Cyclotomic Curve Families over Elliptic Curves with Complete Picard-Einstein Metric

Cyclotomic Curve Families over Elliptic Curves with Complete Picard-Einstein Metric

R.-P.Holzapfel

March 28, 2000

Abstract

According to a problem of Hirzebruch we look for models of biproducts of elliptic CM-curves with Picard modular structure. We introduce the singular mean value of crossing elliptic divisors on surfaces and determine its maximum for all abelian surfaces. For any maximal crossing elliptic divisor on an abelian surface ^Awe construct innite towers of coverings of^Awhose members, inclusively^A, are contracted compactied ball quo- tients. On this way we nd towers of Picard modular surfaces of the Gauss number eld including ^{E E} blown up at six points (^E =^{C= Zi}]), the Kummer surface of the rational cuboid problem (3-dimensional extension of congruence number problem) and some interesting rational surfaces together with the corresponding congruence subgroups of^U((21)^Zi]).

Contents

1 Introduction 2

2 Numerical ball quotient criterion for abelian surface models 3

3 Cyclic coverings of general type 8

4 Bisectional proportional elliptic divisors 14

5 Explicit cyclotomic brations 20

6 Going down to rational and Kummer surfaces 22 7 The Kummer surface of rational cuboid problem and other quo-

tients are Picard modular 24

01991 Mathematics Subject Classication: 11G15, 11G18, 14H52, 14J25, 32L07

Key words: algebraic curve, elliptic curve, algebraic surface, Shimura variety, arithmetic group, Picard modular group, Gau numbers, congruence numbers, Kahler-Einstein metric, negative constant curvature, unit ball

1 Introduction

LetE be an elliptic curve with complex multiplication eldK. In Hir] Hirze- bruch posed the following problem: Has the abelian surface E E a model which is Picard modular ? Starting fromE E he constructed for the eldK of Eisenstein numbers covering models of general type, which are compacti - cations of ball quotient surfaces, see also BHH], I.4.A. In Ho] we proved that they are Picard modular. This means that the corresponding uniformizing ball lattices are commensurable with the (full) Picard modular group^U((2 1) ^OK).

For other CM- eldsK the problem remained open.

In section 1 we de ne elliptic divisors D. For abelian surfaces B we give a simple counting criterion (see 2) in Theorem 2.5), which is necessary for the components of such divisor to bound a (neat) open ball quotient model ofB. The model is constructed by blowing up all intersection points ofD-components.

With the method of cyclic coverings we prove that the criterion 2) is also suf- cient (Theorem 2.5). For the proof in section 2 we combine the Miyaoka-Yau criterion for neat ball quotient surfaces with the Cyclic Covering Theorem. We use the theory of orbital heights on orbital surfaces developed in BSA]. An important role plays a quotient of two special orbital heights, which appears as singular mean value of elliptic divisors on abelian surfaces. From the construc- tion it is easy to see that all the coverings support (Zariski-locally) a bration of explicit equation typeYⁿ=f, wheref = 0 is a (local) equation of the divisorD onB, over an elliptic base curveEB. The bres aren-cyclic covers of an el- liptic curve (with moving branch loci). That's what we call a cyclotome-elliptic bration.

For a neat 2-ball lattice ; the invariant (Bergmann) metric on the ball ^B goes down to a complete Kahler-Einstein metric on ^B=; with negative con- stant holomorphic sectional curvature. Such metrics on surfaces we call Picard- Einstein because Picard was the rst who discovered the role of ball lattices (in connection with Picard-Fuchs systems of partial dierential equations), see Pic], EPD], Yo]. The cusp points (or their resolving cusp curves) appear as degeneration locus of the Picard-Einstein metric.

On this way we discover new "Picard-Einstein surfaces" by nite quotients and coverings ofE E,Eelliptic CM-curve with Gauss number multiplication.

In a forthcoming paper we will show that all these models are quotients of Picard modular groups of the eld of Gauss numbers, which can be determined precisely. Among them the K3 (Kummer) surface (E E)= < ^;

1

^{> is most}

interesting because it is closely connected with rational cuboid problems: Find rational cuboids with (some) rational diagonals. For details and new starts we refer to NS], BvG], Ha]. There is a modular approach to the congruence number problem (dedicated to rational rectangular triangles with rational area) due to Tunnell Tu], see also Koblitz⁰book Ko]. I think that a Picard modular approach to the rational cuboid problems is now possible and could be fruitful.

2 Numerical ball quotient criterion for abelian surface models

LetBbe an abelian surface,D²Div⁺Ba reduced curve onBandY⁰=B^0;! B the blowing up of all intersection points of the irreducible components ofD. The proper transform ofD onY⁰ is denoted byD⁰. We look for curvesD⁰such that the open surface Y := Y⁰ⁿsuppD⁰ is a neat ball quotient surface^B=;, where

B =^fz= (z¹ z²)^{2C2 j}z^j2=^jz^1j2+^jz^2j2<1^g is the two-dimensional complex unit ball and

;Authol^B =^PU((2 1) ^C) =: ^G

is a neat ball lattice. A ball lattice is a discrete subgroup of Aut_hol^B with fundamental domain of nite volume with respect to a ^G-invariant hermitian metric on ^B. ; is neat , i the eigenvalues of each element ² ; generate a torsion free subgroup of ^C. In this case the analytic quotient morphism

B ;!B=; is the universal covering of^B=; and the Baily-Borel compacti cation

d

B=; is a (projective) algebraic surface with nitely many cusp singularities compactifying ^B=;. The cusp singularities are of simple elliptic type, which means that they have an elliptic curve as singularity resolution. For details and proofs we refer to BSA], Ch.IV.

In order to getY⁰ as (smoothly compacti ed) neat ball quotient surface, it is clear that the irreducible components ofDhave to be elliptic curves. Its proper image D⁰ on Y⁰ must be a disjoint sum of elliptic curves. It follows that the intersections of two components ofD have to be transversal. Fortunately, this condition is automatically satis ed. Namely, assume that two dierent elliptic curvesF,F⁰ onB meet in P. Then the embeddings F F⁰ ,^!B can be lifted via universal coverings to embeddings of linesL L⁰ ,^!C2. So the tangent lines ofF, F⁰ at P, henceF,F⁰ themselves, cross each other inP.

Moreover, it follows that the abelian surface B splits up to isogeny into a product of two elliptic curves. Namely, the existence of only one elliptic curve on B induces such a splitting.

Alltogether we found the following (necessary) basic conditions:

(i) all irreducible components ofD are elliptic curves

(ii) these components have (at most) transversal intersections with each other (iii) the irreducible components ofD⁰ have negative sel ntersection

(iv) B is isogeneous to a product of two elliptic curves.

On abelian surfacesB the third property is equivalent to

(iii⁰) each irreducible component of D intersects properly with at least one other component.

Namely, the adjunction formula

;e(C) = (C(C+KX)) (1)

C a smooth curve on a smooth (compact) surface X,KX a canonical divisor, e(C) = 2^;2g(C) the Euler number ofC, yields

0 = (E²) + (EO) = (E²) (2)

for elliptic curvesE on any abelian surfaceB because the canonical class ofB is trivial. It becomes negative after blowing up some points ofB if and only if at least one of these points lies onE.

Denition 2.1

A reduced eective divisorDon an abelian surfaceB with only elliptic components is called elliptic divisor. It is called an intersecting elliptic divisor if and only if (additionally) there are (at least two) components inter- secting each other properly.

It is clear that the properties (i),(ii),(iii) (iii⁰) are satis ed for intersect- ing elliptic divisors. They could be used as de nition. Namely, looking at the simultaneous universal covering of the abelian surface B and the embedded el- liptic curveE ,^! B via tangential spaces it is clear thatE does not intersect another elliptic curveE⁰ if and only if the ane tangential linesTE andT_E⁰ at points onEorE⁰, respectively, are not parallel in the ane tangential planeTB. The intersection must be transversal, so property (ii) is satis ed automatically.

Moreover, if there are two components of D intersecting each other properly, then each third component has to intersect at least one of these two rst com- ponents, because its universal covering line cannot be parallel toTE andT_E⁰ at the same time. So, also the properties (iii⁰)(iii) are satis ed. It follows also that intersecting elliptic divisors are connected.

LetY⁰ =B^{0 ;!}Y^{^} be the contraction of all components of D⁰. The image D^ of D⁰ is considered as set (or cycle) of cusp points . We consider (Y⁰ D⁰), Y or (^Y D^{^}) as orbital surfaces in the sense of BSA]. There we de ned orbital Euler and signature heightsHe(Y),H (Y) of open orbital surfaces, namely:

He(Y) =e(Y⁰) =Euler number of Y⁰ H (Y) =(Y⁰)^;1

3(D⁰²) (Y⁰) =signature of Y⁰: We set

Prop(Y) =Prop(B D) :=He(Y)^;3H (Y): In BSA], see Ch. IV, (4.8.1), (4.8.2) we proved

Proposition 2.2

If Y is a ball quotient, thenProp(Y) = 0.

Denition 2.3

An intersecting elliptic divisor D on the surfaceB satisfying Prop(B D) = 0 is called proportional.

LetS =S(D) be the set of intersection points of all pairs ofD- components ands := #S its number of elements. For abelian surfacesB we know that

e(B) = 0 = 13((K_B²)^;2e(B)) =(B) hence

e(Y⁰) =He(Y) =s (Y⁰) =^;s Prop(Y) = 4s+ (D⁰²): (3)

Going back toB we writeD=_i^PN

=1

Di,Di irreducible, and set Si=S(Di) =SD(Di) :=S^\Di si:= #Si:

Then we get with (1) for the proper transformsD⁰_i onY⁰ the sel ntersections (D⁰_i²) =^;s_i, hence

(D⁰²) =^X(D_i⁰²) =^;Xsi Prop(Y) = 4s^;(s¹+:::+sN) (4)

and the

Corollary 2.4

If B is an abelian surface with intersecting elliptic divisor D such thatY is a ball quotient, then

4s=s¹+:::+sN: (5)

The basic result of this paper is the following

Theorem 2.5

LetA be an abelian surface,C =^PCj, an intersecting elliptic divisor onA,s= #S(C), sj= #S(Cj) dened as above, A^0;!Athe blowing up of A at all points of S(C), C⁰ the proper transform of C and A⁰_fin :=

A⁰ⁿsuppC⁰. Then it holds that 1) 4s^>Psj.

2) A⁰_fin is a neat ball quotient surface (with smooth compactication A⁰) if and only ifC is proportional, or, equivalently

4s=^Xs_j:

3) If the properties ofC in 2) are satised, thenAis isogeneous to the square E E of an elliptic curve E.

We start the proof with

Proposition 2.6

Let f : B ^;! A be an isogeny of abelian surfaces, C an intersecting elliptic divisor on A and D := f^;1(suppC) the preimage of the curve C identied with its reduced inverse image. Then D is an intersecting elliptic divisor onB. If C is proportional, then alsoD is.

Proof. Let E be an elliptic curve onB. By the base change property for etal morphisms (see e.g. Mil], I, Prop. 3.3) the restriction f^;1(E) ^;! E of f is etal, too. Especially, f^;1(E) is smooth, hence this preimage is a disjoint nite union of smooth irreducible curves. These curves have to be elliptic because this is the only possibility of unrami ed covers of elliptic curves by Hurwitz genus formula.

We proved that property (i) lifts fromC toD. The lift of the intersection property (iii⁰)(iii) toD is obvious.

Now let : X^{0 ;!}A be the blowing up of S = S(C) and : Y^{0 ;!} B the blowing up of S(D) = f^;1(S) with proper preimages D⁰, C⁰ of D orC, respectively. ContractingD⁰ andC⁰ we get a commutative diagram

B Y⁰ Y^{^} Y =Y⁰ⁿD⁰ A^? X⁰ X^{^} X =X⁰ⁿC⁰

f

-

q^

?

f⁰

?

f^

?

f

p^{^ -}

(6)

with vertical Galois coverings of order d, say. Counting preimage points, it is easy to see, that together with f also f⁰ is unrami ed. Namely, over the exceptional rational curve M_P = ^;1(P), P ² S, lie precisely d exceptional rational curvesL_Q,Q²f^;1(P). Therefore eachR²M_Phas at leastdpreimage points, each in oneLQ. But it cannot have more, because its number is restricted by the degreed of f⁰. Therefore f⁰ is unrami ed everywhere. This property restricts to f. This means that the orbital quotient surfaceY=G, G=Kerf, coincides withX. HenceY ^;!X is a nite orbital morphism. By de nition of orbital heights we get the relations

He(Y) =dHe(X) H (Y) =dH (X)

(see BSA], III, Prop. 3.7.6). Therefore the proportionality relationHe(X) = 3H (X) lifts toHe(Y) = 3H (Y).

Corollary 2.7

If an abelian surface A has a proportional elliptic divisor C, then each abelian surfaceB isogeneous toAhas innitely many of them. More precisely, for arbitrary N ^{2 N} there exist on B a proportional elliptic divisor with more thanN components.

Proof. Forn^{2 N}, n > 1, we consider the isogenyn : A^;! A multiplying each point with n. Let E be a component of C such that O = OA ² E. There is a unique addition on E with zero point O. The embedding E ,^! A is a homomorphism, this means the addition on A restricts to the addition on E. The multiplication morphism with n on E is denoted by nE. Since nE : E ^;!E is an isogeny of degreen² = #Ker nE, each pointP ²E has preciselyn² preimages onEbutn⁴ preimages onA. Therefore^;1_n (E) consists ofn² disjoint components consisting of the translatesE+t ofE byn-division pointst²A.

More generally, we need not assume that E goes throughO. Then for any pointQ²Ewe haveE=Q+E⁰with an elliptic curveE⁰throughO. Counting preimages it is easy to see now, that also^;1_n (E) =^;1_n (E⁰) +^;1_n (Q) consists of n² components. Its number of components is greater than N, if ^pn > N. With notations and implication of Proposition 2.6 we know that f^;1(^;1_n (C)) is a proportional elliptic divisisor onB. Obviously, its number of components is also greater thanN.

Corollary 2.8

If an abelian surfaceB supports a proportional elliptic divisor, then it is isogeneous toE E for a suitable elliptic curve E.

Proof. With the assumption of the corrollary we know that B is isogeneous to E¹ E² for two elliptic curves E¹ E² (see iv). There exists an isogeny E¹ E^{2 ;!} B. By Proposition 2.6 it suces to show that E¹ E² has no proportional elliptic divisor, ifE¹ andE² are not isogeneous. We assume this latter property. Each elliptic curveF onE¹ E² must be a bre of one of the natural projections ofE¹ E² ontoE¹ orE², becauseF cannot be a covering ofE¹ and E² at the same time. OtherwiseE¹ andE² would be isogeneous to F, hence to each other, in contradiction to our latter assumption. Therefore each elliptic divisorD²Div E¹ E²is a sum of horizontal bresHn=E¹and vertical bresVm=E²:

D= ^XM

m⁼¹Vm+^XN

n⁼¹Hn:

We show thatD is not proportional checking the proportionality condition (5) of Corollary 2.4. We have

s= #S(D) =MN #S(V_m) =N #S(H_n) =M hence

4s= 4MN ⁶=MN+NM=^X#S(V_m) +^X#S(H_n):

Remark 2.9

. We have the estimation

2s s¹+:::+sN with s=s(D) sj=sj(D) for arbitrary intersecting elliptic divisorsD=_i^PN

=1

Di on abelian surfaces B. Namely, on the right hand side we count each intersecting point ofD at least twice because of (iii⁰). So a sum of bres onE E takes the minimal value 2 of the (relative) singular mean value

(D) = (^XN

i⁼¹si)=s

ofD. By the way we proved statement 3) of Theorem 2.5.

3 Cyclic coverings of general type

We want to prove that abelian surfaces with proportional elliptic divisors D become neat ball quotient after blowing upS(D). For this purpose we look rst for nite cyclic coverings of general type satisfying the (neat) proportionality conditionH_e= 3H . The strategy is given by the following two general results.

Ball Uniformization Theorem 3.1

(see HV], Th. 0.1 or HPV], Introduc- tion). For an orbital surface

X

^{= (X}

Z

) the following conditions are equivalent:

(i)

X

has a ball uniformization (ii) The proportionality conditions (Prop 2) He(

X

^{) = 3H (}

X

^)>0

(Prop 1) he(

C

) = 2h (

C

)<0 for all orbital curves

C

Z

are satised, and there exists a nite uniformization Y of

X

, which is of general type.

Cyclic Cover Theorem 3.2

(cit. in EPD], proof e.g. in Liv]). LetV be a smooth algebraic variety,d^>2 a natural number, a reduced eective divisor onV whose linear equivalence class is divisible bydin PicV. Then:

(a) There exist d-sheeted cyclic coverings V() ^;! V with branch locus and totally branched there.

(b) These cyclic coversV() are in one-to-one correspondence with the "d-th roots" (tensor language) of in PicV, that means with all ²PicV satisfyingd= .

We start with an abelian surfaceB and a reduced divisorD=^PDk onB with properties (i), (ii), (iii)(iii⁰). As in the upper row of diagram (6) we blow up the intersection point setS=S(D). We use the notations there and assume that the class ofD is divisible by n > 1 inPicB. Then also the class of the proper imageD⁰=^PD⁰_kisn-divisible inPicY⁰. By the Cyclic Cover Theorem there exists an-cyclic covering⁰ : W^{0 ;!}Y⁰ (totally) branched overD⁰. The surfaceW⁰is smooth becauseD⁰ is a disjoint sum by (ii). The normalization of Bin the function eld^C(W⁰) along⁰ is denoted by W. The components of the preimage ofD⁰_k inW⁰ are contractible because they have together with(Dk) negative sel ntersection. The latter is equal to n(D²_k), which is negative by (iii). Alltogether we get a commutative diagram with verticaln-cyclic coverings

W W⁰ W^{^} W

B^? Y⁰ Y^{^} Y

-

?

⁰

?

^

?

-

q^

(7)

In contrast to W⁰, the surfaces W and ^W are not smooth. We use orbital heights for calculating the Chern numbers ofW⁰. For this purpose we consider the Galois quotientY⁰ofW⁰as support of the orbital surface

Y

^{0= (Y0}

Z

^{0) with}

orbital cycle

Z

^{0 =P}

D

⁰k, where

D

⁰k is the orbital curvenD⁰_k (without orbital points, because the curvesD⁰_k do not intersect each other). Each component D⁰_k has a unique preimageD_k^{0 0} onW⁰ with identical restriction_k⁰ : D_k^{0 0 $}D⁰_k of⁰. According to BSA], chapters II, III, we have the following orbital curve heights

h (D^{0 0}_k) = (D^{0 0}_k²) he(

D

⁰k) =e(D⁰_k) =e(Dk) = 0 h (

D

⁰k) = 1n⁽D_k⁰²) = 1n⁽D²_k^;sk) =^;sk

n s^{k = #}S(Dk): and the orbital relation (degree formula)

h (D^{0 0}_k) = (deg _k⁰)h (

D

⁰k) =h (

D

⁰k):

becauseW^0;!

Y

⁰ is a nite orbital covering. It turns out that (D^{0 0}_k²) =^;sk

The orbital heights ofW⁰,

Y

^{0 are} n :

He(W⁰) =e(W⁰) H (W⁰) =(W⁰) H_e(

Y

⁰) =e(Y⁰)^;X(1^; 1

n⁾h_e(

D

⁰k) =e(Y⁰) =s= #S H (

Y

⁰) =(Y⁰)^;1

3

X(n^;n¹⁾h (

D

⁰k) =^;s_{+ 13(1}^;n¹²⁾

Xsk

with relations

He(W⁰) = (deg ⁰)He(

Y

^{0) =nH}e(

Y

⁰⁾

H (W⁰) = (deg⁰)H (

Y

⁰) =nH (

Y

⁰):

We assumen >1. Using the Riemann-Roch formulas (K_W^{2 0}) = 2e(W⁰)+3(W⁰) for the sel ntersection of canonical class, (W⁰) = ¹²¹(e(W⁰) + (K_W⁰²)) for the arithmetic genus, ands ^;s+^Psk by Remark 2.9 it follows that

e(W⁰) =ne(Y⁰) =ns >0 (W⁰) =^;ns_{+ 13(}n^;1n⁾

Xsk

(K_W²⁰) =^;ns+ (n^;n¹⁾

Xskns^;1n

Xsk

(W⁰_{) = 112(}n^;n¹⁾

Xsk >0: (8)

Most interesting is the Chern quotient c²¹

c²⁽W⁰) = (K_W^{2 0})=e(W⁰) =^;1 + (1^; 1 n²⁾¹s

Xsk: (9)

Denoting the singular mean value by =(D) := 1s

Xsk

we can write

e(W⁰)=s=n

(W⁰)=s=^;n_{+ 13(}n^;1n⁾(D) (K_W^{2 0})=s=^;n+ (n^;n¹⁾(D)n^;n² (W⁰)=s_{= 112(}n^;n¹⁾(D)

c²¹

c²⁽W⁰) =^;1 + (1^; 1 n²⁾(D): (10)

The estimation comes from(D)2, see Remark 2.9. For proportional divisors Dwe have(D) = 4 by Corollary 2.4, hence

3(W⁰)=s=n^;n⁴ (K_W²⁰)=s= 3n^;n⁴ 3(W⁰)=s=n^;n¹

c²¹

c²⁽W⁰) = 3^; 4 n²: (11)

Proposition 3.3

LetB be an abelian surface with intersecting elliptic divisor D, which is n-divisible in PicB, n > 1. Then each n-cyclic cover W⁰ of Y⁰ totally branched over D⁰ is a smooth surface of general type. The contraction W^{0 ;!}W is the minimal singularity resolution. Moreover, W⁰ is the unique minimal model in its birational equivalence class.

Proof. We already mentioned that W⁰ is smooth. Now we show that there is no exceptional curve of rst kind (^;1 line) onW⁰. Assume there is one, denote it by M. Then its ⁰-image L is rational too. On the abelian surfaceB there is no rational curve. ThereforeL=LQ is the blowing up of a pointQ²S(D).

The -preimageP ofQis a unique point becauseQis the intersection of some components ofD, sayQ²Dk, and ^;1(Dk)^;!Dk is bijective. The pointP is the contraction ofM =: MP. We have an orbital Galois coveringM ^;!L with Galois groupG := GP = Gal(W⁰=Y⁰) = ^Z=n^Z. The number of branch points coincides with the numbert(Q) =tD(Q)2 of elliptic components of DthroughQ. We calculate orbital heights of

L

^{= (L}Q t(Q)smooth curve germs of weight n crossing LQ) : he(

L

^{) =e(L);t(Q)(1;}_n^{1) = 2;t(Q)(1;}_n¹⁾

h (

L

) = (L²) =^;1: Therefore

e(M) =he(M) =nhe(

L

^{) = (2;t(Q))(n;1) + 2}

genus g(M) = (2^;e(M))=_{2 = 12(}t(Q)^;2)(n^;1) (M²) =h (M) =nh (L) =^;n ^;2:

The curveM is rational if and only ift(Q) = 2, but (M²)<^;1. ThereforeM is not exceptional of rst kind. We proved thatW⁰ is minimal in its birational class, henceW^0;!W is the minimal singularity resolution.

The Kodaira dimension ^{(Y⁰) is not negative because B is abelian. For any non-constant morphismX ^;!Y⁰, X an irreducible compact complex al- gebraic surface, it holds that ^{(X) ^{(Y⁰). SinceW⁰ coversY⁰ nitely, we

get^{(W⁰)0. Surfaces with non-negative Kodaira dimension have a unique minimal model. This proves the last statement of the proposition.

From (10) we know that the sel ntersection of the canonical class ofW⁰ is positive. But for minimal surfacesX of Kodaira dimension 0 and 1 one knows that (K_X²) vanishes (see e.g. BPV]). Therefore the Kodaira dimension ofW⁰ is equal to 2. This means thatW⁰ is of general type.

Now letAbe an abelian surface with proportional elliptic divisorC=^PCj. It de nes birational morphisms

A X^{0 p^ -} X^{^} X =X⁰ⁿsuppC⁰

as described in the bottom of Diagram (6) for B instead of A. Consider the isogeny =n: A^;!A of multiplication withn >1 of degreen⁴. Following the proof of Corollary 2.7 we know that each componentE=Q+E⁰ ofC has preimage

^;1(E) = ^;1(E⁰) + ^;1(Q)

consisting ofn² components, which are translations of each other. The corre- sponding sheaves on E are isomorphic (via the translations ). So all of them represent the same element inPicA consisting of isomorphy classes of invert- ible sheaves (line bundles). Therefore ^;1(E) and also D = Dn := ^;1(C) isn-divisible inPicA (evenn²-divisible). Moreover, ^;1(C) is an elliptic pro- portional divisor by Proposition 2.6. We use it for the construction ofn-cyclic coverings as in Diagram (7) with (A D) instead of (B D). Together with Di- agram (7) we get the following tower of birational morphism triples (for each

xed n).

W W⁰ W^{^} W

A Y⁰ Y^{^} Y =Y⁰ⁿD⁰ A X⁰ X^{^} X =X⁰ⁿC⁰

?

-

?

⁰

?

^

?

?

-

q^

?

⁰

?

^ⁿ

?

-

p^

(12)

Now we are well-prepared for the

Proofof 2.5 1). By the above diagrams - choose one for each natural number

n > 1 - we dispose on a series of minimal surfaces W⁰ = W_n⁰ = W⁰(⁰_n ) of general type. The well-known Miyaoka-Yau Theorem says that the Chern quotientc²¹=c² is not greater than 3 for smooth compact algebraic surfaces of general type. Combined with the quotient formula in (10) we get

c²¹

c²⁽W_n⁰) =^;1 + (1^; 1

n²⁾(D) 3

for alln. This is only possible if(D) 4. This relation is the same as(C) 4 by the next proposition. The latter relation coincides with 1) of Theorem 2.5.

Proposition 3.4

. The singular mean value of intersecting elliptic divisors on abelian surfaces is an isogeny invariant.

This means that for isogenies f : B ^;!A, intersecting elliptic divisors C on A, D = f^;1(C) considered as reduced intersecting elliptic divisor on A (see Proposition 2.6), the singular mean values(C) and(D) coincide.

Proof. We use the notations of Diagram (6). From (4), (3) and the de nition ofProp(Y) before follows that the mean value

(D) =^;(D⁰²)=s= (Prop(Y)^;4s)=s

= (He(Y)^;3H (Y)^;4He(Y))=He(Y) =^;3(He(Y) +H (Y))=He(Y) is a quotient of orbital heights. Butf : Y ^;!X is a^B-orbital unrami ed nite morphism. For each orbital height H the degree formula H(Y) = dH(X), withd=deg f, holds. Therefore

(D) =^;3(He(Y) +H (Y))=He(Y) =^;3(He(X) +H (X))=He(X) =(C)

Corollary 3.5

. The Chern-quotients of the minimal surfaces W⁰ = W_n⁰ = W⁰(⁰_n ) of general type constructed in Diagram (12) approach the extreme value 3 forn^!1if and only if the intersecting elliptic basic divisor C on A is proportional.

Proof. This is now an immediate consequence of the last formula of (10):

c²¹

c²⁽W_n⁰) =^;1 + (1^; 1

n²⁾(Dn) =^;1 + (1^; 1 n²⁾(C): with limit^;1 +(C).

Proofof 2.5 2). One direction has already been proved before the statement 2), see Corollary 2.4. Now assume that C is a proportional divisor on the abelian surface A. For an arbitrary xed natural numbern >1 we construct diagram

(12). The cyclic covering : W ^;! Y is unrami ed because we omitted the branch locus (Y = Y^{0 n}suppD). We consider again as morphism in the category of open^B-orbital surfaces because we omitted elliptic curves with negative sel ntersections. Together with C also D is proportional elliptic by Proposition 2.6. So we have the relation

Prop(Y) =He(Y)^;3H (Y) = 0

by De nition (2.3) and (4). Multiplication withn=deg yields Prop(W) =nHe(Y)^;3nH (Y) =He(W)^;3H (W) = 0:

The theorem of Miyaoka-Kobayashi-Yau (MKY) for open surfaces (generalizing the compact version, see e.g. KoR]) says that an open surfaceZ with negative elliptic curve compacti cation Z⁰ of general type satisfying Prop(Z) = 0 is a neat ball quotient. This theorem is now part of the most general Ball Uni- formization Theorem 3.1 (proved also by R. Kobayashi KoR] in the case of surfaces of general type). The MKY-theorem is applicable toZ =W, because W⁰ is of general type, see Proposition 3.3. ThereforeW is a neat ball quotient, with Baily-Borel compacti cation ^W.

Both andnare unrami ed coverings. ThereforeXhas the same universal covering asY andW, namely the two ball^B. It follows that Y and X themselves are neat ball quotient surfaces. The proof of Theorem 2.5 is nished.

4 Bisectional proportional elliptic divisors

It is not easy to nd proportional elliptic divisors on abelian surfaces. Theorem 2.5, 3) and Corollary 2.7 reduce the existence problem to abelian biproduct surfacesE E,E an arbitrary elliptic curve. The endomorphism algebra is

EndE E=Mat²(End^circE) =Mat²(^Q)or Mat²(K)

K an imaginary quadratic number eld. We concentrate our attention on the latter (decomposed CM-) case, which happens iEhas complex multiplication.

Then we dispose on the matrix ringMat²(^O) acting on E E, EndE = ^O,

Oan order ofK, which is enough to produce a few special, but arithmetically important, examples.

As in linear algebra the action of G= ²Mat²(^O) can be described by

E E^3;_PQ^{7! ;}_PQ := ^P_P^++Q_Q= ^((P_P⁾⁺⁾⁺⁽⁽_Q^Q))

G : E E ^;! E E is an isogeny i detG = ^{; 6}= 0. It is an automorphism i G ^{2 G}L²(^O). The multiplicative semigroup of isogenies is denoted byIsog E E. We identify

EndE E =Mat²(^O) Aut_OE E=: EndE E=^Gl²(^O) (unit group):

The isogeniesGapplied to bres produce elliptic curves onE E, e.g.

E¹(G) :=G(E O) =^f;P_PP ²E^g E²(G) :=G(O E) =^{f Q}_Q Q²E^g

Transposing columns we get the same class of elliptic curves onE E through O:

(Isog E E)(E O) = (Isog E E)(O E): IdentifyingE withE Othe isogenyGinduces an isogeny

g: E^$E O^;!G(E O) P ^7!(P O)^{7! ((P}_P⁾⁾ with kernel

Ker g=g^;1(O O) =E^;tor^\E^;tor=Ker ^\Ker : (13)

For each ideal^Iof^Owe setE^I;tor :=^fT ²E ^IT =O^g:

Lemma 4.1

^{. For any G2 Mat2(O}) as above, the restrictiong to E O is an isomorphism ontoG(E O) i

(a)Ker^\Ker =O:

This condition is satised if (b)^I:=^O+^O=^O:

In the principal case^O=^OK, both properties (a) and (b) are equivalent.

Proof. The rst statement follows from (13). It is clear that Ker^\Ker =E^I;tor

(14)

hence (a) is a consequence of (b).

In any case we haveE=E(^C) =^C=^a, ^aan ideal of^Owith ^a:^a]K :=^fc²K c^aag:

The (natural) torsion points ofE are represented byK, more precisely,Etor = K=^a. In the principal case^O is a Dedekind domain. Then we know for ideals

I$Othat

^a:^I]K=^aI;1%a (15)

hence there is an element c ² K^na such that c^{I a}. The class c mod^a is a non-trivial ^I-torsion point of E. By (14) condition (a) is not satis ed. We proved the implication (a)⁾(b) in the principal case.

Let p¹, p² be the projections of E E onto the rst or second factor, respectively. By abuse of language, the curve C E E is called a hor- izontal (vertical) section i p¹ (p²) induces an isomorphism C ^! E. It is called a bisection, i C is simultaneously a horizontal and vertical section.

The image curve g(E) =G(E O) is a horizontal section i the implication (P) =(Q)⁾(P) =(Q) holds for all pairsP Q²E. Now the rst three statements of the following corollary are immediately clear.

Corollary 4.2

With the notations of the lemma it holds that:

The image curve G(E O) is a horizontal section i Ker Ker . It is a vertical section i Ker Ker . The curve G(E O) is a bisection i E^I;tor=Ker =Ker . The morphismg is an isomorphism onto a bisection if and only ifand are units in^O.

Proof. We have only to check the last statement. The if-direction is trivial. To- gether with (a) and (13) it is easy to see now that the isomorphy and bisectional assumptions are equivalent with

O=Ker ^\Ker=Ker=Ker :

Therefore the E-endomorphisms and are invertible because they are also surjective.

We want to count intersection points ofEnd(E E)-induced elliptic curves.

It is immediately clear that for G= , G⁰ = ⁰⁰⁰⁰ we have surjective homomorphisms

KerEE ^;;00

;!E¹(G)^\E¹(G⁰) KerEE ^;0

^;0

;!E¹(G)^\E²(G⁰) (16)

with kernelsKer^\Ker ^0\Ker ^\Ker ⁰andKer ^\Ker ^0\Ker^\Ker ⁰, respectively. For instance, the surjection in the rst row sends^;_PQto ^((P_P⁾⁾=

⁰⁽Q^{) 0(}Q⁾

.

Lemma 4.3

. Assume that these kernels in (16) are nite. The number of intersection points are

#(E¹(G)^\E¹(G⁰)) =N(det^;00)

#(E¹(G)^\E²(G⁰)) =N(det^;00) whereN =NK=^Q denotes the absolute norm.

Proof. Along the uniformizing exact sequence

0^;!"^{;!C2 ;!}E E^;!0

we lift, for instance, the curvesE¹(G),E²(G⁰) to the universally covering lines

C 2

L¹(G) : Z^1;Z²= 0or L²(G⁰) : ⁰Z^1;Z²= 0: (17)

The number of intersection points ofE¹(G),E²(G⁰) coincides with the norm of the determinant of the coecient matrix of the system of two linear equations in (17). For this result we refer to BHH], I.5.G (8), or originally, to Ho], Lemma II.5. This proves the second equality of the lemma. The proof of the rst is the same.

Example 4.4

(Hirzebruch Hir], see also BHH], I.4.A). Let K =^Q( ), = e²ⁱ⁼³ primitive third unit root, the eld of Eisenstein numbers,E=^C=^OK and G =^;11;1. Then D = E O+O E+E¹(G) +E²(G) is a proportional elliptic divisor onE E. After blowing up the zero point of E E one gets a D- compactied neat ball quotient surface.

Proof. The elliptic curvesE¹(G),E²(G) are bisections by Corollary 4.2. There- fore they intersect each horizontal and vertical bre in one point only. Since detGis a unit, the curvesE¹(G),E²(G) have also onlyO=O_EEas intersec- tion point by Lemma 4.3. SoD is an intersecting elliptic divisor with

s= #S(D) = 1

s(E O) = #S_D(E O) =s(O E) =s(E¹(G)) =s(E²(G)) = 1: The proportionality condition (41 = 1 + 1 + 1 + 1) is satis ed. Now Theorem 2.5, 2) yields the conclusion.

Fail Example 4.5

(BHH], I.4.G,H). For the ring ^O = ^Z+^Ziof Gaussian integers and the elliptic curveE=^C=^O the authors of BHH] present onE E the intersecting elliptic divisor

D=E¹(F) +E²(F) +E¹(G) +E²(G) +E¹(H) +E²(H)

withF =^;011;1i,G= (¹¹¹_i),H =^;101+1_i,E¹(F) =O E,E¹(H) =E O, s(D) = 4

s(E¹(F)) =s(E²(F)) =s(E¹(G)) =s(E²(G)) =s(E¹(H)) =s(E²(H)) = 2: The proportionality condition of Theorem 2.5 2) is not satis ed:

44>2 + 2 + 2 + 2 + 2 + 2:

So the example fails to be a Picard modular (after blowing up intersection points). The authors of BHH] used this example for the construction of a smooth compact surface withc²¹= 3c²by means of a special Kummer covering of small degree. Knowing proportionality relation 2) of Theorem 2.5 we are able to construct a proportional elliptic divisor on this surface.

Main Example 4.6

.

Take the same abelian surfaceE E as in the previous (fail) example. The matrices G = ^;11;11 , H = ^;1i;1i de ne four bisectional (see Corollary 4.2) elliptic curves

E¹ :=E¹(G) E² :=E²(G) E³ :=E¹(H) E⁴:=E²(H)

onE E. With the formulas of Lemma 4.3 it is easy to calculate the numerical intersection matrixN(number of intersection points as entries) for these curves:

N =

0

B

B

@

1 4 2 2

4 ¹ 2 2

2 2 ¹ 4

2 2 4 ¹

1

C

C

A: For a matrixA²Mat²(^O),detA⁶= 0, we set

(E E)A^;tor :=KerEEA:

Since the adjoint matrixA^{0 2}Mat²(^O) ofAsatis esAA⁰ =A⁰A= (detA)(¹⁰⁰¹), we have the inclusions

(E E)A^;tor(E E)detA^;tor = EdetA^;tor EdetA^;tor

j T

EN⁽detA^);tor EN⁽detA^);tor = (E E)N⁽detA^);tor

(18)

EdetA^;torE_N⁽_detA^);_tor = (^Z=N(detA)^Z)²:

The latter relations transfer to our elliptic curvesEj, j= 1 2 3 4. Restricting diagonal endomorphisms ofE E toEj we get

Ej^;tor =Ej^\(E E)^;torfor all ^2O: (19)

For A = G or H we have ^jdetA^j = 2, N(detA) = 4. Therefore the four intersection points ofE¹ E²or ofE³ E⁴coincide with the four 2-torsion points of these curves, respectively. For example, according to (16) we have

E¹(G)^\E²(G)= (E E)G^;torE^2;tor E^2;tor= (E E)^2;tor: (The minus sign in the second column of G⁰ in (16) can be omitted if only 2-torsion points appear in the kernel). Therefore, by (19),

E^1\E²(E E)^2;tor^\Ej =Ej^2;tor j = 1 2:

The inclusion is the identity because the number of elements is 4 on both sides.

To be more explicit we setTmn := (Tm Tn)²EEwith the vector (T⁰ T¹ T² T³_{) = (0 12} ^{1 +}i

2 i

2)mod^O

of 2-torsion points ofE and get

(E E)^2;tor=^fTmn 0⁶m n⁶3^g

(E E)⁽¹⁺i^);tor=^fO T⁰² T²⁰ T^22g= (^Z=2^Z)²

E^12;tor=^fO T¹¹ T²² T^33g=< T¹¹> < T³³>= (^Z=2^Z)²: becauseE¹ is the diagonal curve onE E. We proved that

E^1\E²=< T¹¹> < T³³>

E¹⁽¹⁺i^);tor=E²⁽¹⁺i^);tor=^fO T^22g=< T²²>=^Z=2^Z

For further intersections one needs only to look at the inversesA^;1 of matrices Aconstructed by pairs of two dierent columns taken fromGandH. Namely, the columns^cofA^;1satisfyA^{c2O O}, therefore^cmod^O2E=^C=^O belongs to (E E)A^;tor. This allows already to ll the numerical intersection matrix N to get the following point intersection schemeP forE¹ E² E³ E⁴:

P =

<T¹¹<T<T>E²²²²¹<T>>³³><T¹¹<T<T>E²²²²²<T>>³³><T¹³<T<T>E²²²²³<T>>³¹><T¹³<T<T>E²²²²⁴<T>>³¹>

!

The elliptic divisorC :=E¹+E²+E³+E⁴ is not proportional:

S(C) =^fO T¹¹ T²² T³³ T¹³ T^31g #S(E_k) = 4 k= 1 2 3 4 4#S(C) = 46>4 + 4 + 4 + 4:

But we can enrich it by adding some horizontal and vertical bres. We take H¹ :=E T¹ H³ :=E T³ V¹ :=T¹ E V³ :=T³ E and consider the elliptic divisor

D :=E¹+E²+E³+E⁴+H¹+H³+V¹+V³=C+F (20)

Since the elliptic curvesEk are bisections, they have only one intersection point with each bre. The intersection indices are equal to 1. Identifying divisors with supports we have

S(F) =^fT¹¹ T³³ T¹³ T^31g=C^\F S(C) hence

S=S(D) =S(C) S(Ek) =SD(Ek) =S(Ek) k= 1 2 3 4 S(Hm) =SD(Hm) =SF(Hm) =^fT¹m T³m^g m= 1 3

S(Vm) =SD(Vm) =SF(Vm) =^fTm1 Tm3^g m= 1 3:

Counting the intersection points of the components we get the proportionality relation

4#S= 46 = 4 + 4 + 4 + 4 + 2 + 2 + 2 + 2 (21)

we looked for.

With Theorem 2.5 we get