Math. Nachr. 200 (1999), 21-58

Hyperbolic D -Planes

By BRUCE HUNT of Leipzig

(Received March 15, 1996)

(Revised Version February 6, 1997)

0. Introduction

A plane is a two-dimensional right vector spacc! V ovoI 11 rilvlniori dgebra D, which we assume in this paper has an involution; a hermltlnrr form Ir : V x V --t D is hyperbolic, if, with respect to a properly choosen b ~ h , 11. IH givan by the matrix ( y A); the pair (V, h) is called a hyperbolic plane. If D In wiitriil nirrrpla over a number field K, a hyperbolic plane (V, h) gives rise to a rcductlvc~ I( group G” = V(V, h) and a simple K-group SGD = SU(V,h) . In thi8 pajw wv Htudy 1 , h case where the real Lie group G(R) is of hermitian type, in other word# tliirt. tho Rymmetric space of maximal compact subgroups of G(R) is a hermitian flyiirtiiotsrlc H ~ I L W . Let d = deg D (that is, d i m ~ D = 8); we consider the following CMYIRH:

1) d = 1: k is a totally real number field of dcgrcw j o v w 42, K l k is an imaginary quadratic extension, and D := K, V := K2. Thc hypc!rl)oll(: pliiirc i R denoted (K2, h).

2) d = 2: k as in l), D a totally indefinite quateriiion (livhioir nlgcbra, central simple over k, V := D2. The hyperbolic plane is denoted (D’, 1 1 ) . 3) d 2 2: k as above, Klk an imaginary quadralic. i!xlmtiioii, D a central simple

division algebra over K with a Klk-involution of lh! Hi!(:Olld kind, V := 0’; this is again denoted (D2, h).

Of course, 1) is the same case as 3), if we view K M c:~!iitriiI Himple over itself; the Klk involution is just the Galois action of K over k. Our HrAt observation (Proposition 2.3) is that the case d = 1 is none other than the cam fitniililtr from the study of Hilbert modular varieties. If K = k ( f i ) , with 7 E kt, then the isomorphism between

1991 Mathematics Subject Closrifimtion. 32M 15. Keywords and phroscs. Arithmetic group, arithmetic quollurilk+ division algebras.

22 Math. Nachr. 200 (1999)

SL2 ( k ) and SU ( K 2 , h) ') is simply

SL2(k) A S U ( K 2 , h )

a b ( c d ) c, (b&/2 "I?)'

The following properties of Hilbert modular varieties are well-known. i) the domain V CY fj x . . . x A is a product of f copies of the upper half plane, and

the rational boundary components are zero - dimensional, i. e., reduce to points. ii) For the maximal order ok C k , the arithmetic subgroup r := SL2(0k) acts on

V, the quotient Xr = r\V being non-compact, with a natural singular compactifi- cation (r\D)* obtained by adding a point at each boundary component (the cusps). Furthermore, X r has exactly h(k) cusps, where h(k) denotes the class number of k.

iii) There are modular subvarieties, known in the case of Hilbert modular surfaces as Hirzebruch - Zagier cycles.

These are some of the properties it would be reasonable to consider for hyperbolic. planes over higher - dimensional D. The main results of this paper contribute towards that end. First, in analogy to i), we find that for a hyperbolic plane ( D 2 , h) , the domain V is a product of f copies of an irreducible domain V1, and 2)1 is a tube domain2). It is the disc for d = 1, Siege1 space of degree two (type 1112) in case 2), and of type Id,d in case 3), respectively. The rational boundary components for the groups Go are points. It already follows from this that the Baily-Bore1 compactification X; of the arithmetic quotient Xr = r\V, C Go arithmetic, is obtained by adding finitely many isolated cusps to Xr, much as in the situation for Hilbert modular surfaces. We remark that although we will not study the smooth compactifkations of Xr, SATAKE in [lo] has considered these, and in particular has given a formula for a "cusp defect".

To introduce arithmetic subgroups one fixes a maximal order A C D, and considers r A = M2(A) n Go and S r A = Mz(A> n SGD. S r A is the generalisation of the Hilbert modular group (d = 1) to the higher d case, and ra is the generalisation of the extended Hilbert modular group. The arithmetic quotients we study are the rA\V = xr, and SrA\V = XSr,,. The first surprise is that one can easily calculate the class number, that is, the (finite) number of cusps on X r , . We prove (Corollary 4.11)

Theorem 0.1. The number of cusps of r A i s the class number of K an case 3) and the class number of k in cases 1) and 2).

This result is just a straightforward generalisation of the known d = 1 result, as mentioned in ii). We then turn to a generalisation of iii) above, and describe a rough analogue of the Hirzebruch-Zagier cycle F1 to the case of higher d. This results from the enrichment of the algebraic theory, which arises from subfields of the divisioii algebra D . Let us briefly explain this in the simplest case, case 3). Then D is a cyclic algebra D = ( L / K , 0, -y), -y E K * , and the subfield L c D is of degree d over K ; if.

l ) In this paper we will reserve the notation SL2(k) for that group, and otherwise will utw

2, This is certainly well-known to experts. notations like SL(n, K ) , etc.

Ilunt, Hyperbolic D- Planes 23

IH a splitting field for D. These matters are recalled in the first paragraph for the convenience of the reader, as they are an essential part of the remainder of the paper. The subfield L C D defines a subspace L2 C D2, and +La =: h~ is again hyperbolic. 111 other words, in a hyperbolic plane over a degree d cyclic division algebra, we have H. subhyperbolic plane (L', h ) ; thia is of the type d = 1, but with f = d. This results In corresponding subgroups GL C Go, subdomains DL c 2) and arithmetic subgroups roL c rA; finally these yield modular subvarieties @L c Xr,, which are natural images of the arithmetic quotients Mt of DL by ro,. By the result mentioned above, this means: on an arithmetic quotient X r , for d 2 2, there are Hilbert modular Hubvarieties of dimension d. For the general statement, see Theorem 4.14 below. An d&tional analysis leads to the following strengthening (Theorem 4.16):

Theorem 0.2. Given a cusp p E X;,\Xr, there is a modular subvariety EL,^ c - Xr,,, such that p E ML,p, that as, p i s also a cusp for MI,,^.

It is now quite straightforward to apply Shirnurak t,hoory to give the moduli inter- pretation of the arithmetic quotients X r , . A second application of this theory gives the moduli interpretation of the varieties ML. The moduli interpretation of the Xr, is summarized as follows:

1) d = 1: abelian varieties of dimension 2f with complex multiplication by K , of signature (1, l), OR: abelian varieties of dimension f with real multiplication by k.

2) d = 2: abelian varieties of dimension 4 f , with multiplication by D. 3) d 2 2: abelian varieties of dimension 2 8 f , with multiplication by D. We note that the two possibilities for d = 1 are related by the fact that, given A / , an

abelian variety of dimension f with real multiplication by k, we get one of dimension 2f with complex multiplication as follows: if the field K = k ( f i ) , let K' = Q (6). Then if Af % C! /A, we have a lattice in C! 2f given by A' := A 82 OK#. The abelian variety A2f := CZf/A' is the desired abelian variety with complex multiplication. Conversely, if A2f is given, it is easily seen that the lattice A' has a sublattice A such that A' % A @z O K I , and C f /A is an abelian variety of half the dimension with real multiplication.

Our third main result then is the following description of the moduli interpretation of the modular subvarieties.

Theorem 0.3. With the notations as above, let G; denote one of the modular sub- varieties occuring in Theorem 0.2. Then this subvariety pammeterizes abelian varieties

2) d = 2: the abelian variety A'/ with multiplication by D splits, A'f is isogenous to u product A2f x A2f , where A2f has complex multiplication by an imaginary quadratic extension field L c D of the center of D , k .

3) d 2 2: the abelian variety AZdaf with multiplication by D splits, AZdaf is isogenous to a product A 2 4 x - - x A'T where A 2 4 has complex multiplication by the splittiny

field L C D.

as follows:

- d factors

24 Math. Nachr. 200 (1999)

This result is of particular interest in relation to the general result in [15] about, the isogeny - splitting of abelian varieities in algebraic points. Here we have this kind of splitting along a subvariety, a perhaps somewhat unexpected result. In the fifth paragraph we discuss in more detail a central eimple division algebra D of degrev three over an imaginary quadratic extension KIQ , which defines unitary groups which it appears may be related to Mumford's fake projective plane.

1. Cyclic algebras with involution

We will be considering matrix algebras Mn(D), where D is a simple division algebra with an involution. There are two kinds of involution to be considered, Fix, for thr rest of the paper, a totally real number field k of finite degree f over Q, and assumc D is a division algebra over k with involution. The two cases are:

(i) D is central simple over k, and k is a maximal field which is symmetric with respect to the involution on D (involution of the first kind).

(ii) D is central simple over an imaginary quadratic extension K ( k , and k is thc maximal subfield of the center which is symmetric with respect to the involution on D; the involution restricts on K to the Klk involution (involution of the second kind). If one sets K := k in case (i), then in both cases one speaks of a Klk- involution. With these notations, let d be the degree of D over K (i. e., dimKD = 8) . Specifically, wc will be considering the following cases:

1) d = 1, D = K with an involution of the second kind, namely the Klk - involution.

2) d = 2, D a totally indefinite quaternion division algebra over k (1.1) with the canonical involution (involution of the first kind).

3) d 2 2, D a central simple division algebra of degree d over K with a KIk-involution (involution of the second kind).

Next recall that any division algebra over K is a cyclic algebra ([l], Thm. 9.21, 9.22). These algebras are constructed as follows. Let L be a cyclic extension of degree d ovei K and let u denote a generator of the Galois group GalLiK. For any 7 6 K', on(' forms the algebra generated by L and an element e,

A = ( L / K , o , ~ ) := L e e L $ . . . e e d - ' L , ed = 7 , e . z = z ' . e , for all J E L . (1.2)

This algebra can be constructed as a subalgebra of Md(L), by setting: . . . . . . 0 1 0 z 0

e = ... ) . ;), z = [ ... J'

@d-I . . . . . . ...

One verifies easily that these matrices fulfill the relations (1.2), and letting B be tho algebra in Md(L) generated by e and J 6 L, we have (L/K,u,?) E B , and clearly

Ijunt, Hyperbolic D-Planes 25

LI @K L 2 Mn(L), giving the explicit splitting. It is known that A = ( L / K , a , r ) is split if and only if 7 E JVL~K(L') (111, Thm. 5.14 that A is a division algebra; we will denote it in the sequel by D.

Now assuming d 2 2, we consider the question of involutions on D

he cyclic algebra . We will assume

It is well-known that for d = 2, an involution of the second kind is the b a ~ c change of an involution of the first kind ([l], Thm. 10.21). Mhermore, if d 2 3, thcre are no involutions of the first kind, since an involution of the first kind gives an iRomorphism A rY AOP (AOP the opposite algebra), implying that the class [A] in the Brauer group BrK is of exponent one or two. Hence, for d 2 3 the restriction that tlic itivolution is of the second kind Is no restriction at all. Finally in case d = 1, an involutioii of the first kind on D is trivial, so this case is not hermitian, and we will riot corisidcr it,.

Let us now consider the quaternion algebras D. ltecall thltt D is said to be totally definite (respectively totally indefinite), if at d l real ))rimen v , the local algebra D, is the skew field over R of the Hamiltonian quatcrnionn W (ronpectivcly, if for all real primes v, the local algebra D, is split). Rscall also that D rimiif ie~ at a finite prime p if D, is a division algebra; the isomorphism class of D i N deterniilicd by the (finite) set of primes p at which it ramifies and its isomorphism clam at, tho~c> primes ([l], Thm. 9.34). As a special case of cyclic algebras, quaternion algebrw (:a11 he displayed as algebras in M z ( l ) , where l / k is a real quadratic extension, l = k(Jd). Then there is some b E k' such that

where t' = (21 + = 21 - f i z z . In other words, we can write for any a E D,

Then the canonical involution is given by the involution

(: :) - (-"e -.") on Mz(e) , and the norm and trace are just the determinant and trace of the matrix (1.4). We will use the notation (a, b) to denote this algebra ( . f /k ,a , b), l = k ( J ; E ) , if no confusion can arise bom this. We remark also that for quaternion algebras we have

where the trace and norm are the reduced traces and norms. All traces and norms occuring in this paper are the reduced ones unless stated otherwise.

In the case of involutions of the second kind, note first that the Klk-conjugation extends to the splitting field L; its invariant subfield t is then a totally real extension of k, also cyclic with Galois group generated by 6. We have the following diagram:

26 Math. Nachr. 200 (1999)

Diagram 1

and the conjugations on L and K give the action of the Gdois group on the extensiona LIL and Klk; these axe ordinary imaginary quadratic extensions. There axe precitw relations known under which D admits a Klk-involution of the second kind.

Theorem 1.1. "11, Thm. 10.18.1 A cyclic algebm D = ( L / K , CJ, 7) has a n involution of the second kind i f f there is an element w E e such that

d - 1 77 = N K , & ( Y ) = Nt,&(W) = w . w " . * . W U .

If this condition holds, then an involution is given explicitly by setting:

where x I+ Z denotes the LIL-involution. In particular for x E L we have

x J = 5, and x = xJ X E e .

Later it will be convenient to have a description for when x + xJ = 0. This resulbn from the following.

Theorem 1.2. "11, Thm. 10.10.1 Given a n involution J of the second kind 011

an algebm A , centml simple of degree d ouer K , there are elements u l , . . . , Ud, with Ui = u:, such that A is generated over K by ul, . . . , ud. f irthennore, there is uii

element q E A , qJ = -9, q2 E k, such that, as a k -vector space,

A = A + + q A + ,

where A+ = { x E AJx = x J } .

If x E A is arbitrary, then i ( x + x J ) E A+, while & ( x - x J ) E A - . For examplo, we have e' + (e ' ) =: E' E A+, and then we have an isomorphism J

(1.7) A+ p! eeEee. . .eEd-le . If, as above, K = k ( f i ) , L = e ( f i ) , then we may take fi = q in the theoreill above, and for elements in

(1.8) QA+ CY f i m E f i e e . . . e E d - l f i e ,

liint, Hyperbolic D - Planes 27

m have y = -uJ . In particular, the dimension of qAS is dZ aa ti k - vector space, and ti(# dimension of A is 2 8 . An involution of the second kind is related to a hermitinn form defined over L as

idlows. Under the embedding i : D c) M&) described abovc!, tho involution J on D nil be extended to one J L on &(L) by identifying M,,(L) with D o o ~ L aiid setting

( d @ r n ) J L := d J @ K ,

vlrcre X denotes the L/e conjugation. A calculation drown t l r s l . 11ic* iiivolution J I , is [Ivcm by

t = (z,,) H z J ~

,d-2 *I ' W ) P1.d-1

. . .

. . .

...

. . .

...

28 Math. Nachr. 200 ( 1 9 1 ~ )

By a general theorem (see [12], Theorem 7.4), JL is conjugate to the standard involii. tion (M'= conjugate transpose) by a unique hermitian element a = a' in &(I,),

i ( z ) J L = ai(z)'a-1 .

A computation shows that a = diag (1, w', w'wuz, . . . , w' . . . w U ' - l ) . The elemenf, II can be considered to be the matrix of a hermitian form +a on Ld, and the correspondiri~

G = U(+, , ,Ld) = {z E Md(L) I zaz' = a } unitary group

is defined. Note that

DltJ := {Z E D I z - x J = 1) { i ( ~ ) E Md(L) I i ( ~ ) * i ( ~ ) ~ " = 1 ) = {i(z) E M ~ ( L ) 1 i(z)ai(z)*a-' = 1)

{i(z) E Md(L) I i(z)ai(z)' = a } , =

so i(D'iJ) c U ( @ , , L d ) . Linearizing theequation x . z J = 1 (i.e., replacingz by 1-1 ( and disregarding terms of second and higher order in <) yields

< + t J = 0,

the set of J - skew symmetric elements.

2. Algebraic groups

Let D be a central simple K-division algebra with a KIk-involution z ct Z IM discussed above (denoted x I+ zJ above, but unless it is a cause of confusion we fix the notation z I+ f for the remainder of the paper), with Klk imaginary quadratic, for cases 1) and 3) and K = k in case 2). Consider the simple algebra Mz(D); i I is endowed with an involution, M C) 'm =: M'. The group of units of M z ( l ) ) is denoted GL(2,D) , its derived group is denoted by SL(2,D). Consider a noii degenerate hermitian form on D2 given by

( 2 . 1 ) h(X,Y) = z1V2 + z251,

where x = (21, z2), y = (31, 32) E D2. This means the hermitian form is given by 1 , I i v

matrix H := ( y i). The unitary and special unitary groups for h are

U ( D 2 , h ) = (9 E GL(2,D) I gHg' = H } , S U ( D ~ , ~ ) = U ( D ~ , ~ ) n S L ( 2 , D ) . (2.2)

The equations defining U ( D 2 , h) are then

Iiiint, Hyperbolic D-Planes 29

SU(1 , l ;K) = {(; g ) E G L ( ~ , K )

'I'lic additional equation d e h q SU(D2, h) can be written in terms of determinants, iinliig Dieudonnb's theory of determinants over skew fields, (see [3], p. 157)

@/I)

'J'tic center of U ( D 2 , h ) , which we will denote by C, is givc:rl by

det(g) = NDlik(ad - aca-'b) = I .

C U ~ T - @ = I 1 .

wlrere we view K C 6: as a subfield of the complex nurrrbors. In particular:

Lemma 2.1. For d = 2, K = k a real field, we have C = { f l ).

In other words, for the case 2 ) , there is no essential diffcrciicc: between the unitary and rpecial unitary groups. Otherwise, U ( D 2 , h ) i R a rcductivc K - group, and SU(D2,h) In the corresponding simple group.

Proposition 2.2. Let SGD denote SU ( D 2 , h ) , the simple ulgebruic K - group de- lined by the relations (2 .3) and (2.4). Then SGD i s simple with the following index (cl. [171)

1 ) d = 1, 'A:,,. 2) d = 2, C&. 3, 2 2 J 2 A i d - l , l *

Proof. This is immediate from the description in [17] of these indices. 0

The unitary group for d = 1 is familiar in different terms. If we set A = ( 1:;; -: ) ,

I n particular, these two hermitian forms are equivalent. For the form HI := llie corresponding unitary group is customarily denoted by U(1, l ) . Hence for K c 6: Imaginary quadratic over a real field, we have an isomorphism

(2.5) U ( K 2 , h ) 7 U ( 1 , l ; K ) 9 H AgA-';

tlic group U(1, l ; K) is given by the familiar conditions

niid the special unitary group is given by

30 Math. Nachr. 200 (1991,

Recall that this latter group is isomorphic to a subgroup of GL(2, R), in fact we haw the well - known isomorphism

SU(1, l ;C) A SL(2,R) 9 c-) BgB-' ,

where B = ( -f l ) is the matrix of a fractional linear transformation mapping thr disk to the upper half plane. If, instead of B, we use

B, = ( - i 2 ) K = k ( f i ) , f i f i ' then we have in fact

Proposition 2.3. The matrix B, gives an isomorphism

SU(1,l;K) -7 SL2(k) 9 I--) B,gB;'.

In particular, we have an isomorphism

SU(K2 ,h) 7 SL2(k) 9 ts B,A~A-' B;'.

Proof. We only have to check that B,gB;' is real, BS it is clearly a matrix It

SL(2 , K), and SL(2, K) n S L ( 2 , R ) = SL2(lc). If g = (; E) then

J = ; ; i ( - c r - Z + p + p ) - ( r - P + E + P

q(a - p + P - E ) - 6 ( a + /iJ + & + P )

which is clearly red. I

Note that the inverse SL2(k) + S U ( K 2 , h) is given by conjugating with

c = (B,A)-' =

so we may describe the latter group as:

Remark 2.4. For the theory of Shimura varieties (which we do not pursue in IM paper) it is important to have a reductive subgroup G, whose derived group is tlii

liiint, Hyperbolic D- Planes 31

poup SGD above. These groups G, are the groups of unitary similitudes derived from f i g := Reskl~U(V, h), which are defined as follows. G , is the algebraic group over Q whose points in any commutative Q- algebra R are

G,(R) = GU(V,h)(R) := { g E GLD(V) I ghg' = Ah, X E R X } .

In particular, in case 1) we get an exact sequence

1 -+ SU(K2,h) + G U ( K 2 , h ) -+ k X -+ 1 (2.9) I1 I) II

1 -+ SLz(k) -+ GL(2,k) --t k X -+ I,

which shows that there are two equivalent ways of forinulating the required reductive uroup.

Next we describe some subgroups of GD = U(D2, / I . ) ririd SGII . For this, recall that w(! are dealing with right vector spaces, and matrix mult,iplication is done from the

First, the vectors (1,O) and (0,l) are isotropic; the stabilizers of the lines they span 111 D2 are maximal (and minimal) parabolics. For example, the stabilizer of (0 , l )D is

P = ( 9 E GD I ( 0 , l ) g = (0, I ) A , X E D'} (2.10)

Itideed, it is immediate that for g = ( g :) E P, we must have c = 0, while the first

tclation of (2.3) reduces to aa = 1, or d = F1. Since a parabolic is a semidirect product of a Levi factor and the unipotent radical, to calculate the - latter we may wsume a = 1. Then the second relation of (2.3) is a6 + bE = b + b = 0. The corresponding parabolic in SGD takes the form

(2.11) SP = { g E S G o / 9 = (i a E D " , b + Z = O

Note that g in (2.11) has K-determinant

detK(g) = NDIK (4E-l) = N D l K ( a ) NDIK(Z)- ' i

wo the requirement on g is that a must satisfy NK~~(NDIK(~)/NDII((E)) = 1. Thisis otltisfied for all a E K", where a is viewed as an element of D'; if a 6 K, then the condition becomes N q ~ ( a ) = N q ~ ( a ) , i.e., N D ~ K ( ~ ) E k. It follows that the Levi component is a product

32 Math. Nachr. 200 (1999)

Now recall from (1.8) that in case 3) for d 2 2, the set of elements fulfilling b+6 = 0 in D is f i D + E fit!@E,Eje@...@Ed-lfiel where E' = ei+z. In particular, the unipotent radical of SP, generated by (i !) with b + 6 = 0, has dimension d2 over k .

We also have non - isotropic vectors ( 1 , l ) and (1, - 1). Any element g E Go pre- serving one preserves also the other (as they are orthogonal). Hence the stabilizer of ( 1 , l ) is . .

(2.12)

For d = 1, this can be more precisely described as

In case 2), the relations give, viewed as equations over k , four relations, so t'he group K is four-dimensional. Finally, for d 2 3, we get (over k ) d relations (as elements ~ 7 i are in e, so diagonal) from the first condition and dL - d + 1 conditions from thr second, leaving dL + d + 2 parameters for the group K.

If D has an involution of the second kind, we have the splitting field L (= K for d = l), which is an imaginary quadratic extension of the totally real field t! (= k for d = l), see the Diagram 1. Suppose then d = 2, D quaternionic, say D = (t!/k, 0, b) , where l = k ( & ) , e2 = b E k'; in this case l is a splitting field. Recall the conjugation r~ is given by

so the element c = diag(JSi, -&) representing & E t satisfies c' = -c, while e' = e. Consequently, the relation (1.2) for c is ec = cue = -ce, and

(2.13) (ec)' = (ec)(-ce) = -ec2e = -ab,

so k(ec ) CY k ( G ) . If we assume, as we may, that a > 0, b > 0 (otherwise replace -ab in what follows by the negative one of a, b), then L := is an imaginary quadratic extension of k which is a subfield of D. So in all cases (l), 2) and 3)) wv have an imaginary quadratic extension of l (cases 1) and 3)) or k (case 2)), L, whicli is a subfield of D. We now claim that U ( L 2 , h) is a natural subgroup of U(D2, h). I n all cases U ( L2 , h) and U ( D 2 , h) are defined by the same relations (2.3) , so we haw

(21 + &ZJ = 21 -Jazz,

(2.14) u ( L ~ , h) = u(02, h) n M ~ ( L ) ,

which verifies the claim. In the case d = 1, where we took L = K , this is in fact t h whole group. But even in this case we can get subgroups in this manner. Considcr taking, instead of L = K , L = Q ( 6 ) if K = k ( 6 ) . More generally, fot any Q k' k, we have the subfield k ' ( 6 ) C K , and we may consider t h corresponding subgroup:

u(k'(J_?)2,h) = U ( K 2 , h ) n h f z ( k ' ( J _ ? ) ) .

33 Hunt, Hyperbolic D - Planes

So, setting L = k ’ ( f i ) in (2.14), these subgroups are defined by the same relations (2.3). Now recall the description (2.8) for p u p s of the type (2.14).

Proposition 2.5. We have the following subgroups of U ( D 2 , h ) . 1) d = 1 , for any Q E k’ C k we have the subpup

2 ) d = 2 , for the field L = k ( m ) above, we have the subgroup

3) d 2 2 , for the degree d cyclic extenaion LIK we have the following subgroup

These are interesting subgroups, whose existence is a natural part of the description of D as a cyclic algebra. In all cases we may view these subgroups as stabilizers. Let L denote one of the fields k ’ ( f i ) , k ( G ) , l ( f i ) as in Proposition 2.5; let L2 c D2 denote the k-vector subspace of D2, viewing D as a k-vector space, and consider the stabilizer in GD. Noting that L2 C D2 may be spanned over L by two non-isotropic vectors (1 , l ) and ( 1 , - l ) , it is clear that g E U ( L 2 , h ) preserves the L span of these two vectors, that is, L2, hence U(L2 ,h) is contained in the stabilizer. The converse is also true: if an endomorphism of D2 preserves the L span, then its matrix representation is in Mz(L), so the stabilizer is contained in the intersection U(D2,‘h) n M2(L) = U(L2 , h ) . Let us record this as

Observation 2.6. The subgroups of Proposition 2.5 are stabilizers (with determi- nant 1 ) of k - vector subspaces of the k - vector space D2.

Remark 2.7. Considering the reducitive subgroup GU(V, h ) of Remark 2.4, we have subgroups GL(2, k’), GL(2, k) and GL(2,1), respectively. Again this is important for the theory of Shimura varieties.

3. Domains

Recall that given an almost simple algebraic group G over k , by taking the R points one gets a semisimple real Lie group G(IR). For the three kinds of groups above we determine these now.

Proposition 3.1. The neal p u p s G D ( R ) am the following.

34 Math. Nachr. 200 (1999)

1) d = 1, GD(R) = (U(1, I))’, SGD(R) ( S L ( 2 , R ) ) f . 2) d = 2, GD(R) = SGD(R) = (Sp(4,IR))f. 3 ) d 1 2, GD(R) = (U(d,d))’, SGD(IR) = (SU(d,d))’.

Proof. The d = 1 case is obvious. For case 2), note that since D ..i totally indefinite, D, CY Mz(R) for all real primes, and consequently ( G o ) , is a simple group of type Sp for each v, not compact as G o is isotropic, of rank 2, hence Sp(4,R). Consider now case 3). By Proposition 2.2 we know the index of G o is 2A&-l,l, and the index, as displayed in [17], is

T *- ......

0- ...... do Diagram 2

where the number of black vertices is 2d - 2, d - 1 on each “ranch. This shows in particular that the isotropic root is the one farthest to the right. If a root is isotropic over $, then all the more over IR. Hence, considering the Satake diagram of the real groups of type 2 A , we see that the only possibility is SU(d,d) , as this is the only real group of this type for which the right most vertex is R-isotropic. Actually, thifi only proves that at least one factor (GO)” is SU(d,d) ; but our hyperbolic form has signature (d, d ) at any real prime, so it holds for all (non - compact) factors. That. there axe no compact factors follows from the fact that GD is isotropic. In all weti there axe f factors, as this is the degree of klQ, 80 the Q -group, ReskIQGD, has f

0

It is a consequence of this proposition that the symmetric spaces associated to th(8 real groups GD(R) are hermitian symmetric; indeed, this is why we have choosen those D to deal with. More precisely, one has the following statement.

factors over R. This verifies the proposition in all cases.

Theorem 3.2. The hennitian symmetric domains defined by the G D ( R ) as h

(a) are tube domains, products of irreducible components of the type3 I ~ J , 1112, Id,(,

(b) For any rational parabolic P c Go, the corresponding boundary component 1 1 ’

Proposition 3.1

in the cases l), 2 ) and 3 ) , respectively.

such that P(R) = N ( F ) is a point.

Proof. The first statement follows immediately from Proposition 3.1. The second is well known in the case d = 1 from the study of Hilbert modular varieties. In case 2) the statement clearly holds for k = Q (just look at the index, from which one sees t h v boundary components are points, not one - dimensional), and the general statement. follows from this: all boundary components are of the type ( c l , . . . , c f ) E D; x - - - x ID;, where c, E DT is a point, which is a general fact about Q -parabolics in a simple Q - group (this applies directly to ResklQSGD, but G o and SGD define the same domain and boundary components). Finally, in w e 3), we point out that in the diagram 2, thcb isotropic vertex (farthest right) corresponds to a maximal parabolic which stabilizcw

Hunt, Hyperbolic D-Plonea 35

M point, establishing the statement for k = $. Then the general statement follows as nbove from this. 0

We now consider the Btructure of the real parabolics for d 2 2 in more detail. For this we refer to SATAKE'S book [Q] gIII.4.

i ) Case 2): Here we have the case denoted b = n in SATAKE'S book, and the parabolic In Sp(4,lR) is

P(R) = (G(') * G(2) ) M U * V = 1 * GL(2, R) M S y m z ( R ) , '

where Syrnz(R) denotes the set of symmetric real 2 x 2 matrices. Refering to (2.10), we may identify the Levi component of P with

md, as D is indefinite, D*(R) 2 GL(2, R); this is thc factor GL(2, R). The unipotent radical is

U fy { ( t :> I n(b) = 0 ) E Do (= totally imaginary elements),

which is three-dimensional, and an isomorphism U(R) E Symz(R) is given by

This describes the real parabolic completely in this w e .

is

where GLo(d ,C) = {g E G L ( d , C ) I det(g) E R') and ?fd(C) denotes the space of tiermitian d x d matrices. We must now consider the parabolic SP as in (2.11). The Levi component is again

ii) Case 3): Here we have b = d in SATAKE'S notation, d = p = q. The real parabolic

P ( R ) = (G(') * G ( 2 ) ) M U * V = (GLo(d, C ) ) M ?fd(C),

!,he second condition because of determinant 1. horn the multiplicativity of the norm, I,he second relation can be written N q K ( a ) = N q ~ ( i z ) . Since the norm is given by the determinant, this means, in D'(R) ,

L(R) PL {a E D'(lR) I det(a) E R'} fy GLo(d, C ) .

Similarly, the unipotent radical takes the form

U E {(o 1 b I ) l b E D , b + 6 = 0 } fy G O + ,

36 Math. Nachr. 200 (1999)

which is d2-dimensional. An explicit isomorphism U(R) CY 3td(C) is given by th(1 identity on D+(R), since x E D+ =$ x = 5, hence as a real matrix, this element i N

hermitian. The maximal compact subgroups of GD(R) are the groups K(R), where K: is thc

group of (2.12). Indeed, it is easy to see that the stabilizer of the non-isotropic vector (1 , l ) is the stabilizer, in the domain V, of the base point, which is by definition thc maximal compact subgroup. Now consider the subgroups U ( L 2 , h ) c G o ; let us introduce the notation G,, for

these subgroups. Then, since GL is of the type d = 1 in Proposition 3.1, it follow8 that Theorem 3.2 applies to GL(R) as well as to GD(R).

Proposition 3.3. We have a commutative diagmm

where DL and DD are hennatian symmetric domains, and the maps G(R) + V are the natural projections. Moreover, the subdomaim V L are as follows:

1) d = 1; We now wsume that the eztemion k/k' i s Galois. Then, if degQk' = j ' , f / f' = m, we have V L CU (4)" and 'Do CU ((fi)")f ' and the embedding V L c 'DD ia given by 4 C) (4)" diagonally, and the product of this f ' times.

b(;rl ) , where (, : k + R denote the distinct nal embeddings of k .

2) d = 2; V L E (: 6cprl ) x - . x (

3) d > ~ ; V L (0 ... ;I. Proof. Considering the tube realisation of the domain VD, we have the base point

0 = (diag(i,i ,..., i ) ) f ,

and we just calculate the orbit of the R-group GL(R) of this point; this gives tliv corresponding subdomain. The commutativity of the diagram follows from this.

1) d = 1: Since k/k' is Galois, we have that GL C GD is the subgroup fixed und(-r the natural Gdklk' action; hence lifting the -groups to R, we get the identificatioii

In terms of the domains, if V D = (a)', then Galalp acts by permuting m copies at, 11

time, and the invariant part is the diagonally embedded 4) L) (A)", z I+ (2,. . . ,z) . Since 231, CY (fj)f', the result follows in this case.

2) d = 2: First we remark that Gm is conjugate to the standard Sp(4,lR) (by whicli we mean the symplectic group with respect to the symplectic form J = (-! :) undcr

h n t , Hyperbolic D - Planes

the element

37

Q = (+) -1 0

that is QGRq-' = Sp(4,R). To see this, we recall how (Iiermitian) form is related to alternating forms. Since Dw. CY M2(R), our vector space Dz over R is Vm = (Mz(R))~; the hyperbolic form gives a Mz(IR) -valued form on VR. We have matrix units eij, i , j = 1,2 in M2(R), and setting W1 = ellV, W2 = e22V, the spaces Wi are two-dimensional vector spaces over R. The hermitian form h(x, y), restricted to W1, turns out to have values in IRelz: if

9

the ( D - valued) hyperbolic

then x, y E W1 .C 22, = zii = y2i = 9ii = 0. Then we haveS)

Then the alternating form (x,y) is defined by

W , Y ) = (X,Y)*e12, X , Y E W l . In other words, viewing the symplectic group as a subgroup of GL(2,Mz(]R)), the Rymplectic form will be given by J1 = ( Js 2 ) , where J2 and Js are symplectic forms

on R2. It is clear that q( 2 ) ' q = J1 is of this form, and this shows qGRq-' =

Next, recalling that we view D as a cyclic algebra of 2 x 2 matrices, an element A = ( s f ) E &(D) is actually a 4 x 4 matrix; each of a, P, 7, 6 are of the form (1.4), and this 4 x 4 matrix must be conjugated by g, then the element acts in the well known manner on the domain VD:

Sd4, w.

Finally we note that the subgroup U ( L 2 , h) c U (D2, h) consists of the elements of the form as in Proposition 2.5 2):

a, P , y ,6 E k , a

x = ( 7 G j 2 6

The involution on Mz(k) is a8 in (1.5).

38 Math. Nachr. 200 (199!~)

and the element = ec as in (2.13), so that (-)-' = (ec)-'. In other words, as a 4 x 4 matrix, the element x above is

and after conjugating with q this becomes

From the form of the matrix (3.2) we see that the corresponding subdomain is con-

tained in the set of diagonal matrices . Now calculating, setting g = qxq-' , we have

(3.3)

So in particular, GL maps the set of matrices

(3.4)

into itself. (In the expression (3.3), the bracket ( . . .)ci means (, is applied to all matrix elements). This proves the theorem in the case d = 2.

Remark 3.4. The description of the subdomain V in terms of b, depends on tliv way we described D as a cyclic algebra. We took l = k(&), D = (l /k,o, b ) . Bur, we could also consider lf = k(&), D' = ( l ' /k ,o ' ,a ) ; these two cyclic algebras an' isomorphic, an isomorphism being given by extending the identity on the center k by

D D' e I-+ cf

c c) e',

Hunt, Hyperbolic D - Planes 39

a 0 % a"

where

0

5

Indeed, as one seea immediately, (ec)' = -a6 = (e'c')', so the map above is a mor- phism, which is clearly bijective. Consequently, it, would be better to denote (3.4) ' D L , ~ , since we also have a subdomain

DL,o = (" O ).....(" O a c f q O ) , O aclq

neither of which is a priori privileged.

3) d 2 2: Here the situation is simpler; the matrix H i R the matrix of the hemitian form whose symmetry group acts on the usual unbounded realisation of the domain of type Id,d, hence no conjugation is necessary. Let GL be the subgroup we wish to consider, and recall from Proposition 2.5 that i t consists of mtttrices

Thinking of D itself as d x d matrices, this element is

(3.5) 5 =

ritid the orbit of the base point diag(i, . . . , i)' under elements as z is clearly the diagonal robdomain (diag(T1,. . . , 7d))', and the action is, for k = $,

40 Math. Nachr. 200 ( 1 9 9 ~ )

and for general k, letting 88 above (1,. . . , (f denote the f embeddings of k,

This establishes all statements of the theorem. Q

Finally, let F be the rational boundary component E Vb Corresponding to tlio isotropic vector (0, l), of which the parabolic P of (2.10) is the normalizer, P ( R ) = N ( F ) . Note that by construction, (0,l) is also isotropic for GL; let FL be the rational boundary component E Dof, corresponding to it, so the parabolic in GL, PL of (2.10) is the normalizer, PL(R) = N(FL) . Then clearly

(3.8) pL = G L n P .

Also we have

Proposition 3.5. Let VL C V D , FL E 732, F E Vb be ad above, and let a : Dof, v Db be the induced inclusion of Satake wmpactafications of the domaina. Then ~(FL) = F .

We will refer to this aa “the subdomain VL contains F as a rational boundary component.” The proposition itself is, after unraveling the definitions, nothing but, (3.8).

4. Arithmetic groups

It is well-known how to construct arithmetic groups I? C GD(K); we recall this briefly. View D2 aa a D-vector space, and let A c D be a maximal order, that is ail O~-module (where D is central simple over K , K = k in case 2)), which spans D as a K-vector space, is a subring of D , and is maximal with these properties. Then scb

(4.1) r A := G D ( K ) ~ M ~ ( A ) .

If we view A2 c D2 as a lattice, then this is also described aa

Fkom the second description it is clear that r A is an arithmetic subgroup. We will denote the quotient of the domain VD by this arithmetic subgroup by

Ilunt, Hyperbolic D-Planes 41

If r A were fixed -point free on DD this would be a non - compact complex manifold; If r A has fixed points (as it usually does), these yield singularities of the space Xa, which is still an analytic variety. It iS a V-variety in the language of SATAKE. In fact It has a global finite smooth cover: let C be a normal subgroup of finite index which does act freely (for example a principle congruence subgroup of level N 2 3). 'l'tien we have a Galois cover

Xr -+ Xrn, wid Xr is smooth. Hence the singularities of Xr, arc encoded in the action of the finite Galois group of the cover.

The Baily-Bore1 compactification of Xr (r C rA) i H an embedding

Xr c X; c P" where X; is a normal algebraic variety. Since thc boundrrry components of VD are points (Theorem 3.2), it follows that the singularities of X,? contained in the boundary X; - Xr are also isolated points. In particular, if I' c has no elements of finite order, then X; is smooth outside a finite set of isolated points, all contained in the compactification locus.

The singularities of X; may be resolved by means of toroidal compactifications; let us denote such a compactifkition by Xr, for which we make the assumptions:

i) Xr - Xr is a normal croseings divisor, ii) xr X; is a resolution of singularities and Xr is projective algebraic.

Such a compactification Xr depends on a set of polyhedral cones, and one can make clioices such that i) and ii) are satisfied, by the results of [14]. We will not need these In detail; the mere existence will be su5cient.

Recall that a CUSP of an arithmetic group r is a maximal flag of boundary components of I?; in the case at hand this is just a point in X; - Xr. The number of CUSPS is the tiumber of points in X; - Xr, and may be defined in the following ways:

a) It is the number of r-equivalence classes of parabolic Q-subgroups of Go or SGD; here r acts on the parabolic subgroups by conjugation.

b) It is the number of I?- equivalence classes of isotropic Q- vectors of the hermitian glsne ( D 2 , h); here l? is acting as a subgroup of Go on the vector space.

In the d = 1 case we actually have two Merent arithmetic groups, SU(O&,h) atid SL2(Ok), In the latter case we are talking about the number of CUSPS of a Illlbert modular variety. Let US recall how these numbers are determined. For this wc consider fist the group SL2(k) (instead of SU(K2,h)), which acts on a product of upper half planes, VD = 3'. The boundary is the product (B'(R))f, and the rrtional boundary components are the points of the image P'(k) L) (IP'(R))' given by 2 t+ (& , . , . , x( f ) . Hence we may denote the boundary components by ( = (<I :

(, E 01. Let a€ denote the ideal generated by (1,(2. Then one has

Proposition 4.1. Two boundary components ( = (€1 : ( 2 ) and q = (ql : 112) an: Equivalent under SL2(0b) iff the ideals at and a, are in the same class. In particular, llrc number of cwps w the dass number of k .

42 Math. Nachr. 200 (1999)

Proof. One direction is trivial: if (: :) (::) = (;:) , g = (z :) E SL2(0k), then the ideals a€ and a, are the same. This is because g is unimodular, so affects just a change of base in the ideal at. Conversely, suppose and q are given, and suppose that a( and a, have the same class. After multiplication by a.n element c E k', we may assume a€ = a, = a. Furthermore, writing Ot = aa-' = C1a-l + &a-1, we see that 1 E Ok can be written iw

(4-3) 1 = = mrli - rhrlI, ti', rl: E a - l .

But this means the matrices (acting from the left)

(4.4)

are in SL2(k) , and

(4.5)

Of course Mt, M, are not in SL2(0k), but from the fact that .$,q: E a-l it follows that McM;' E SLz(O1.). Hence

and the cusps are conjugate under SL2(0k). 0

We now translate this into the corresponding statement for the hyperbolic plane (K2, h), where Klk is an imaginary quadratic extension. First note that if (::) E K 2 is isotropic, then

In fact, the converse also holds, TrKIk(tlE2) = t lC2 + t2c1 = 0.

L e m m a 4.2. A vector (ii) E K 2 is isotropic with respect to h iBTr(<1c2) =

~ r ( f , ( 2 ) = o i 8 < 2 = o or ~r([l<;l) = 0.

Proof. By definition, (::> is isotropic iff &f2 + <2Vl = 0, but this is 71([lf2) =

0. Noting that (F1 = & f 2 , if & # 0, we get the second equivalence, from the linearity of the trace, Tr(&{;') = Tr(& h e 2 ) = &Tr(t1f2). If (2 = 0, then

I3

Let KO denote the purely imaginary (traceless) elements of K , KO Ei f i k for K = k(J-?j). It follows from Lemma 4.2 that any z E KO of the form z = (l<T1 or

z = <1C2 determines an isotropic vector (::) of K 2 . Consider in particular the caso

( ::) = (2 ) is isotropic anyway, establishing both equivalences as stated.

Ilunt, Hyperbolic D - Planes 43

that (Ei) E 0; is integral. Recall that (non-zero) (1 and (2 are relatively prime iff IOiere are 2, y E OK such that s& + yt2 = 1. Define the set of relatively prime integral lwotropic vectors,

and if (, # 0, i = 1,2, then 3 2 , , , ~ ~ K z(1 + V(Z = 1 . 1 llhllowing the proof of Proposition 4.1, we consider when two isotropic integral vectors iirc conjugate under SrA = SU(O;, h) . Let now a( be the ideal generated by tl, t2 It1 OK.

Proposition 4.3. Two isotropic vectors ( = (::) and (' = under SU(Og, h) ajj the ideals at and a(! are equivalent in O K .

I E 3 are conjugate (2:

Proof. If g( ::) = ($), g E SU(O$, h ) , then since g is unimodular, the ideal

classes coincide. Conversely, suppose (::) and ($ ) are equivalent; again we may

wsume at = a0 = a. Then, as above, there are p, p' E such that (4.3) holds. Consequently we have Me, Me!, but now in SU(KZ, h), such that (4.4) and (4.5) hold. I t follows that Mt MF' E SU(O;, h) maps t' to (. This completes verification of the Proposition. 0

Note that the two cases, while being similar, do not coincide. For example, if k = Q , then SLz(Z) has a single cusp, while SU(Og, h) has 2'-' cusps, where t denotes the number of prime divisors of the discriminant of K (see (181, I 3.10). In general, SU(O$, h) is a subgroup of finite index in SLz(Z).

We now derive an analogue of the above for general D as in (1.1). We first note that 1,emma 4.2 is true here also. Since the hermitian form is given by h(( , () = (1t2 +(2z1,

we have

Lemma 4.4. ( i s isotropic with respect to h i f f (1c2 + ( z t , = 0.

Next we note that any isotropic vector is conjugate in Go to the standard isotropic vector (0,l).

Lemma 4.6. Let ( = ((1,G) E D2 be isotropic. Then there is a matrix Me E GD uuch that (0,l)Mc = (.

Proof. If Mc = ($ z ) , then (0,l)Mt = (. So we must show the existence of such

iin Me E Go. The equations to be solved (for (:) are

Math. Nachr. 200 (1SUD) 44

Since ( is isotropic, 3) is fulfilled by Lemma 4.4. If t1 # 0, then setting (i = 0, 4; D-

t1 , t f fulfills both 1) and 2). If t1 = 0, then t2 # 0 and we set similarly (i = (2

and = 0. r.1

Now we consider as above integral isotropic vectors. Let A C D be a maximal ordai , and set

3L3 = ( ( ( 1 , b ) E A' - {(o,o)) I (lcz +&!ti = o , (11€2 # o * 32,vEA h z + < Z y = 1) 8

-- I -1

Lemma 4.6. Let ((1, (2) E 3a. Then there exists a matrix

+I Proof. If (2 = 0, then <[ = 1, any (r with (r + t1 = 0 gives such a matrix,

similarly, if €1 = 0, then ( y = 1 gives a solution. So we assume ( i # 0; then Iy assumption we have ( q g ) E A' such that (lz + €29 = 1, hence ZZl + jjcz = 1. So setting = E , ( i = jj, the first relation of (4.7) is satisfied. Again 3) is satisfied hy assumption, so we must verify 2). It turns out that we may have to alter (: to achicvv this. The relation 1) can be expressed as h ( f , () = 1, where € = (€1, (z), €I = (ti, ti), Since ( is isotropic by Lemma 4.4, h(( , <) = 0, and with respect to the base ((, (') of D', h is given by a matrix H ( , o = (! ), where c = h((' ,<') = 6 + 5, 6 = ( i f L # 0; otherwise we are done), in particular 6 E A. Now setting

€I1 = (€;,€:) = ( - (iJ+ ti, -€2T+ ti) E A 2 , we can easily verify h((", < I f ) = 0, h((, ( ' I ) = h(t'f, () = 1, so this vector ( I1 gives II

0

We require the following refinement of Lemma 4.5. For this, given (€1 , €2) E A2, lct.

matrix Mc E r A , as was to be shown.

a€ denote the left ideal in A generated by the elements ( I , & .

Lemma 4.7. Given ( = (&, (2) E A2 isotropic, the e n t ~ e s of the matrix Mt o/ Lemma 4.5 am in a:'.

€i * 0 Proof. This follows from the proof of Lemma 4.5, as we took just inverses of e1emenl.N

We can now consider when two integral isotropic vectors are equivalent under F A

Proposition 4.8. Let ( = ((I,('), r ) = (71,~) E A' be two integrul isotropte vectors, at, a,, the left ideals generated by the components of ( and r), respectivelu. Then < and q are equivalent under an element of r A iff a,, Ez at a8 A -modules.

Proof. If there is an element g = (: :) E r A with (€l,&)g = (ql ,*) , then q l = ( l a + & c , m = ( 1 b + & d , h e n c e a , ~ a ~ a n d t h e m a p

4 : a,, + a€ c) ~ 1 x 1 + 712x2 <i(aAi + bA2) + G ( c h + dA2)

Ilunt, Hyperbolic D - Planes 46

IN 1111 isomorphism of A-modules. The inverse is given is a similar manner by g-] . [:orrversely, if a, c* at we may assume at = h, hence also a;' = a;'. Applying hmma 4.7 and making use of aCaF1 = A, which holds in central simple division rlgebraa over number fields, we h d for the matrices Mc, M, associated by Lemma (,ti to < and q, respectively, that McM;' E rA. For example, if (1 # 0, ql # 0, then

'l'his verifies "e" and completes the proof. 0

Finally, as above we can now express this in terms of ideal classes. The ideals a€ nnd ~ l l of Proposition 4.8 are equivalent, according to the usual definition. For general dlvision algebras, one requires a weaker notion of equivalence than isomorphism as A -modules, the notion of stable equivalence. However it is standard that for central slmple division algebras over number fields, excluding definite quaternion algebras, the rlronger notion coincides with the weaker one (see [8], 35.13). This is: A-modules A4 and N are stably equivalent if

M + A r E N + A r ,

for some r 2 0, "En being isomorphism of A -modules.

denote the set of infinite primes which ramify in D, and eet A t any rate, one defines the class ray group (depending on D) as follows. Let S c C,

e l ~ ( 0 ~ ) := {ideals}/ {aOK, a E K', ay > 0 for all v E S } . (Note that if S # 0, then necessarily D is quaternion and K = k is real, so this condition makes sense). Furthermore, let ef(A) denote the set of equivalence classes of left ideals in A. The two are related by

ef(A) 7 &?,(OK) [MI - "DlK(M)I.

For details on these matters, see [B], $35.

&?(OK) (where again K = k in case 2)). Hence Now notice that for the division algebras of (l.l), for d 2 2, we have e l ~ ( 0 ~ ) =

Proposition 4.9. For the division algebras D we are considering, if d _> 2 and b c D is D maximal order, then

@(A) 7 @(OK). We can now proceed to carry out the program sketched above for the division alge-

bras D.

Theorem 4.10. Let D be a centml simple division algebm over K as in (I.]), D2 Llte right vector space with the hyperbolic form (2.1). Let € = ((1, (2) and 6 = (711, r l z ) l e two wotmpic vectors in 3 ~ . Then

46 Math. Nachr. 200 (1991))

(i) In case 2), < and ?,I are equivalent under

(ii) In case 3) , < and q are equivalent under

i f the ideals N D l k ( ( l ( ) and NDlk(a,,)

i f f the ideals NDlK(ac) and NDlK(a,,)

Proof. By Proposition 4.8, < and q are equivalent under r A iff the ideals at and a,, N D ~ K ( ~ , , ) (where K = k fot

0

are equivalent in O k .

are equivalent in OK.

are equivalent. By Proposition 4.9 at E a,, NDlK(ac) d = 2). This is the statement of the theorem.

As a corollary of this (due to ARAKAWA [2] for d = 2 and k = Q)

Corollary 4.11. The number of cusps of is the class number of K (case 3)) 01'

the class number of k (cases 1) and 2)).

Proof. It follows from Theorem 4.10 that the number of cusps is at most the clil~b number in question; it remains to verify that it is at least the class number. If a( denotes the ideal in A, and NDIK(at) denotes the corresponding ideal in OK, thcii 51, <2 form a basis of at, while N D ~ K ( < ~ ) , N ~ l ~ ( t 2 ) form an OK-basis of the noriii ideal. So the question here is, given an ideal (01, az) c OK generated by two elements, is there an ideal (b1 ,bz) in the same ideal class of (al,a2), such that bi = N q ~ ( a ; ) for some a: E A? But this is just the statement of Proposition 4.9: given any ideal a, there is an ideal b C [a], and an ideal 6' C A such that b = N q K ( b ' ) . We furthar require that 6' defines an isotropic vector in A2, in the above sense. There should 1~ b, E 6' which generate b', such that the vector (b1,h) E A2 is isotropic with respecl to h. For this we may assume that (NDlK(bl), N D ~ K ( ~ ) ) is isotropic; this means thcit

N(bl)NO+N(b2)N(bl) = 0,

b& + = 0 .

and since the norm is zero only for the element 0, if follows from this that in fact

This just says that (bl, b2) is isotropic in the hyperbolic plane D2. In sum, for m y isotropic vector u E K2, the vector is also an isotropic vector v E D2. Any isotropic! vector of D2 yields by the norm map an isotropic vector of K 2 . It follows from P r o p sition 4.9 that this gives an isomorphism on equivalence classes, or in other words, 111~ isomorphism of Proposition 4.9 can be represented geometrically by isotropic vectorn.

Ll

Remark 4.12. We have not been very precise about the groups, but the numbrr of cusps of and S r A are the same, as is quite clear.

Now assume d 2 2 and recall the subfield L c D and subgroups GL C GD of Proposition 2.5. By Proposition 3.3 these give rise to subdomains of the domain DJI, Consider, in G L , the discrete group ro, c GL(K).

Lemma 4.13. We have for any maximal order A C D, r A n G L = roL.

I{ant, Hyperbolic D - Planes 47

Proof. This follows from the definitions and the fact that A n L = OL, 0

Let ML denote the arithmetic quotient

(4 4 ML = FOL\DL.

I 0 follows from Proposition 3.3 that we have a commutative diagram

(4.9)

Ewh ML has its own Baily-Bore1 compactification MI, which is also affected by wdding isolated points to ML. Moreover, from Proposition 3.5, we see that the map M L -+ X r , can be extended to the cusp denoted FI, E Vi, respectively F E DY, there. Ily Theorem 3 of [ll], we actually get (unique) holomorpliic maps of the Baily-Bore1 rimbeddings of ML and X r , , respectively.

Theorem 4.14. Let Mi C IF”, X; , C P” be Baily - Bore1 embeddings. Then there is a linear injection lF” L) PN‘ making the diagram

Mi c) PN 1 I x;, v B”

commute and making the image gi of Mi in X;A an algebraic subvariety.

Proof. We have an injective holomorphic embedding VL L) ’DO which comes from h Q -morphism p : (GL)c L) ( G D ) ~ (for this one takes the restriction of scalars from k to Q yielding an injection ResklQGL v R e s k l ~ G ~ , then lifts this to C ) such that p(roL) C FA. Hence we map apply [ll], Thm. 3, and there is a unique holomorphic

0

As a matter of notation, we will refer to the images GL of ML in Xr, and zi of

map Mi + X;, extending that of (4.9), and the theorem follows.

Mt in X;, as modular subvarieties.

Corollary 4.15. In case 2), there are modular curves Zi on the algebraic threefold A’;, such that the cusps of @i are cusps of X;, . In case 3), we have Hilbert modular sorieties of dimension d, Mi C X;, in the d 2 -dimensional algebraic variety XFA, ruch that the cusps of @i are cusps of X;,.

The previous Theorem 4.14 applies to the cusp of X;A which represents the equiv- dence class of the isotropic vector (0,l). We now consider the others. Given (6, G ) , RII isotropic vector representing a class of cusps, let Mt be the matrix in GD of Lemma 4,5 which maps it to (0,l). Then

-

rA,( := Mt * rA ’ Mc’ c u(02, h)

48 Math. Nachr. 200 (1999)

is a discrete subgroup of Go, and the cusp ((1 , &) of r A is mapped by Mt to the cusp ( 0 , l ) of rA,(. Letting, as above, ro, C rA denote the discrete subgroup defining the modular subvariety z~ above, we have

(4.10) rOLx := M€. ro, . iql c FA,( , and without difficulty this gives a subdomain

(4.11) v L , ( c DO i

such that, if Ft denotes the boundary component corresponding to <, then Ft E V:,(. More precisely, G L , ~ := Mc .GL .M;' is the k - group whose R- points G L , ~ (R) define D L , ~ . Then the parabolic subgroup of rA,r is FA,( f l P (P 85 in (2.10) the normalizer of (0, l)), and similarly for r O L , € . The corresponding modular subvariety

(4.12) MLX = ~ o L . , c \ ~ L , c 1

is, as is easily checked, a Hilbert modular variety for the group SL2(Ot, b2), where for any ideal c one defines

and where b is the intersection of the ideal at with Of. The same arguments as above then yield

Theorem 4.16. Given any cusp p E XFA\XrA, there is a modular subvariety - C X r , such that p E Gi,p.

Proof. If the cusp p is represented by the isotropic vector ( = ((l ,(z)l then the modular subvariety is M L , ~ as in (4.12). Just as in the proof of Theorem 4.14, we get an morphism of the Baily-Bore1 embeddine, and as mentioned above, Ft E V;,(, and p is the image of Ft under the natural projection A : Vi,{ + Mi,€ = Mi,p. 0

Corollary 4.17. In case 2), there are modular curves %i,p C XF, passing througli each cusp of XcA. In case 3), there are Hilbert modular varieties (to groups 0s in (4.13)) of dimension d passing through any cusp of X;, .

5 . An example

The theory of quaternion algebras is quite well established, and the corresponding arithmetic quotients have already been studied in [2] and [5). However the case 3) seems not to have drawn much attention as of yet, so we will give an example to illustrate the theory. Curiously enough, I ran across this example in the construction of Mumford's fake projective plane. Recall that this is an algebraic surface S of general

Hunt, Hyperbolic D-Planes 49

type with = ~ C Z , c: = 9, c2 = 3, just as for the projective plane. It then follows from Yau’s theorem that S is the quotient of the two- ball & by a discrete subgroup,

(5.1) s = r \ ~ ~ , where r is cocompact and acts without fixed points. Mumford’s construction involves lifting a quotient surface from a 2 - adic ficld to the complex numbers, and while the 2-adic group is clearly not arithmetic (as a subgroup of SL(3 ,Q2) , its elements are unbounded in the 2 -adic valuation), it i R not rlesr from the construction wliether r is arithmetic. Now the arithmetic cocompact group are known: these derive either from anisotropic groups (over $) which are of the form U(1, D) or SU(l,D), where D is a central simple division algebra over an imaginary quadratic extension K of a totally real field k, such that if the corresponding hermitian symmetric domain is irreducible, then k = $, and such that D has a KlQ -involutioti, or from unitary groups over field extensions kl$ of degree d 2 2, such tlinb, rot nll In i t one infinite prime, the real groups G, are compact. The strange thing i N l l i~l 111 Mumford’s construction such a D comes up (implicitly) naturally, a11d i t . i8 tliiH 11 wc will Introduce. After this article was completed, it was brought to the illlthOt‘H attmtioii that it has been known since ten years that this group does, indeed, givo r i w lo M~~ttiford’~ fake projective plane (personal communication from T. ZINK), althou~li it W ~ L N IIOVOI writtan up. It follows for example from Theorem 3.2 of [7], 0s followH. W(@ Iiiivo 11 Iiarinilhi form

$ : D x D ---t Q

given by $(z,y) := n0(yz) , and our group is

G = { Z € D ~ I ~ ? E ~ ) ,

Further we have an isomorphism

which results from the fact that D is only ramified at 2. Tli(!ro i H tlicii another Q-group H, such that G(A(2)) E H ( A ( 2 ) ) , but at thc priiric 2 wo Iiavc! H ( Q 2 ) fy P G L ( Q ) . If the compact open subgroup C is defined by

C = (9 E G(A,) I y( A 63 A,) C A 00 A/ ) ,

where A is a maximal order in D, then we 1ra.v~ t1iv c!xprcwion for the Shimura variety

Mc(G, X) = G(Q)\A‘ x G(Aj) /C.

Let d2) c G(A(2)) denote the part outside of tlic prime 2 of C, then one forms a product

where c H , 2 C H ( Q ) , and similarly form the quotient

cp = c(‘) x CN,2 ,

50 Math. Nachr. 200 (1999)

Here R3 denotes Drinfeld's upper half space over Zz. It is then clear from Mumford's construction that, writing his surface over Q as r\R3, we have an inclusion

r \ ~ 3 ~1 H ( Q ) \ ~ ~ x H(A(~))/C;).

Finally, we have the description

(5.2) r = { ~ E H ( Q ) I ~ E c ~ ) }

(5.3) = { h E H(Q) I hL@) c L@)}

(5.4) = { h E H(Q) 1 hL [i] c L [i] } . The last description is precisely that given by MUMFORD, see [6], p. 240, definition of r1.

Now to the example. This is a cyclic algebra, central simple over K , with splitting field L. The fields involved are

(5.5)

Lemma 5.1. L is a cyclic extension of K , of degrve three. The Galois group is generated by the transformation

40 = c 2 , 43 = c41 a(?) = c , where 1, c , C2 genemte L over K

Proof. Let 7 = E K , so OK = Z fB yZ. The ring of integers of L is generated by the powers of c,

Or, = L G 3 ~ Z @ * . . G 3 ~ ~ Z l

(remember that cg = -1 - c - . - . - c6), with the inclusion Z c OK, Z C Or, on the first factors. The key identity is the following

(5.6) -y = < + c 2 + c 4 , 7 = c3+e6+C. Fkom this relation we see that Or, is generated over OK by c and c2,

or, = O K ~ ~ O K G ~ C ' O K .

The conjugation on K extends to (complex) conjugation on L, which we will denote by z t) 2. Its action on OK is clear, y t-) 7, and on Or, it affects

- c = ( 6 , T' = (6, c3 = c'.

1 - 1 , c c--) c2, c2 - r-C-C21

It is clear that LJK is cyclic, with a generator of the Galois group

nrid by (5.6)) this is the statement of the Lemma. 0

Now consider the fixed field in L under conjugation, t c L.

Lemma 5.2. 1 w a d q m thne cyclic extension of $, with a generator of the Galow p u p being

o : C+? I-+ c2+c6 - p t ( 4 ct c + p . Proof. It is clear that Q = c + $, rh = C2 + (', t/:j = (' + C4 are in e (they are

r/l = nqt(C), rh = nLIt (C2) , rn = Tr~11(5~)). Moreover they generate the integral closure of Z in f, hence

and there is a relation qs = -171 - 172 - 1. Now observc that, u of Lemma 5.1 acts 88 follows

Of = Z$~1Z@7p~Z!,

n(vl) = U(C+C6) = u(C)+o(C') = c 2 + C 6 = 1721 b ( r ) 2 ) = 7/3 9 o(173) = 91,

Rnd this is the map specified in the lemma. 0

Next consider the cyclic algebras D = ( L / K , o , r ) and D = ( L / K , o , 2 7 ) for the element 7 above. Then D will be split iff 7 (resp. 27) is the L / K norm of some element. However, it seems difficult to verify this condition explicitly, so we show directly that these are division algebras.

Proposition 6.3. D = ( L / K , u , y ) and D = (L/K,u,27), with 7 = *, are

Proof. Note first that since the degree of D over K is three, a prime, D is a division algebra iff D is not split. To show that D is not split, it sufEcea to find a prime p C OK for which the local algebra D, is not split. This, it turns out, is easy to find. Quite generally we know that D, is split for almost all p, and non-split at most at the divisors of the discriminant of L/$ . Note that

division algebras, centrrrl simple over K .

NKlQ(7) = 77 = 2 , NKIQ(27) = 8

while K ramifies over Q only at the prime fl. Hence the primes 7, 7 and J-'5 are the primes where D, might ramify. To see that for p = (7) D, actually does ramify, we determine the Hasse invariant inv, (D) E Q /Z of D, .

To do this we must first understand the action of the Frobenius acting as generator for the maximal u n r d e d extension L, /K, . But Fkobenius in characteristic two is just a squaring map, so is clearly the mod (p) reduction of the o of Lemma 5.1. We denote this by 0,. Finally, since we are at the prime p = (7)) we may take the image of 7 as local uniformising element rp. Then (for both the algebras mentioned)

D, = ( L p / ~ p r ~ p , r p T p ) 9

which is the cyclic algebra with invp(D) = LIK is 3, invp(D) =

(this is standard; since the degree of for some s = 1, 2 or 3. The number s is the order of 7 with

52 Math. Nachr. 200 (1999)

respect to T,, i.e., the cyclic algebra (LP/Kp,ap,~~) has invariant f , see [12], 10.2.1. iii.) From this it follows that D, is not split. That D is central simple over K is clear from construction.

Finally we require a KIQ -involution on D. It is necessary and sufficient for the existence of such an involution J that there exists an element g E 1, such that

0 oa (5.7) NKIQ(27) = 4 7 5 = NfIQ(9) = 99 9 .

Given such an element g, the involution is given by (see (1.6))

An alternative to finding an explicit g is to use a theorem of LANDHERR ((121, Thm. 10.2.4): D admits a KJQ -involution 8

1) invp(D) = 0 , for all p = F , and

2) invp(D) + invg(D) = 0 , for all p # ji . (5.8)

This condition is satisfied for all p for which D, splits, so must be verified only for those primes which divide the discriminant. For us, this means at most n, 7 , 7, which we will denote by q, p, F, respectively. We showed above that invp(D) = 5 . The same argument shows that invp(D) = -i. The negative sign occurs because at the prime ji, =j is a local uniformising element TF, so localising 7 at ji gives 5’. This verifies (5.8) for the primes p # ji lying over 2. Consider the prime q. Here is where the difference between the two division algebras of Proposition 5.3 occurs. Since 27 z - l ( f l ) , it is invertible, and consequently the algebra for 27 is split at q, which implies the condition above is satisfied. On the other hand, a computation shows that the division algebra defined by 7 actually ramifies at q and hence does not

0 possess an involution of the second kind.

This completes the proof of the following

Theorem 5.4. The cyclic algebra D = (L/K,a,27) constructed above is a central sample division algebra over K with a K($ -involution. Zt ramifies at the two primes y and 7, and is split at all others.

This algebra gives rise to an anisotropic group GL(1,D) CY D*, and D*(IR) is a twisted real form of GL3; since it cannot be compact, it must be U(2, l ) . A maximal order A C D gives rise to an arithmetic subgroup of D*(R), and the quotient is then a compact ball quotient. As described above, this is related with Mumford’s fake projective plane.

We can consider the hyperbolic space (D2, h) , and the corresponding Q -groups (here k = $) G o and SGD as in (2.2) and (2.4), respectively, as well as the arithmetic subgroups r A and S r A . By Corollary 4.11, we see that r A has h ( K ) cusps, where h(K) is the class number of K; this is known to be 1. So the arithmetic quotient

Hunt, Hyperbolic D-Planes 53

liaa only 1 cusp. We also have the subgroups roL as in Lemma 4.13, as well as the modular subvarieties of X;,. We note that these are Hilbert modular threefolds coming from the cyclic cubic actension el$. Such threefolds have been considered in 1161. We also remark that on these six- folds, there are surfaces of the type considered ribove (fake projective plane), arising from embeddings of the group denoted G there rmd the hyperbolic plane group.

6. Moduli interpretation

The moduli interpretation of the arithmetic quotients X r , , as well as of the modular nubvarieties ZL, by which we mean the description of these spaces as moduli spaces, is a straightforward application of Shimura's theory. Fix a hyperbolic plane (D2, h) , and consider the endomorphism algebra M2(D) of D2, endowed with the involution 3: I+ %, z E M2(D), where the bar denotes the involution in D. Now view D2 as a $-vector space, of dimension 4f, 8f and 4d2f, in the cases l), 2) and 3), respectively. The data determining one of Shimura's moduli spaces (with no level Btructure) is (0, a, *), (T, M), where D is a central simple division algebra over K , ip is a representation of D in g((n, C ) , * is an involution on D, T is a * -skew hermitian form (matrix) on a right D vector space V of dimension rn, with gl(n, C!) % End(V, V ) R , and finally, M C D" is a Z-lattice. Then for suitable xi E V,

is a lattice and 6: "/A is abelian variety with multiplication by D. The data (D, a, t ) will be given in our cases as follows. D is our central simple division algebra over K with a Klk-involution, and the representation ip : D + M N ( C ) is obtained by base change from the natural operation of D on D2 by right multiplication. Explicitly,

where N = dimq D = 2 f, 4 f , 2d f in the cases d = 1, d = 2 and d 2 3, respectively. The involution * on D will be our involution, which we still denote by x c) L. Then a - -skew hermitian matrix T E Mz(D) will be one such that T = -T*, where ( t , j ) * = ( jji), the canonical involution on M2(D) induced by the involution on D. Note that

For any c E D* such that c = 4, the matrix T = cH H our hyperbolic matrix

( y i)) has this property. To be more specific, then, we set

(

1) d = l : T = &H 6 0

(6.3)

54 Math. Nachr. 200 (1999)

Tb = e H = (: i) , where e = (: k ) .

Since two such forms T are equivalent when they are scalar multiples of one another, assuming T of the form in (6.3) is no real restriction. Finally the lattice M will bc A2 c 0’. Then D2 @Q R E C N , N as above, and for “suitable” vectors 2 1 , x2 E D2, the lattice

(6.4)

gives rise to an abelian variety A, = C N / A z . SHIMURA has determined exactly what “suitable” means; the conditions determine certain unbounded realisations of hermitian symmetric spaces, in our cases just the domains ’DO. The Riemann form on A, is given by the alternating form E(x , y) on C

= { Qi(ai)si + @(az)zz I (ai , az) E A’}

defined by:

2

E C *(ai)zi, C a(Bj)zj = n D I Q ( igl a i t i j p j ) 7

for ol i ,B j E DR, and ( t i j ) = T is the matrix above. In particular, in all cases dealt. with here the abelian varieties are principally polarized.

SHIMURA shows that for each z E V D , vectors 21 , 2 2 E D2 are uniquely determined, hence by (6.4) a lattice, denoted A(%, T, M). The data determine an arithmetic group, which for our cases is just defined above (not S r A ) , cf. [13] (38). The basic result,, applied to our concrete situation, is

( : 1 (6.5)

Theorem 6.1. “131, Thm. 2.1 The arithmetic quotient X r , is the moduli space oj isomorphism classes of abelian Varieties determined by the data:

where Q, is given in (6.2), T in (6.3).

The corresponding classes of abelian varieties can be described as follows: 1) d = 1. Here we have two families, relating from the isomorphism of Propositioii

2.3 and the two arithmetic groups SU(O&,h) and SL2(0h). The first, for D = k, D2 = k2 yields Dk E C’, and we have abelian varieties of dimension f with rcnl multiplication by k. Secondly, for D = K , D2 = K 2 , we have abelian varieties of dimension 2f with complex mulitplication by K , with signature (l,l), that is, for each eigenvalue x of the differential of the action, also occurs. If K = Ic(J-rl), then setting K‘ = Q (6) we have k @Q K‘ CY K , hence k2 @Q K’ CY K2 and

llaiit, Hyperbolic D- Planes 66

kb, 8 K k points. Moreover, ok C3z

K k , giving the relation between the Q -vector spaces and thcir rcal p! OK, and if

IW t i lattice giving an abelian variety with mulitiplication by k,

A, := C / A z , k ,

I h n Az,k QDz OK' = A z , ~ is a lattice in 2 f , and determines an abelian variety

)>om the fact that I'oK is a subgroup of finite index in rob, we have a h i t e cover

Remark 6.2. It turns out that this case is one of the exceptions of Theorem 5 in 1131, denoted case d) there. The actual (rational) endomorphism ring of the generic member of the family is larger than K:

Theorem 6.3. "131, Prop. 18.1 The endomorphism ring E of the generic element v j the family (6.8) is a totally indefinite quaternion algebra over k , having K as a quadratic subfield.

In our situation, the totally indefinite quaternion algebra E over k is constructed 08 the cyclic algebra E = ( K / k , 0, A), where X = -u-lu, if the matrix T of (6.3) is diagonalized T = (1; ). So in our case we have X = 1 and hence the algebra E is aplit; the corresponding abelian variety is isogenous to a product of two copies of a nimple abelian variety B with real multiplication by k, as has been described already iibove. The conclusion follows from our choice of T, i. e., of hyperbolic form. It would ncem one gets more interesting quaternion algebras by choosing different hermitian forms (which, by the way, will also lead to other polarizations).

2) d = 2. D = ( e l k , 0, b ) = (a, b) is a totally indefinite quaternion algebra, central nimple over k, with canonical involution. For each infinite prime v, the localization then

Mz(D QDQ R) = Mz(lR4f) 2 M4 (C?). Let A c D be B inuxiiriul order, c GI, thc corresponding arithmetic group. Two vcctors z1, 22 E D' arisirig froiri H poiiil, i n the domain 8; (82 the Siege1 space of degrec 2) determinc! H lubt,ic:r AT ntv in (G.G), with (a1 , az) E A2, and A, = C4' /A, is the corresponding abolian variety.

3) d 2 2. In this case D is the cyclic algebra of dqrcn! d ovcr I < , arid the abelian varieties are of dimension 2d2 f .

Now we come to the most interesting point of tlie whole Htory - tlie iiioduli inter- pretation of the modular subvarieties GL C Xr,, of (4.9). The moduli interpretation

natisfies D, iz Mz(L@Q IR) = M2(IR), and D @ Q R p! m4', and IICIKO M ~ ( D ) Q o Q R E

56 Math. Nachr. 200 (1999)

of the EL is as stated in Theorem 6.1, d = 1 case. Disregarding the case d = 1, the subvarieties have @e following interpretations.

2) d = 2: As ML arises from the group U ( L 2 , h ) , where L = k(@) in our notations above, this is the moduli space of abelian varieties of dimension 2f with complex multiplication by L. On the other hand, the space Xr, parameterizes abelian varieties of dimension 4f with multiplication by D. The relation is given as follows. By definition we have D = e @I et, which in terms of matrices, is

0 0

e! { ( a o +0a1& a0 - a 1 4 ) @ (b(a2 -a3&)

Now our subfield L is the set of matrices of the form

If we let c = diag(J;i, -6) be the element representing J;; and e = (i i) , then c(ec) = c(-ce) = - 2 e = -ae, so we can generate D over Q by c and (ec). Recall that L eY k(ec), hence

(6.9) D G! L03cL.

Now consider the representation @; we have @(D) = @ ( L @ cL) = 9 1 ~ ( L ) @I @p(cL) . The lattice A2 c: D2 gives rise to a lattice A, as in (6.4), and we would like to determine when the splitting (6.9) gives rise to a splitting of the lattice A,, hence of the abelian variety A,. Consider the order

(6.10) 4' := O L $ C O L C A ;

A' is in general not a maximal order, but it is of finite index in A. Note that A' and a point z (consisting of two vectors 2 1 , ~ E D2) determine a lattice

which is also of finite index in A,. Therefore A; = Q: f A; and A, are isogenous.

@(ai) = 9(af + ca;) = @loL ( a t ) + c @ p t (a:). Then we have We now assume xi E L2. h o m (6.10) we can write ai = af + U# for ai E A', henco

and each of A' has complex multiplication by L. It follows from this that

J ~ u I I ~ , Hyperbolic D-Planes 67

~ i r l each abelian variety A:, of dimension 2 f , has complex multiplication by L. Sin(:(! A: -+ A, is an isogeny, we have

Proposition 6.4. In m e 2), the abelian varieties pammeterized by the modular. rtrbvariety ZL are isogenow to products of two abelian varieties of dimensions 2f aillr complex multiplication by L.

3) d 2 2: Again the situation is somewhat simpler here. As above, the subvariety fi,, parameterizes abelian varieties of dimension 2tCf with complex multiplication by 1,. Again we consider the order

A' := Or,@eOL@...@eed-'Or, c A ,

where we assume e is as in (1.3) and -y E Ok; then as above we can write the lattice A: for 21, 2 2 E L2, in terms of @ I L . If we write a, = at + ea: + .- . + ed-lat with (1.1 E OL, then @(ai) = 91~ (a t ) + e@lL (a:) + - - + eed-'@IL (at), and consequently

@(al)zl + @(a2)22 = (@IL(a;)zl + @ , L ( a : ) 2 2 )

+e (@IL (+I + @ , L ( 4 2 2 )

+ . . . + ed-l (+p(af)zl + + I L ( 4 ) 2 2 ) ,

{ @(ai)zi 1 (a1 , a2) E AI2} = A' @ eA2 @ - @ ed-lAd,

find each sublattice A' has complex multiplication by L. Again, the abelian variety A; so determined is isogenous to A,, and hence

Proposition 6.5. In case 3), the abelian varieties pammeterized by the modular eubvariety EL are isogenous to the product of d abelian varieties of dimension 2df with complex multiplicntion by the field L.

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Moz - Plonck - Inrtitui fir Mothemotik in den Noturwissenschaften Imclstr. 22 - 26 D - 04109 Lcipzig Germany c -moil: bhuni Omis. mpg. de