3) ty¼yt so tðUðnÞÞ¼UðnÞ
(4) If C¼AþiB2UðnÞ then tðCÞ¼C(cid:2) ¼A(cid:8)iB.
It follows that t passes to an involution (which we denote by the same letter)
t:h !h on the Siegel space such that
n n
tðg(cid:10)ZÞ¼tðgÞ(cid:10)tðZÞ
for all g2Spð2n;RÞ and for all Z2h . In fact t:h !h is the antiholomorphic
n n n
involution given by tðZÞ¼(cid:8)Z(cid:2). Its fixed point set is the orbit
iC ¼GLðn;RÞ(cid:10)iI(cid:11)M ðCÞ ð3:2:2Þ
n n(cid:25)n
THEMODULISPACEOFREALABELIANVARIETIESWITHLEVELSTRUCTURE 7
of the general linear group. Here, C ffiGLðn;RÞ=OðnÞ denotes the cone of positive
n
definite symmetric real matrices. (See also Proposition 4.6.)
NOTATION3.3. Forg2GandZ2h wewillusuallywriteg(cid:2) ¼tðgÞandZ~ ¼tðZÞ.
n
For any g2Spð2n;RÞ, define the locus of g-real points
htg ¼fZ2h jgZ¼(cid:8)Z(cid:2)g ð3:3:1Þ
n n
asinSection2.1.IfG(cid:11)Spð2n;RÞisanarithmeticsubgroupsuchthattG¼G,define
the set htG of G-real points to be the union S htg.
n g2G n
LEMMA 3.4. Let g2Spð2n;RÞ and let hg be the set of points in h which are fixed
n n
under theactionofg.Thenhg\iC isaproperreal algebraicsubvariety ofiC unless
n n n
g¼(cid:27)I2GLðn;RÞ.
Proof 3:5: Suppose (cid:2)AB(cid:3)(cid:10)iY¼iY. Then AY¼YD and B¼(cid:8)YCY. These are
CD
algebraic equations for Y. Suppose every Y2C is a solution. Taking Y¼tI (with
n
t>0)givesA¼DandB¼(cid:8)t2C.SoB¼C¼0andAY¼YAforallY2C ,which
n
implies that A is a scalar matrix and that D¼tA(cid:8)1. So the scalar is (cid:27)1. &
4. The Variety Gð4mÞnh
n
4.1. We shall be concerned with the following arithmetic groups:
. Gð1Þ¼Spð2n;ZÞ
(cid:4) (cid:5)
. GðNÞ¼ g2Gð1Þj g(cid:6)Iðmod NÞ
(cid:9)(cid:6)A B(cid:7) (cid:8) A;D(cid:6)I ðmod2Þ (cid:10)
. G ð2Þ¼ 2G(cid:8)
2m C D (cid:8)B;C(cid:6)0 ðmod2mÞ
. G ð1Þ¼GLðn;ZÞ
‘
. G ðNÞ¼fg2G ð1Þjg(cid:6)Iðmod NÞg
‘ ‘
(The analogous construction of G ð2Þ does not yield a group unless m is even.)
m
Throughout the remainder of the paper we will be interested in the case N¼4m
and m51. The main result in this section is the following.
THEOREM 4.2. For all m51, the mapping H1ðhti;Gð4mÞÞ!H1ðhti;G ð2ÞÞ is
2m
trivial.
4.3. LetX¼GðNÞnh .ThenXhasthestructureofaquasi-projectivecomplexalge-
n
braic variety. The involution tðZÞ¼(cid:8)Z(cid:2) passes to an anti-holomorphic involution
on X and defines a real structure on X. (It is a theorem of Shimura [Sh2] that the
Baily–BorelcompactificationVofXevenadmitsthestructureofanalgebraicvariety
defined over the rational numbers. See Section 8.) Let S¼htG be the set of all
n
Gð4mÞ-real points in h . Denote by Gð4mÞnS the image of S in X¼Gð4mÞnh . Note
n n
that Gð4mÞ is normal in Gð2Þ and, in the notation of (2.4.1),
G ð2Þ¼GðGð4mÞ;Gð2ÞÞ:
2m
So Lemma 2.5 gives the following:
8 MARKGORESKYANDYUNGSHENGTAI
COROLLARY4.4. ThesetX ofrealpointsofXispreciselyGð4mÞnS.Itconsistsof
R
the disjoint union
a
X ¼ gG ð4mÞngiC ð4:4:1Þ
R ‘ n
g
of finitely many copies of G ð4mÞnC , indexed by elements
‘ n
g2Gð4mÞnG ð2Þ=G ð2Þ¼H1ðhti;Gð4mÞÞ: ð4:4:2Þ
2m ‘
4.5. The following tower of groups contains Gð4mÞ:
G ð2Þ(cid:11)Gð2Þ(cid:11)Spð2n;ZÞ(cid:11)Spð2n;RÞ:
2m
WewillshowthattheimageofH1ðhti;Gð4mÞÞinthecohomologyofeachoftheseis
trivial, starting with the rightmost.
PROPOSITION 4.6. With respect to the above action ð3:2:1Þ, the ðnon-AbelianÞ
cohomology sets H1ðhti;UðnÞÞ and H1ðhti;Spð2n;RÞÞ are trivial.
Proof 4:7: Let g¼(cid:2)A B(cid:3)2UðnÞ and suppose f is a 1-cocycle, that is, gg~ ¼I.
(cid:8)BA g
Then A2þB2 ¼I and AB¼BA. Comparing with (3.1.1) gives At ¼A and tB¼B.
Such commuting symmetric matrices may be simultaneously diagonalized by an
orthogonal matrix h2OðnÞ. In other words, hðAþiBÞh(cid:8)1 ¼L2UðnÞ is a diagonal
pffiffiffiffi
matrix. Choose a square root, m¼ L(cid:2) 2UðnÞ (by choosing a square root of each
diagonal entry) and set d¼h(cid:8)1mh. Since h~¼h and m(cid:2) ¼m(cid:8)1, we find
d~d(cid:8)1 ¼h(cid:8)1tðmÞm(cid:8)1h¼h(cid:8)1m(cid:2)m(cid:2)h¼h(cid:8)1Lh¼AþiB
so f is a coboundary, hence H1ðUðnÞÞ¼0. From this we may also conclude:
g
if g2Spð2n;RÞ and if f is a 1-cocycle then
g
f is a coboundary iff htg 6¼f: ð4:7:1Þ
g n
For, suppose Z2htg. Write Z¼hðiIÞ for some h2Spð2n;RÞ. Then h~(cid:8)1ghðiIÞ¼
n
h~(cid:8)1gZ¼h~(cid:8)1Z~ ¼tðiIÞ¼iI hence a¼h~(cid:8)1gh2UðnÞ. Moreover, aa~ ¼h~(cid:8)1gg~h~¼I so
by the preceding paragraph there exists d2UðnÞ with a¼d~d(cid:8)1. Therefore
g¼tðhdÞðhdÞ(cid:8)1.
At this point we remark that if g¼(cid:2)AB(cid:3)2Spð2n;RÞ and if gg~ ¼I, then
CD
A¼tD; C¼tC; B¼tB; A2(cid:8)BC¼I; CA¼tAC; and AB¼B tA: ð4:7:2Þ
Nextweconsidertwospecialcases.Letg¼(cid:2)AB(cid:3)2Spð2n;RÞandsupposegg~ ¼I.
CD
Suppose either (1) A¼D¼0 and C¼(cid:8)B(cid:8)1 is diagonal, or (2) C¼0. Then f is a
g
coboundary. For in case (1), let B¼diagðb ;b ;...;b Þ and let Z¼iY where
1 2 n
Y¼diagðjb j;jb j;...;jb jÞ.Theng(cid:10)Z¼(cid:8)BZ(cid:8)1B¼(cid:8)Z(cid:2) ¼tðZÞsof isacobound-
1 2 n g
ary by (4.7.1). In case (2), the cocycle f is equivalent to f where
g g0
(cid:2) (cid:3)
A 0
g0 ¼ ¼h~gh(cid:8)1
0 tA
THEMODULISPACEOFREALABELIANVARIETIESWITHLEVELSTRUCTURE 9
and
(cid:2) (cid:3)
I 1A(cid:8)1B
h¼ 2
0 I
using (4.7.2). Then A2 ¼I so A is GLðn;RÞ-conjugate to a diagonal matrix L (with
eigenvalues (cid:27)1). Therefore f is equivalent to the 1-cocycle defined by the element
g0
(cid:2)L 0(cid:3)2UðnÞ. It follows that f is cohomologically trivial.
0L(cid:8)1 g0
Nowconsiderthecaseofageneral1-cocyclef whereg0 ¼(cid:2)A0B0(cid:3).By(4.7.2),C0 is
g0 C0D0
symmetricsothereexistsk2OðnÞsothatC¼kC0tk¼(cid:2)C0100(cid:3)whereC1 2Mr(cid:25)rðRÞis
diagonal and nonsingular. Let h ¼(cid:2)k0(cid:3). Then f is cohomologous to f where
1 0k g0 g
(cid:2) (cid:3)
A B
g¼h~ g0h(cid:8)1 ¼
1 1 C D
(cid:13) (cid:14)
and (4.7.2) holds. Write A¼ AA14AA24 where A1 2Mr(cid:25)rðRÞ and A4 2Mðn(cid:8)rÞ(cid:25)ðn(cid:8)rÞðRÞ,
and similarly for B and D. Then CA¼tAC implies that A ¼A ¼0.
2 3
(cid:13) (cid:14)
Set h ¼ Ix where
2 0I
(cid:2) (cid:3)
C(cid:8)1 tA 0
x¼ 1 1 :
0 0
Then A2(cid:8)BC¼I implies that f is cohomologous to f where
g g
Since gg~ ¼I use case (1) and (2) above to find Z 2h and Z 2h so that
r r n(cid:8)r n(cid:8)r
(cid:2) (cid:3) (cid:2) (cid:3)
0 (cid:8)C(cid:8)1 A B
1 Z ¼(cid:8)Z(cid:2) and 4 4 Z ¼(cid:8)Z(cid:2) :
C 0 r r 0 tA n(cid:8)r n(cid:8)r
1 4
(cid:13) (cid:14)
Set Z¼ Zr 0 . Then gZ¼(cid:8)Z(cid:2) hence the corresponding cocycle is cohomologi-
0 Zn(cid:8)r
cally trivial. &
LEMMA 4.8. Let g2Spð2n;ZÞ. Fix m51.
ð1Þ If g2GðmÞ then g~g(cid:8)1 2Gð2mÞ.
ð2Þ If g2Gð2mÞ then g~g2Gð4mÞ.
ð3Þ If g~g(cid:8)1 2Gð4mÞ then g¼bu for some b2G ð2Þ and for some u2GLðn;ZÞ.
2m
Proof 4:9: Let g¼(cid:2)AB(cid:3). Since g~ ¼g(cid:8)(cid:2)0 2B(cid:3) we have
CD 2C 0
(cid:2)(cid:8)2B tC 2B tA (cid:3)
g~g(cid:8)1 ¼I(cid:8) : ð4:9:1Þ
2C tD (cid:8)2C tA
10 MARKGORESKYANDYUNGSHENGTAI
This gives part (1). If g¼Iþ2mg0 then g~þg is even, and g~g¼Iþ2mðg~0þg0Þþ
4m2g~0g which proves part (2). Now suppose that g~g(cid:8)1 2Gð4mÞ so that B tC(cid:6)
B tA(cid:6)C tD(cid:6)0 mod2mÞ. Then
B¼BðtAD(cid:8) tCBÞ¼ðB tAÞD(cid:8)ðB tCÞB(cid:6)0 ðmod2mÞ
tC¼ðtAD(cid:8) tCBÞ tC¼ tAðD tCÞ(cid:8) tCðB tCÞ(cid:6)0 ðmod2mÞ:
Moreover, since C tB(cid:6)0 ðmodð2mÞÞ and A tD(cid:8)B tC¼I we see that A tD(cid:6)
I ðmod2Þ. Hence A is invertible ðmod2Þ. But reduction ðmod2Þ is a surjective
mapping GLðn;ZÞ!GLðn;Z=ð2ÞÞ hence there exists U2GLðn;ZÞ so that U(cid:6)
A ðmod2Þ from which it also follows that U(cid:8)1 (cid:6) tD ðmod2Þ. Let u¼
(cid:2)U 0 (cid:3). Then
0tU(cid:8)1
(cid:2) (cid:3)
AU(cid:8)1 B tU
b¼gu(cid:8)1 ¼ 2G ð2Þ: &
CU(cid:8)1 D tU 2m
The following proposition is a consequence of the theorem of Silhol [Si] (Theorem
1.4) and Comessatti, and will be proven in Section 7.
PROPOSITION 4.10. Let g2Gð2Þ and suppose that Z2h is not fixed by any ele-
n
ment of Spð2n;ZÞ other than (cid:27)I. Suppose that Z~ ¼gZ. Then there exists
h2Spð2n;ZÞ such that g¼h~h(cid:8)1.
4.11. PROOFOFTHEOREM4.2.
Let g2Gð4mÞ and suppose f is a cocycle. Its image in H1ðhti;Spð2n;RÞÞ is a
g
coboundary so there exists h2Spð2n;RÞ with g¼h~h(cid:8)1. This implies that
htg ¼hiC is a copy of the cone of positive definite matrices. By Lemma 3.4 there
n n
exist points Z2htg which are not fixed by any element of Spð2n;ZÞ other than
n
(cid:27)I and in fact the set of such points is the complement of a countable union of
proper (real) algebraic subvarieties of htg. Applying Proposition 4.10, we conclude
n
that g¼h~h(cid:8)1 for some h2Spð2n;ZÞ. By Lemma 4.8 (3) we may write h¼bu for
some b2G ð2Þ and some u2GLðn;ZÞ. Then h~h(cid:8)1 ¼b~b(cid:8)1 ¼g which proves that
2m
f is trivial in H1ðhti;G ð2ÞÞ. &
g 2m
5. Upper Half-plane
Fortheupperhalf-planeh itispossibletodoalittlebettersincethegroupGð2Þ=(cid:27)I
1
acts freely.
LEMMA 5.1. A point Z2h is Gð2Þ-real if and only if gZ is Gð2Þ-real, for every
1
g2SLð2;ZÞ.
Proof 5:2: Suppose tðZÞ¼gZ for some g2Gð2Þ. Then
tðgzÞ¼tðgÞtðzÞ¼ðtðgÞg(cid:8)1Þðggg(cid:8)1ÞgZ2Gð2Þ(cid:10)Gð2Þ(cid:10)gZ: &