Abelian approach to modular forms of neat 2-ball lattices

Abelian approach to modular forms of neat 2-ball lattices: Dimension formulas

R.-P.Holzapfel

October5, 2001

Abstract

In previous papers Ho86], Ho00] we found neat Picard modular sur- faces with abelian minimal model and, conversely, a divisor criterion on abelian surfaces A for such a situation. For the corresponding ball lattices

; we prove dimension formulas for modular forms depending only on the intersection graph of the image on A of the compacti cation divisor of the

;-quotient surface.

1 Introduction: Main results and motivations

We look for explicite structures of ringsR(;) of modular forms for Picard mod- ular groups ; , especially in cases when the corresponding Picard modular sur- faces are well determined by explicitly known algebraic equations. The quotient surface ; ^B, ^B the complex two- dimensional unit ball, can be compactied by means of nitely many cusp singularities to a (normal complex projective) alge- braic surface;^B, the Baily-Borel compactication. By Baily-Borel's theorem B-B] one has

;^B =Proj R(;):

Generators of R(;) and relations between them dene a projective model of

;^B. It is not a simple problem to discover the ring structure in connection with the algebraic equations assumed to be known.

If, moreover, ; is a neat ball lattice, then we are in a comfortable situation.

Namely, there is a natural ring isomorphism R(;) =^M1

n⁼⁰H⁰(X^;02(logT⁰)) (1)

01991 Mathematics Subject Classication: 11G15, 11G18, 14H52, 14J25, 20H05, 22E40, 32M15, 32L07

Key words: algebraic surfaces, Shimura varieties, arithmetic groups, Picard modular groups, unit ball, modular forms

onto the ring of logarithmic pluricanonical forms of the (smooth compact) Picard modular surfaceX^;0 with compactication divisor

T⁰=^Xh

j⁼¹T_j⁰ (2)

which is a disjoint sum of elliptic curves. For a conite group extension ;¹ of

;, dened by an exact sequence of groups

1^;!;^;!;^1;!G^;!1 (3)

with nite groupG, we get isomorphisms R(;¹) =R(;)^G=^M1

n⁼⁰H⁰(X^;02(logT⁰))^G (4)

Assume, we know the ring structure in the neat case and the representation of GonR(;). Then it is "only" a matter of invariant theory for nite groups to get the structure ofR(;¹). For this latter step good software as "SINGULAR"

or "GAP" should be used. For the rst step, it is necessary to determine the dimensions

h⁰(X^;02(logT⁰)) := dimH⁰(X^;02(logT⁰))

Knowing some important cases (e.g. Picard modular surfaces of Gauss and Eisenstein numbers, see Ho00]) we concentrate our attention in this paper to abelian ball quotient surface modelsX^;0 with neat ball lattices ;, which are not cocompact.

De nition 1.1

. A ball lattice is called coabelian i the corresponding com- pactied quotient surface is abelian up to birational equivalence.

Remark 1.2

. Neat coabelian ball lattices are not cocompact, because the quo- tient surfaces of neat cocompact lattices are known to be of general type.

An abelian surface is the (unique) minimal model in its birational equivalence class (of smooth surfaces). Therefore, for any neat coabelian ball lattice ; there exist birational morphisms

A A⁰ :=X^{;0 -} X^{^;} X^; := ; ^B (5)

where ^X^; is the (normal projective) Baily-Borel compactication ofX^; with (minimal) singularity resolutionX^;0,the natural embedding andAis an abelian surface. In Ho00], Cor. 2.8, we proved the rst part of

Proposition 1.3

. The abelian surface A, which is a contracted ball quotient as described in (5), is isogeneous to EE for a suitable elliptic curve E. If, moreover, E has complex multiplication, e.g. in the case of a Picard modular surface, thenA is isomorphic to EE.

The second part follows from the rst by a theorem of Shioda-Mitani SM], see also BL], X, Corollary (6.3). So the determination of the structure of the ring of Picard modular forms in the neat abelian case can be reduced to the theory of elliptic functions. Namely, looking back to (1) we get

R(;) =^M1

n⁼⁰H⁰(A⁰²(logT⁰)) (6)

where A⁰ =X^;0 is a blown up abelian surface A EE, where " " means isogeny. Using obvious notations (omitting⁰) the image divisor of the compact- ication divisorT⁰

T :=(T⁰) =^Xh

j⁼¹Tj

(7)

is an elliptic divisor onA. This means thatT is a reduced divisor with elliptic curves as components. On the universal covering ^C2 of A they are lifted to ane complex lines. Therefore the components Tj intersect each other (at most) transversally. The set of all intersection points is the singular locus

S=S(T) :=

j⁶⁼kSj^\Sk

(8)

ofT. We consider also the subsets ofS on the components Sj =Sj(T) :=S^\Tj:

(9)

Remark 1.4

. The morphism =: S in (5) is nothing else but the blowing up of all points ofS.

Surprisingly, abelian ball quotient surface models (AT) can be recognized by an intersection property of the elliptic divisorT. Namely, we proved

Theorem 1.5

(Ho00], Theorem 2.5). Let(AT) be an abelian surface with an elliptic divisor T and : A^{0 ;!} A the blowing up of A at the singular locus S=S(T) of T with proper transformT⁰ of T onA⁰. The following conditions are equivalent:

(i) (A⁰T⁰) is a neat (coabelian) ball quotient surface with compactication divisor T⁰

(ii)

4s=^Xs_j

with cardinalitiess := #S,sj := #Sj andSj dened in (9).

Let

L=L¹+:::+Ls

(10)

be the exceptional divisor of : A^{0 ;!}A. It is a disjoint sum ofsprojective lines onA⁰ with selntersection index^;1.

Theorem 1.6

. Let ; be a neat coabelian ball lattice with smoothly compacti- ed quotient surface A⁰ =X^;0 EE,E a suitable elliptic curve. With the notations around Theorem (1) the dimensions of spaces;n] of ;-automorphic forms of weight nare

dim;n] =

(h⁰(A^0OA⁰(L+T⁰)) if n= 1 3^;n2s+h if n >1

For the dimensions of spaces ;n]⁰ of ;-cusp forms of weight n we get the following explicit formulas:

Proposition 1.7

. In the situation of Theorem 1.6 it holds that dim;n]⁰= 3

n 2

s+n¹ n^2N wheren^12f01^gis the Kronecker symbol.

Example 1

(Hirzebruch Hi] and Holzapfel Ho86], Ho00]). Neat coabelian Picard modular group of Eisenstein numbers:

A=EE,E elliptic CM-curve withK=^Q(^p;3) =^Q(!)^; multiplication,! primitive 3-rd unit root, ; commensurable with the full Picard modular groups

U((21)^OK) of Eisenstein numbers

T⁰=T¹⁰+:::+T⁴⁰ h= 4 L=L¹ s= 1 onA⁰= (EE)⁰

dim;1] =h⁰(^O(_EE⁾⁰(L¹+T¹⁰+:::+T⁴⁰)) dim;n] = dim;n]⁰+ 4 = 3

n 2

+ 4 n >1:

Example 2

(Ho00]). Neat coabelian Picard modular group of Gauss numbers:

A=EE, E elliptic CM-curve withK=^Q(i)- multiplication, ; commensu- rable with the full Picard modular groups^U((21)^OK) of Gauss numbers,

T⁰=T¹⁰+:::+T⁸⁰ h= 8 L=L¹+:::+L⁶ s= 6 onA⁰= (EE)⁰

dim;1] =h⁰(^O(_EE⁾⁰(L¹+:::+L⁶+T¹⁰+:::+T⁸⁰)) dim;n] = dim;n]⁰+ 8 = 9n^2;9n+ 8 n >1:

Example 3

(Vladov). Neat coabelian Picard modular group of Gauss numbers:

A = E E, E elliptic CM-curve with K = ^Q(i)- multiplication, ; group extension of index 2 of the ball lattice in Example 2, hence also commensurable with the full Picard modular groups^U((21)^OK) of Gauss numbers,

T⁰=T¹⁰+:::+T⁶⁰ h= 6 L=L¹+:::+L³ s= 3 onA⁰= (EE)⁰

dim;1] =h⁰(^O(_EE⁾⁰(L¹+:::+L³+T¹⁰+:::+T⁶⁰)) dim;n] = dim;n]⁰+ 6 = 9

n 2

+ 6 n >1:

In the forthcoming article Ho] we compose lifted quotients of elliptic Jacobi theta functions to abelian functions on hyperbolic biproducts of elliptic curves.

We are able to transform them to explicit Picard modular forms. Basic alge- braic relations of basic forms come from dierent multiplicative decompositions of these abelian functions in simple ones of same lifted type. Especially, for Vladov's example we can show that the explicitly constructed basic modular forms yield a Baily-Borel embedding into

P

²² together with explicit relations (homogeneous equations) for the Picard modular image surface.

2 Proof of dimension formulas

For the sake of clearness we remember to precise denitions. By ^U((21)^C) we denote the unitary group ^U(V) of a hermitian vector space (V<>) with dim^C(V) = 3 and a hermitian form<>of signature (21). The ball^B appears as subspace

B =^PV^;:=^Pfv²V< vv ><0^gPV =^P2(^C)

of all complex lines in V generated by a "negative" vector v. The group

U((21)^C) acts on^B via the natural composition

U((21)^C)^Gl(V)^;!PGl(V) =Authol(^PV) =^PGl³(^C) =Authol^P2(^C): (11)

LetK=^Q(^p;d) be an imaginary quadratic number eld,da square-free pos- itive integer, and^OK the ring of integers inK. A Picard modular group (of the eldK) is, by denition, commensurable with the the full Picard modular group

U((21)^OK). All Picard modular groups are ball lattices. This means that they act proper discontineously on^B and the volume of a ;-fundamental domain with respect to the^G(^R)-invariant hermitian (Bergmann) metric on^B (uniqely deter- mined up to a non-trivial constant factor) is nite. The quotient surface ; ^B can be compactied by means of nitely many cusp singularities to a (normal complex projective) algebraic surface ;^B, the Baily-Borel compactication.

Now let ;^U((21)^C) be a ball lattice. It acts via Authol^B =^PU((21)^C) on the^C-vector spaceH⁰(^BOB) of holomorphic functions on^B corresponding to eachf(z¹z²) the function*(f)(z¹z²) =f((z¹z²)). For each n one gets a representation

n : ;^;!AutH⁰(^BOB) ;³:f ^7!j^;n (f) (12)

with the Jacobi determinants

j(z¹z²) =det(@(z¹z²)

@(z¹z²) )

Then ;n] H⁰(^BOB) is dened to be the eigensubspace of n(;) of the eigenvalue 1, that means

;n] =^ff ²H⁰(^BOB)(f) =j_n f for all ²;^g (13)

;-cusp forms are ;-automorphic forms which vanish at innity, this means at the cusps. To be more precise, let us rst interprete automorphic forms as holomorphic sections of sheaves of higher dierential form bundles^K=^KnB :=

K n

B with the sheaf ^KB of holomorphic dierential forms on ^B. The canonical action of ; on^B is dened by

: !=fdz^{1^}dz^{2 7!} (!) =(f)(dz^{1^}dz²) =(f) j^;n dz^{1^}dz²: The embeddings

H⁰(^BOB)^;!H⁰(^BKn) f ^7!f (dz^{1^}dz²)ⁿ (14)

are compatible with the corresponding ;-actions ( n on the preimage space) and

;n_{] =}H⁰(^BKn)^;: (15)

The latter space has the advantage to go down to the quotient space ; ^B: H⁰(^BKnB)^;H⁰(; ^{BKn; B})

(16)

if we assume that ; acts freely on ^B, that means ^{B ;!} ; ^B is a universal covering. The space of cusp forms ;n]⁰ ;n] is dened by corresponding

to forms ! ²H⁰(; ^{BKn; B}) which can be extended to zero at all boundary (cusp) pointsP ²;^{B n}(; ^B).

Now let ; be a neat ball lattice and X⁰ = X^;0 the corresponding minimal smoothly compactied ball quotient surface as in (5) (forgetting the arrow on the left-hand side there) with compactication divisor T⁰, which has disjoint elliptic curve components. The link between sections of line bundles onX⁰with

;-automorphic forms used in (1) is:

;n] =H⁰(X(^KX^0T0)ⁿ) ;n]⁰=H⁰(X^K_nX^0T0n;1) (17)

where^KX⁰ =^OX⁰(K) is the canonical bundle ofX⁰,Ka canonical divisor, and

T

0 =^OX⁰(T⁰) the line bundle corresponding to T⁰. We refer to (33) of Ho98]

or, more originally to Hemperly Hem]. The Riemann-Roch formula expresses the Euler characteristics for arbitrary line bundles^VonX⁰ as

(^V) :=^X

j (^;1)^jh^j(X^0V_{) = 12(}^V (^VK;1_X⁰)) +(X⁰) where

(X⁰) =^X

i (^;1)ⁱhⁱ(^OX⁰) =^X

i (^;1)ⁱdimHⁱ(X^0OX⁰)

is the arithmetic genus ofX⁰. Using intersections of divisors we want to calculate the Euler characteristics of

Gn:= (^KX^{0 T})ⁿ and ^Fn :=^K_nX^0Tn;1: (18)

(^Gn) =(^Fn) =

n 2

((K+T⁰)²) +(X⁰) (19)

Namely, by the above Riemann-Roch formula we have (^Gn) = 12(n(K+T⁰) ((n^;1)(K+T⁰) +T⁰))

= 12n(n^;1)(K+T⁰)²+ 12n((K+T⁰) T⁰) (^Fn) = 12((n(K+T⁰)^;T⁰) (n^;1)(K+T⁰))

= 12n(n^;1)(K+T⁰)^2;1

2(n^;1)(T⁰ (K+T⁰))

For each neat ball quotient surfaceX⁰with (elliptic) compactication divisorT⁰ it holds that (T⁰ (T⁰+KX⁰)) = 0, see the proof of (iii) in the next propostion.

So the second summands of both identities vanishes. This proves (19).

Now we concentrate our attention to neat coabelian ball lattices ; and the corresponding quotient surfaces.

Proposition 2.1

. Consider a neat coabelian ball quotient surface(X^;0T⁰) = (A⁰T⁰) with compactication divisor T⁰ and exceptional divisor L of . With the notations around Theorem 1.5 it holds that

(i) K=KX⁰ =Lis a canonical divisor of X⁰ (ii) (K²) = (L²) =^;s

(iii) (T⁰ (T⁰+K)) = (T⁰ (T⁰+L)) = 0 (iv) ^;(T⁰²) = (L T`⁰) = (K T⁰) = 4s

(v) ((K+T⁰)²) = ((L+T⁰)²) = (K²)^;(T⁰²) = 3s (vi) ((K+T⁰) K) = ((L+T⁰) K) = ((L+T⁰) L) = 3s.

Proof. (i): The canonical divisor of the abelian surface A is trivial. The cano- nical divisor of a blown up surface is the sum of the exceptional divisor and the inverse image of the original surface, see BPV], I, Theorem (9.1), (vii). This means in our situation: KX⁰ =(O) +L=L.

(ii) follows immediately from (i) and (10).

(iii) needs the adjunction formula (see e.g. BPV], II.11, (16))

;(C (C+KY)) =e(C) (Euler number)

for smooth curvesCon smooth compact surfaces Y. For the elliptic curvesT_j⁰ we get

0 =^;e(Tj) = (T_j⁰ (T_j⁰+K)) hence

(T⁰ (T⁰+K)) =^Xh

j⁼¹(T_j⁰ (K+^Xh

m⁼¹T_m⁰ ) =^Xh

j⁼¹(T_j⁰ (K+T_j⁰)) = 0: (iv) The rst two identities come from (iii) and (i). With the help of Theorem 1.5, (ii), we get

(T⁰²) =^X(T_j⁰²) =^;XSj=^;4s:

(v), (vi) follow immediately from the relations proved just before:

((K+T⁰)²) = (T⁰ (T⁰+K)) + (K (T⁰+K))

= 0 + (K T⁰) + (K²) = 4s^;s= 3s:

The Hodge diamond (h^pq) = (h^p(^q)) of the abelian surfaceAis well-known to be

0

@

h⁰⁰ h⁰¹ h⁰² h¹⁰ h¹¹ h¹² h²⁰ h²¹ h²²

1

A(A) =

0

@

1 q p

q h¹¹ q p q 1

1

A(A) =

0

@

1 2 1 2 4 2 1 2 1

1

A:

Since the geometric genuspand the irregularityqare birational invariants and the Euler number e=^P(^;1)^p+qh^pq increases by 1 after applying a-process at one point, we get the following Hodge diamond forA⁰ =X⁰:

0

@

1 q p

q h¹¹ q p q 1

1

A(X⁰) =

0

@

1 2 1

2 4 +s 2

1 2 1

1

A: (20)

Notice that

(X⁰) = 1^;q+p= 0 (arithmetic genus)

e(X⁰) = 2 1 + 2 p+h^11;4 q=s (Euler number): (21)

Recall that

h^p0=h^0p=h^p(⁰(Y)) =h^p(^OY)

for each compact complex algebraic manifoldY. By Serre duality our Hodge diamond contains also the following dimensions of cohomology groups:

h²(^KX⁰) =h⁰(^OX⁰)=h⁰⁰= 1 h¹(^KX⁰) =h¹(^OX⁰)=h¹⁰=q= 2 h⁰(^KX⁰) =h²(^OX⁰)=h²⁰=p= 1: (22)

With (20) and (v) of Poposition 2.1 we make the relations of (20) more explicit:

Proposition 2.2

For the line bundles ^Fn,^Gn dened in (18) onX⁰ =X^;0,; a neat coabelian ball lattice, it holds that

(^Gn) =(^Fn) = 3

n 2

s for alln^2N+.

Proofof Proposition 1.7.

The casen= 1 is easy because ^F1 is the canonical bundle^K=^KX⁰. With (17) and (22) one gets

;1]⁰=h⁰(^F1) =h⁰(^KX⁰) = 1

in general. Forn >1 we need the following Kodaira vanishing result:

Proposition 2.3

(see Ho98], Prop. 3.6 Hem], Thm. 9.1). For any neat ball quotient surfaceX^;0 the invertible sheaves^Fn,n >1, are cohomologically trivial (acyclic) in the sense that the (higher) cohomology groups H^j(X^Fn), j > 0, vanish.

Together with (17) and Proposition 2.2 it follows that dim;n]⁰=h⁰(^Fn) =(Fn) = 3

n 2

s for alln >1.

Proofof Theorem 1.6.

The second cohomology group of^Gn vanishes because of Serre duality:

H²(X^0Gn) =H⁰(X^0KG;1_n ) =H⁰(X^0O(^;nT^0;(n^;1)K) = 0 n >0: Namely, ^;nT^0;(n^;1)KX⁰ is a negative divisor on X⁰ because T⁰ >0 and alsoK=L >0 by choice, see (i) of Proposition 2.1. Proposition 2.2, (17), the denitions of^Gn and Euler characteristics yield

dim;n] =h⁰(X^0Gn) =(^Gn) +h¹(^Gn)

=(^Fn) +h¹(^Gn) =h¹(^Gn) + 3

n 2

(23) s:

for alln^2N+.

We have to calculate the rst cohomology group of ^Gn. Consider the exact residue sequence (see BPV], II.1, (6)) of sheaves

0^;!KX^{0 ;!K}X^{0 T0;!}!_T^{0 ;!}0

with canonical sheaf!on a smooth curves (written as index). SinceT⁰=^PT0j

is a disjoint sum of s elliptic curves can we identify

!^T0 =^M!T^j0 =^MOT^j0 =^OT⁰

(24)

Tensor products with the sheaves

Gn^;1=^Kn_X^{;10 T0n;1}=^Gn1;1 yield the exact sequences

0^;!Fn ^;!Gn^;!Gn^{;1OT0 ;!}0 (25)

We deduce long exact sequences of cohomology groups:

0 ^;! H⁰(X^0F_n) ^;! H⁰(X^0G_n) ^;! H⁰(T⁰(^G1OT0)^n;1)

;! H¹(X^0F_n) ^;! H¹(X^0G_n) ^;! H¹(T⁰(^G1OT0)^n;1)

;! H²(X^0Fn) ^;! H²(X^0Gn) = 0: (26)

Especially, for n = 1 this sequence coincides with

0 ^;! H⁰(X^0K) ^;! H⁰(X^0G1) ^;! H⁰(T^0OT0)

;! H¹(X^0K) ^;! H¹(X^0G1) ^;! H¹(T^0OT0)

;! H²(X^0K) ^;! H²(X^0G1) = 0

>From (24) it is clear that

H⁰(T^0OT⁰) =^Lh_j⁼¹H⁰(Tj^O_Tj⁰) =^Ch

H¹(T^0OT⁰) =^Lh_j⁼¹H¹(Tj^O_Tj⁰) =^Lh_j⁼¹H⁰(Tj!_Tj⁰)

=^Lh_j⁼¹H⁰(Tj^OT^j0) =^Ch (27)

Together with (22) our long exact sequence looks like 0^{;!C ;!}H⁰(X^0G1)^;!Ch

;!C 2

;!H¹(X^0G1)^;!Ch

;!C ;!H²(X^0G1) = 0:

The alternating sum of dimensions of all vector spaces in an exact sequence vanishes. Thereforeh¹(^G1) =h⁰(^G1) and nally

;1] = 3

1 2

s+h⁰(^G1) =h⁰(X^0G1)

by (23), which proves together with (i) of Proposition 2.1 the casen = 1 of Theorem 1.6.

Forn >1 we remark that the canonical sheaf onT⁰is obtained by restriction

!T⁰ =^KX^{0 T0O}T⁰ =^G1OT⁰

(adjunction formula, see Ha], II.8, Proposition 8.20). This sheaf coincides with

OT⁰ by (24). Taking tensor powers we get identications (^G1OT0)^n;1=^On_T^;10 = (^Mh

j⁼¹

OT^j0)^(n;1)=^Mh

j⁼¹

O (n^;1) T^j0 =^Mh

j⁼¹

OT^j0 =^OT⁰: Taking also into account the vanishing ofH^p(X^0Fn),p= 12 (Proposition 2.3), the exact sequence (26) splits into two short exact sequences

0^;!H⁰(X^0Fn)^;!H⁰(X^0Gn)^;!H⁰(T^0OT0) =^{Ch ;!}0 0^;!H¹(X^0Gn)^;!H¹(T^0OT0) =^{Ch ;!}0

Now use the second row and (23) or the rst row to get

;n] =h⁰(X^0G_n) =h⁰(X^0F_n) +h= 3

n 2

s+h:

which was to be proved.

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