Applications of principal isogenies to constructionsof ball quotient surfaces

Azniv Kasparian

Abstract

Let((B/Γ1)′ , T (1)

)be а torsion free toroidal compactification with abelian

minimal model (A1, D(1)). An arbitrary principal isogeny µa : A2 → A1, a ∈ Cpulls-back (A1, D(1)) to the abelian minimal model (A2, D(2)) of a torsion freetoroidal compactification

((B/Γ2)′ , D(2)

). The present work makes use of the

isogeny pull-backs of abelian ball quotient models, towards the constructionof infinite series of mutually non-birational co-abelian torsion free Galois covers((B/Γn)′ , T (n)

)of a ball quotient compactification B/ΓH of Kodaira dimension

κ(B/ΓH

)≤ 0. It provides also infinite series of birational models of certain

B/ΓH .The first section studies the isogenies µa : E2 → E1 of elliptic curves Ej =

C/(Z+τjZ) with τj from the modular domain F(PSL(2,Z)). It establishes thatif {τn}∞n=1 ⊂ F(PSL(2,Z)) consists of different points then an arbitrary infiniteseries of split abelian surfaces An = En × En with En = C/(Z + τnZ) containsinfinitely many mutually non-birational members. The presence of a principalisogeny µa : A2 = E2 ×E2 → E1 ×E1 = A1 requires A1 to have a decomposedcomplex multiplication (DCM) by an imaginary quadratic number field Q(

√−d)

if and only if A2 has DCM by Q(√−d). For any infinite sequence of isogenies

µan : En → En−1 of non-isomorphic elliptic curves there follow ∩∞n=1π1(En) = Zand ∩∞n=1End(En) = Z. If the sequence {τn}∞n=1 ⊂ F(PSL(2,Z)) has a finitelimit point τ∞ = lim

n→∞τn ∈ F(PSL(2,Z)) \ {∞}, then the series {an}∞n=1 ⊂ C

of the isogenies an : En → En−1 has an infinite convergent subsequence exactlywhen the series {deg(an)}∞n=1 ⊂ N of the corresponding degrees has an infiniteconstant subsequence. If so, then there exist p, q, r, s ∈ Z, such that an =rτn−1 + s, anτn = pτn−1 + q for ∀n ∈ N and E∞ = C/(Z + τ∞Z has complexmultiplication by the imaginary quadratic number field Q(

√(p− s)2 + 4qr).

The second section constructs an infinite isogeny series{(

(B/Γn)′ , T (n))}

of torsion free co-abelian (B/Γn)′, which cover B/ΓH1 = (B/Γ1)′ /H1 for asubgroup H1 of Aut

((B/Γ1)′ , T (1)

)with linear part L(H1) < GL(2,Z). The

sequence{

(B/Γn)′}∞n=1

contains infinitely many mutually non-birational mem-bers. LetAut(Ai, D(i))Rj be the subgroup of the automorphisms h of (Ai, D(i)),whose linear parts L(h) are automorphisms of Aj , L(h) ∈ GL(2, Rj). An ar-bitrary principal isogeny µa : (A2, D(2)) → (A1, D(1)) of abelian ball quotient

models induces a homomorphism ψa : Aut(A2, D(2))R1 → Aut(A1, D(1))R2 ,multiplying the translation parts by a and fixing the linear parts. As a result,an arbitrary subgroup H1 of Aut(A1, D(1))R2 lifts to a subgroup H2 = ψ−1

1 (H1)of Aut(A2, D(2))R1 with quotient H2/ ker(ψa) = H2/ ker(µa) ' H1. That pro-duces infinite series Hn-Galois covers ζHn : (An, D(n)) → (A1/H1, T (1)/H1)

and ζ ′Hn :((B/Γn)′ , T (n)

)→(B/ΓH1 , T (1)/H1

)with strictly increasing de-

grees.The constructions of infinitely many birational models

(B/ΓH,n, T (n)/H

)of(B/ΓH,1, T (1)/H

)in section three makes use of H-equivariant principal

isogenies µa : (A,D(n)) → (A,D(n − 1)) of abelian ball quotient models.After showing the existence of infinitely many integers z ∈ Z, which pro-vide Aut(A1, D(1))-equivariant principal isogenies µz : (A,D(2)) → (A,D(1)),the work elaborates on the not necessarily Galois, eventually ramified coverµa,H : (A/H,D(2)/H) −→ (A/H,D(1)/H). More precisely, if (P,Q) ∈ Aand ζψ−1

a (H) : (A,D(2)) → (A/H,D(1)/H) is the ψ−1a (H)-Galois cover of

(A/H,D(1)/H), then the fibre of µa,H over ζψ−11

(P,Q) is shown to be the Ga-lois quotient of ker(µa) by the linear part LStabH(P,Q) of the H-stabilizerStabH(P,Q) of (P,Q). In particular, if StabH(P,Q) = {IdA} then µa,H isunramified over ζψ−1

a (H)(P,Q). If the fundamental group π1(E) of the ellipticfactor E of A = E × E is the integers ring O−d of an imaginary quadraticnumber field Q(

√−d), then µa,H is unramified over ζψ−1

a (H)(P,Q) if and only ifLStabH(P,Q) is a subgroup of the principal congruence group

GL(2,O−d)(aO−d) = ker[GL(2,O−d)→ GL (2,O−d/aO−d) .

An arbitrary ball lattice Γ < SU2,1 contains a torsion free normal subgroup Γoof finite index [Γ : Γo] < ∞. Thus, any discrete ball quotient B/Γ admits a finiteΓ/Γo-Galois cover B/Γo with fixed point free Γo. According to [3], H = Γ/Γo acts onthe toroidal compactifying divisor T of B/Γo and provides a compactification B/ΓHwith at worst cyclic quotient singularities. Let ξ : (B/Γo)′ → X be the blow-downof the (−1)-curves on the toroidal compactification of B/Γo to its minimal model X.By [2], the pair (X,D = ξ(T )) has finite automorphism group Aut(X,D), containingH. Thus, any fixed point free toroidal compactification (B/Γo)′ has finitely manyfinite Galois quotients (B/Γo)′ /H = B/ΓH , which are compactifications of discreteball quotients B/ΓH . The present note establishes the existence of (B/ΓH , T/H) withinfinitely many non-birational finite Galois covers

((B/Γn)′ , T (n)

), which are torsion

free toroidal ball quotient compactifications. By their very construction, (B/Γn)′ areco-abelian, i.e., have abelian minimal models An. The work constructs also an infiniteseries of birational ball quotient models

(B/Γn,H , T (n)/H

). The considerations are

in the virtue of Uludag’s infinite series of ball quotient orbifolds, supported by theprojective plane P2, ramified along three lines, intersecting in three nodes and givenin [4].

2

Holzapfel has proved in [1] that the abelian minimal models A of torsion freetoroidal compactifications (B/Γo)′ are isogeneous to Cartesian squares E×E of ellipticcurves. For simplicity, we restrict to A = E×E. If ξ : (B/Γo)′ → A is the blow-downof the (−1)-curves and D = ξ(T ), then (A,D) determines uniquely

((B/Γo)′ , T

).

Namely, the smooth elliptic irreducible components Ti of T =h∑i=1

Ti are mapped

biregularly onto Di = ξ(Ti) and (B/Γo)′ is the blow-up of A at the singular locusDsing =

∑1≤i<j≤h

Di ∩ Dj of D. We refer to (A,D) as an abelian ball quotient model

and say that the torsion free toroidal compactification((B/Γo)′ , T

)is co-abelian.

Holzapfel’s article [1] establishes that an arbitrary isogeny α : B → A of abeliansurfaces pulls-back an abelian ball quotient model (A,D) to an abelian ball quotientmodel (B,α−1(D)). Namely, there is a commutative diagram

(B,α−1(D))((B/Γ1)′ , T (1)

)

(A,D)((B/Γo)′ , T

)?

α

�ξ1

?

α′

�ξ

,

where α′ is an unramified Gal(α)-Galois cover, ξ1 is the blow-up of B at α−1(D)sing =α−1(Dsing) and T (1) is the proper transform of α−1(D) under ξ1. The present workrestricts to the principal isogenies α = aI2, a ∈ C∗. It combines the isogeny pullbacks with the finite Galois quotients of the co-abelian ball quotient models (A,D).By [2], any subgroup H of Aut(A,D) = Aut

((B/Γo)′ , T

)lifts to a ball lattice ΓH ,

containing Γo as a normal subgroup with quotient ΓH/Γo = H. Applying Proposition1 from [3], one concludes the H-invariance of the toroidal compactifying divisor T =(B/Γo)′ \ (B/Γo) and obtains a commutative diagram

(A,D)((B/Γo)′ , T

)

(A/H,D/H)(B/ΓH , T/H

)?

ζH

�ξ

?

ζ′H

�ξH

with H-Galois covers ζH , ζ ′H , B/ΓH = (B/ΓH) ∪ (T/H), E(ξ) = ξ−1(Dsing) and thecontraction ξH of E(ξ)/H to Dsing/H. (If p ∈ Dsing and ζH(p) is of multiplicity m,then ζ ′Hξ−1(p) is a smooth rational curve with self-intersection −m.)

3

1 Infinite series of principal isogenies of split abeliansurfaces

Let H = {z ∈ C | Im(z) > 0} be the upper half-plane and

F(PSL(2,Z)) =

=

{z ∈ C

∣∣∣ − 1

2< Re(z) ≤ 1

2, |z| > 1

}∪{eϕi

∣∣∣π3≤ ϕ ≤ π

2

}' H/PSL(2,Z)

be the standard fundamental domain for the PSL(2,Z)-action on H, called briefly themodular domain. Any isomorphism class of elliptic curves has a unique representativeEτ = C/(Z + τZ) with τ ∈ F(PSL(2,Z). The elliptic curves E, isomorphic to Eτare E = Ec,τ = C/c(Z + τZ) for some c ∈ C∗. From now on, we restrict to ellipticcurves of the form Eτ = C/(Z+τZ) and split abelian surfaces Aτ = Eτ×Eτ . For anysequence {τn}∞n=1 ⊂ F(PSL(2,Z)) of pairwise different τi 6= τj, ∀i < j we show thatthe abelian surfaces An = En ×En with En = C/Λn, Λn = Z + τnZ, n ∈ N representinfinitely many birational equivalence classes. Any subgroup H of ∩∞n=1Aut(An) isproved to have a linear part L(H) ≤ GL(2,Z). Further, the existence of Ano withcomplex multiplication by an imaginary quadratic number field Q(

√−d) implies that

all An, n ∈ N have complex multiplication by the same field Q(√−d).

Any sequence of principal isogenies µan : An → An−1 turns {An}∞n=1 into a linearlyordered inverse system of abelian surfaces. Restricting to µan : En → En−1, oneobtains a linearly ordered inverse system {En}∞n=1 of elliptic curves and a linearlyordered inverse system {Λn}∞n=1 of free Z-modules Λn = π1(En) = Z + τnZ of rank 2.By the very definition, the inverse limit

lim←−

Λi = {(λi)∞i=1 ∈ ∩∞i=1Λi | ai+1 . . . ajλj = λi for ∀j > i} =

=

{λ1 ×

(λ1

a2 . . . ai

)∞i=2

∈ Λ1 ×∞∏i=2

Λi

}' Λ1

is the first member Λ1.Any birational map of smooth curves is known to be biregular. The presence of

two mutually transversal foliations of smooth elliptic curves on Eτ ×Eτ extends thisproperty to split abelian surfaces.

Lemma 1. Let

τj ∈ F(PSL(2,Z) =

{z ∈ C

∣∣∣ − 1

2< Re(z) ≤ 1

2, |z| > 1

}∪{eϕi

∣∣∣π3≤ ϕ ≤ π

2

},

Ej = C/(Z + τjZ) and Aj = Ej × Ej for j = 1 or 2. Then any birational mapf : A2 > A1 is biregular.

4

Proof. Let us denote by Df the non-empty Zariski open subset of A, on which fis defined and biregular. In other words, Df is the intersection of the regularitydomain of f withe the image of the regularity domain of f−1. The abelian surfaceA2 = ∪P∈E2P × E2 foliates by elliptic curves P × E2 ' E2. The Zariski closedsubset A1 \ Df contains at most finitely many P1 × E2, . . . , Pk × E2. For any P ∈E2 \ {P1, . . . , Pk} the biregular restriction f : (P × E2) ∩ Df → f((P × E2) ∩ Df )can be viewed as birational maps f : (P × E2) > f((P × E2) ∩ Df ) in the Zariskiclosures f((P × E2) ∩ Df ) of f((P ×E2)∩Df ) in A1. The elliptic curves P ×E2 aresmooth, so that

f : P × E2 −→ f((P × E2) ∩ Df )

are biregular for ∀P ∈ E2 \ {P1, . . . , Pk}. Thus, ∪P∈E2\{P1,...,Pk}(P × E2) ⊆ Df .In order to justify the biregularity of f on A2, let us consider the other naturalfoliation A2 = ∪Q∈E2E2 × Q of A2. The restrictions f : E2 × Q → f(E2 × Q)are biregular for all but at most finitely many Q ∈ E2 \ {Q1, . . . , Ql}. As a result,∪Q∈E2\{Q1,...,Ql}(E2 ×Q) ⊆ Df . The assumption Df 6≡ A2 requires

∅ 6= A2\Df ⊆(∪ki=1(Pi × E2)

)∩(∪lj=1(E2 ×Qj)

)= {(Pi, Qj) | 1 ≤ i ≤ k, 1 ≤ j ≤ l}

to be a finite set. As a result,[∪ki=1Pi × (E2 \ {Q1, . . . , Ql})

]∪[∪lj=1(E2 \ {P1, . . . , Pk})×Qj

]⊂ Df ,

contrary to the choices of[∪ki=1Pi × E2

]∪[∪lj=1E2 ×Qj

]⊆ (A2 \ Df ). The contra-

diction justifies that A2 = Df and f : A2 → f(A2) is a biregular map. The abeliansurface A2 = E2 ×E2 is a projective variety, so that the image f(A2) of f : A2 → A1

is Zariski closed in A1. On the other hand, the morphism f : A2 → A1 of abelianvarieties is a group homomorphism, after an appropriate choice of an origin oA1 of A1.The only Zariski dense abelian subvariety of (A1,+) is A1 itself, so that f(A2) = A1

and f : A2 → A1 is biregular.

Lemma 2. Let Ej = C/(Z + τjZ) 1 ≤ j ≤ 2 be elliptic curves with different endo-morphism rings R1 6= R2. Then the abelian surfaces A1 = E1×E1 and A2 = E2×E2

are not biregular.

Proof. Let us assume that there is a biregular map f : A1 → A2. After moving theorigin oA2 of A2 at f(oA1), one has a linear map

f =

(a bc d

)∈ GL(2,C),

which lifts to the universal covers f : A1 = C2 → C2 = A2. The restriction of f to the

5

elliptic curve E1× oE1 and its universal cover C× 0 provides a commutative diagram

C× 0 f(C× 0)

E1 × oE1 f(E1 × oE1)?

UA1

-f

?

UA2

-f

where UAj : C2 = Aj → Aj stand for the universal covers of Aj. Straightforwardly,

(C× 0,+) ∩ (π1(A1),+) = (π1(E1)× 0,+),

so that UA1 : C× 0→ E1 × oE1 is a (π1(E1),+)-Galois cover. On the other hand,

f(C× 0) = {(at, ct) | t ∈ C}

is an additive subgroup of C2, intersecting the fundamental group of A2 in

f(C× 0),+) ∩ (π1(A2),+) = {(at, ct) ∈ π1(E2)× π1(E2) |t ∈ C} =

= {(at, ct) | t ∈ a−1π1(E2) ∩ c−1π1(E2)}.Bearing in mind the bijectiveness of the maps f : C×0→ f(C×0) and f : E1×oE1 →f(E1 × oE1), one concludes that

π1(E1) = a−1π1(E2) ∩ c−1π1(E2).

As a result, for ∀r2 ∈ R2 = End(E2) there follows

r2π1(E1) ⊆ r2a−1π1(E2) ∩ r2c

−1π1(E2) ⊆ a−1π1(E2) ∩ c−1π1(E2) = π1(E1),

whereas r2 ∈ End(E1) = R1. In other words, the presence of a biregular mapf : A1 = E1 × E1 → E2 × E2 = A2 implies the inclusion R2 ⊆ R1. In a similar vein,the isomorphism f−1 : A2 → A1 requires R1 ⊆ R2, whereas R1 = R2, contrary to theassumption.

Corollary 3. Any infinite sequence {An}∞n=1 of abelian surfaces An = En × En withEn = C/(Z + τnZ) and different {τn}∞n=1 ⊂ F(PSL(2,Z)) contains infinitely manynon-birational members.

Proof. Let us assume that there are finitely many birational equivalence classes ofAn. Then any of them contains infinitely many An with different Rn = End(En),because any Rn corresponds to finitely many τn. But Rn 6= Rm implies that An andAm are not birational. The contradiction justifies the existence of infinitely manynon-birational An.

6

Lemma 4. There exists an infinite sequence µan : An → An−1 of principal isogeniesof mutually non-birational split abelian surfaces An = En × En.Proof. We show the existence of an infinite sequence µkn : An → An−1 of multipli-cations µkn by natural numbers kn ∈ N on split abelian surfaces An = En × En,En = C/(Z+ τnZ) with different τn ∈ F(PSL(2,Z)). Inductively on n, let us assumethat we are given different τ1, . . . , τn ∈ F(PSL(2,Z)) and homomorphisms of abelianvarieties µki : Ai → Ai−1 with ki ∈ N for 1 ≤ i ≤ n. It suffices to establish that for asufficiently large kn ∈ N the lattice

1

knZ +

τnkn

Z

intersects the modular domain F(PSL(2,Z)) in infinitely many points, so that therealways exists

τn+1 ∈[(

1

knZ +

τnkn

Z)∩ F(PSL(2,Z))

]\ {τ1, . . . , τn}.

For Re(τn) > 0 let us consider the real lines

L mkn

=

{m

kn+ t

τnkn

∣∣∣ t ∈ R}

(1)

with sufficiently small m ∈ Z, such that zo = L mkn∩{Re(z) = −1

2

}has Im(zo) > 1.

Namely, if zo = mkn

+ toτnkn

then to = − 12Re(τn)

(kn + 2m) and one chooses to > knIm(τn)

,i.e.,

m < −kn(Im(τn) + 2Re(τn))

2Im(τn)

for an arbitrary kn ∈ N. Then for any kn > Re(τn) there is t(m, kn) ∈ Z withmkn

+t(m, kn) τnkn∈ F(PSL(2,Z)). More precisely, if Re

(mkn

+ t1τnkn

)= 1

2then it suffices

to show the existence of an integer t(m, kn) on the real segment (t0, t1). However,t1 = − 1

2Re(τn)(2m− kn) implies that t1 − t0 = kn

Re(τn)> 1, so that Z ∩ (t0, t1) 6= ∅. As

far as there are infinitely many m < −kn(Im(τn)+2Re(τn))2Im(τn)

, we are done in the case ofRe(τn) > 0.

If τn = 0 then we fix kn = 1 and observe that {tτn | t ∈ Z} ⊂ F(PSL(2,Z)).For Re(τn) < 0 choose kn > −Re(τn) and a sufficiently large m ∈ N, so that the

real line (1) intersects Re(z) = 12in

z1 =m

kn+

(kn − 2m)

2Re(τn)

τnkn

with Im(z1) =(kn − 2m)

2Re(τn)

Im(τn)

kn> 1.

In other words, for m < kn(Im(τn)−2Re(τn))2Im(τn)

one has an integral point t(m, kn) on (t1, t0)

for t1 = kn−2m2Re(τn)

and t0 = − kn+2m2Re(τn)

with Re(mkn

+ t0τnkn

)= −1

2, as far as the difference

t0 − t1 = kn(−Re(τn))

> 1.

7

Lemma 5. If µa : A2 → A1 is a principal isogeny of Cartesian squares Ai = Ei ×Eiof elliptic curves Ei = C/(Z + τiZ) then A1 has (decomposed) complex multiplicationby an imaginary quadratic number field Q(

√−d) if and only if A2 has (decomposed)

complex multiplication by the same field Q(√−d).

Proof. The presence of a principal isogeny µa : E2 × E2 → E1 × E1 requires

a[π1(E2)× π1(E2)] = [aπ1(E2)]× [aπ1(E2)] ⊆ π1(E1)× π1(E1),

which is tantamount to aπ1(E2) ⊆ π1(E1). In the case under consideration, theinclusion a(Z + τ2Z) ⊆ (Z + τ1Z) of Z-modules is equivalent to the simultaneoussatisfaction of

a ∈ Z + τ1Z and aτ2 ∈ Z + τ1Z.

Therefore a and τ2 belong to the extension field Q(τ1) of Q by τ1 and Q * Q(τ2) ⊆Q(τ1). Note that the endomorphisms End(Ai) = End(Ei)2×2 = (Ri)2×2 form thering of the 2× 2-matrices with entries from the endomorphism ring Ri of Ei and Aihas decomposed complex multiplication by Q(

√−d) if and only if Ei = C/(Z + τiZ)

has complex multiplication by Q(√−d). The last condition is equivalent to τi ∈

Q(√−d)\Q. Thus, if E1 has complex multiplication by Q(

√−d) then E2 has complex

multiplication by the same imaginary quadratic number field Q(√−d).

Conversely, suppose that E2 has complex multiplication by Q(√−d). Then τ2 ∈

Q(√−d) \ Q implies the inclusions Q * Q(τ2) ⊆ Q(

√−d) whereas the coincidence

Q(τ2) = Q(√−d), due to [Q(

√−d) : Q] = 2. On the other hand, a ∈ (Z + τ1Z) ∩

τ−12 (Z + τ1Z) amounts to the existence of p, q, r, s ∈ Z with

p+ qτ1 = a =r + sτ1

τ2

and τ2 =p+ qτ1

r + sτ1

.

If f(x) = x2 + αx+ β ∈ Q[x] is the minimal polynomial of τ2 over Q, then

(p+ qτ1)2 + 2(p+ qτ1)(r + sτ1) + β(r + sτ1)2 = 0

for τ1 6∈ Q requires τ1 to be algebraic of degree 2 overQ. However, τ2 = p+qτ1r+sτ1

∈ Q(τ1)\Q forces the inclusions Q * Q(τ2) ⊆ Q(τ1) and reveals the coincidence Q(τ2) = Q(τ1).Thus, τ1 ∈ Q(τ2) = Q(

√−d) and E1 has complex multiplication by Q(

√−d).

Lemma 6. Let µan : En → En−1 be an infinite series of isogenies of mutually non -isomorphic elliptic curves En = C/Λn with lattices Λn = Z+τnZ, τn ∈ F(PSL(2,Z)).Then

∩∞n=1Λn = Z and ∩∞n=1 Rn = Z.

Proof. Assume the opposite. Then for any λo ∈ ∩∞i=1Λi \ Z there exist pi ∈ Z,qi ∈ Z \ {0}, such that λo = pi + qiτi. As a result, τi = p1−pi+q1τ1

qifor ∀i ≥ 2.

8

The isogenies µai : Ei → Ei−1 give rise to isogenies µa′2...ai : Ei → E1, so thata2 . . . aiΛi ⊆ Λ1. Equivalently, a2 . . . ai = ui + viτ1 for some ui, vi ∈ Z and τi = ri+siτ1

a2...aifor some ri ∈ Z, si ∈ Z \ {0}. Altogether,

(p1 − pi) + q1τ1

qi= τi =

ri + siτ1

ui + viτ1

for ∀i ≥ 2

imply

q1viτ21 +liτ1+mi = 0 for li = (p1−pi)vi+uiq1−qisi ∈ Z, mi = (p1−pi)ui−qiri ∈ Z.

If the discriminants Di = l2i − 4q1vimi ≥ 0 is non-negative, then τ1 ∈ R, contrary torkΛ1 = rk(Z+ τ1Z) = 2. Therefore Di = −di for some di ∈ N and τ1 ∈ Q(

√−di). As

a result, all di = d are equal and Ei = C/Λi have complex multiplication by Q(√−d).

Let τi = αi + βi√−d for some αi ∈ Q, βi ∈ Q \ {0}. Then by

p1 + q1τ1 = λo = pi + qiτi for ∀i ≥ 2

there follow q1Im(τ1) = qiIm(τ1) or τi = αi +q1β1qi

√−d. If the sequence {|qi|}∞i=1 ⊂ N

is not bounded then there is a subsequence {|qik |}∞k=1 with limk→∞|qik | = ∞. Bear-

ing in mind that τi ∈ F(PSL(2,Z)), one infers that |τi|2 = α2i +

dq21β1q2i≥ 1 and

|Re(τi)| = |αi| ≤ 12. Limit transition on 1

4≥ αik ≥ 1− dq21β1

qikyields 1

4≥ 1, which is an

absurd. Therefore the sequence {|qi|}∞i=1 ⊂ N is bounded and contains a stationarysubsequence. Without loss of generality we assume that qi = q1 for ∀i ≥ 2. Thenτi = αi + β1

√−d and

Λi = Z + τiZ = Z + (αi + β1

√−d)Z = Z + β1

√−dΛ1 = Λ1 for all i ≥ 2.

As a result, all elliptic curves Ei are isomorphic to E1 and their reduced periodsτi = τ1 coincide for ∀i ≥ 2. This contradicts the choice of pairwise different τi, τj for∀i < j.

For Λn = Z + τnZ note that Rn ⊆ Λn, so that ∩∞n=1Rn ⊆ ∩∞n=1Λn = Z and∩∞n=1Rn = Z.

Corollary 7. Let {An = En × En}∞n=1 be an infinite sequence of Cartesian squaresof elliptic curves En = C/(Z+ τnZ) with pairwise different reduced periods {τn}∞n=1 ⊂F(PSL(2,Z)) and H be a subgroup of ∩∞n=1Aut(An). Then the linear part L(H) ofH is a subgroup of GL(2,Z).

Proof. Note that the lattices π1(An) = π1(En)× π1(En) = (Z+ τnZ)× (Z+ τnZ) aregenerated by (1, 0), (τn, 0), (0, 1) and (0, τn) as Z-modules. Therefore a matrix

M =

(a bc d

)∈ C2×2

9

belongs to the endomorphism ring End(An) of An if and only if[M

(10

)]t= (a, c),

[M

(τn0

)]t= (aτn, cτn),

[M

(01

)]t= (b, d),

[M

(0τn

)]t= (bτn, dτn) ∈ (Z + τnZ)× (Z + τnZ).

A complex number z belongs to the endomorphism ring Rn = End(En) of En exactlywhen z(Z + τnZ) ⊆ (Z + τnZ) or z, zτn ∈ Z + τnZ. Thus, M ∈ End(An) amountsto a, b, c, d ∈ Rn and End(An) = (Rn)2×2 consists of the 2 × 2-matrices with entriesfrom Rn.

If h ∈ Aut(An) is an automorphism of An then the linear part L(h) and its inverseL(h−1) = L(h)−1 belong to End(An) and L(h) is from

Gl(2, Rn) = {M ∈ (Rn)2×2 ∩GL(2,C) | M−1 ∈ (Rn)2×2}

over Rn. Recall that

GL(2, Rn) = {M ∈ (Rn)2×2 | det(M) ∈ R∗n}.

We claim that ∩∞n=1L(Aut(An)) = ∩∞n=1GL(2, Rn) coincides with GL(2,∩∞n=1Rn), inorder to apply Lemma 6 and conclude that L(∩∞n=1Aut(An)) ⊆ ∩∞n=1L(Aut(An)) =GL(2,Z). Indeed, if

M =

(a bc d

)∈ ∩∞n=1GL(2, Rn)

then M ∈ (∩∞n=1Rn)2×2 and det(M) ∈ ∩∞n=1R∗n ⊆ (∩∞n=1Rn)∗, so that ∩∞n=1GL(2, Rn)

is a subgroup of GL(2,∩∞n=1Rn). (Actually, ∩∞n=1R∗n = (∩∞n=1Rn)∗.) Consequently,

∩∞n=1Rn is a subring of Rn for all n ∈ N so that (∩∞n=1Rn)∗ is a subgroup of R∗n andGL(2,∩∞n=1Rn) is a subgroup of GL(2, Rn). Thus, GL(2,∩∞n=1Rn) ⊆ ∩∞n=1GL(2, Rn)and GL (2,∩∞n=1Rn) = ∩∞n=1GL(2, Rn).

We say that the covering f : E ′ → E of elliptic curves is natural if it induces theidentical inclusion f∗ ≡ Id : π1(E ′) ↪→ π1(E) of the fundamental groups. Equiva-lently, f : E ′ → E is a natural covering if it lifts to the identity f : IdC2 : E ′ → E ofthe universal covers E ′ ' C ' E. Lemma 6 asserts that there is no elliptic curve Eo,which is a common natural cover of infinitely many En = C/(Z+ τnZ) with differentτn ∈ F(PSL(2,Z)), but there is a common natural cover (C,+)/(Z,+) ' C∗ of En,∀n ∈ N.

Lemma 8. Let µan : En → En−1 be an infinite series of isogenies of elliptic curvesEi = C/(Z + τiZ) with convergent reduced periods {τn}∞n=1 ⊂ F(PSL(2,Z)), τ∞ =limn→∞

τn ∈ F(PSL(2,Z)) \ {∞}. Then the following are equivalent:

10

(i) {an}∞n=1 ⊂ C has a convergent infinite subsequence;(ii) {deg(µan)}∞n=1 ⊆ N has a constant infinite subsequence.If so, then

an = rτn−1 + s, anτn = pτn−1 + q

for fixed p, q, r, s ∈ Z and E∞ = C/(Z + τ∞Z) has complex multiplication by theimaginary quadratic number field Q(

√(p− s)2 + 4qr).

Proof. Recall that µan : En → En−1 are endomorphisms of elliptic curves if and onlyif an(Z + τnZ) ⊆ Z + τn−1Z. That amounts to the existence of matrices

Cn =

(pn qnrn sn

)∈ Z2×2 with

Cn

(τn−1

1

)=

(anτnan

). (2)

By the structure theory of the finitely generated free abelian groups, there are Z-bases(λ′n−1

λ′′n−1

)= Mn

(τn−1

1

)of Z + τn−1Z

with transition matrices Mn = (mij(n))2i=1

2j=1 ∈ SL(2,Z), such that(

mn 00 mnkn

)(λ′n−1

λ′′n−1

)for some mn, kn ∈ N are Z-bases of Mn(an(Z+ τnZ)).

As a result,(β′nβ′′n

)= M−1

n

(mn 00 mnkn

)(λ′n−1

λ′′n−1

)= mnM

−1n

(1 00 kn

)Mn

(τn−1

1

)are Z-bases of an(Z + τnZ). Let us denote by Ln = (lij(n)) ∈ SL(2,Z) the transitionmatrices from (

anτnan

)to

(β′nβ′′n

),

Ln

(anτnan

)= LnCn

(τn−1

1

)= mnM

−1n

(1 00 kn

)Mn

(τn−1

1

)=

(β′nβ′′n

).

Bearing in mind that

LnCn ∈ Z2×2, mnM−1n

(1 00 kn

)Mn ∈ Z2×2

and (τn−1, 1) are Z-linearly independent, one concludes that

LnCn = mnM−1n

(1 00 kn

)Mn.

11

The determinant of the above relation reads as

pnsn − qnrn = det(Cn) = m2nkn = deg(µan), (3)

where m2nkn stands for the order of

Gal(µan) = Ker(µan) = a−1n−1(Z + τn−1Z)/(Z + τnZ) '

' (Z + τn−1Z)/[an(Z + τnZ)] '(Zλ′n−1 + Zλ′′n−1

)/(Zmnλ

′n−1 + Zmnknλ

′′n−1

).

without loss of generality, one can choose the arguments ϕn = 1i

log(an|an|

)∈ [0, 2π)

of an = |an|eiϕn from [0, 2π). Then the bounded sequence {ϕn}∞n=1 ⊂ [0, 2π) hasa convergent subsequence. After passing to a subsequence, one can assume that{ϕn}∞n=1 converges to ϕ∞ = lim

n→∞ϕn.

Suppose that ϕ∞ 6∈ Zπ. Then the sequence

Re(an)

Im(an)=rnRe(τn−1) + snrnIm(τn−1)

= cot(arg(τn−1)) +

(snrn

)1

Im(τn−1)= cot(ϕn)

has a finite limit, so that{snrn

}∞n=1⊆ Q converges to some real number α1, due to

τ∞ = limn→∞

τn ∈ C \ R. In particular, rn 6= 0, since otherwise rn = sn = 0 and an = 0.Similarly, for ψ∞ = arg(τ∞) and ϕ∞ + ψ∞ 6∈ Zπ, the sequence

Re(anτn)

Im(anτn)=pnRe(τn−1) + qnpnIm(τn−1)

= cot(arg(τn−1)) +

(qnpn

)1

Im(τn−1)

has a finite limit and there exists α2 = limn→∞

(qnpn

)∈ R. In particular, pn 6= 0.

Expressing

pn

(τn−1 +

qnpn

)= pnτn−1 + qn = anτn = (rnτn−1 + sn)τn = rn

(τn−1 +

snrn

)τn,

one derives that

pnrn

=

(τn−1 + sn

rn

)τn

τn−1 + qnpn

converges to some α3 = limn→∞

(pnrn

)∈ R. Moreover, α3 6= 0, according to τ∞ ∈ C \ R.

Now (3) takes the form

p2n

(rnpn

)(snrn− qnpn

)= m2

nkn. (4)

12

We claim that α1 = limn→∞

(snrn

)6= lim

n→∞

(qnpn

)= α2. Otherwise,

anrn− snrn

= τn−1 =anτnpn− qnpn

from (2) yields

limn→∞

{anrn

[1− τn

(rnpn

)]}= 0.

Bearing in mind that

limn→∞

[1− τn

(rnpn

)]= 1− τ∞

α3

6= 0,

one concludes that limn→∞

(anrn

)= 0, whereas

τ∞ = limn→∞

τn−1 = limn→∞

(anrn− snrn

)= −α1 ∈ R,

which is an absurd. Thus, α1 6= α2.If at least one of the sequences {mn}∞n=1 ⊆ N or {kn}∞n=1 ⊆ N is unbounded, then

it contains a subsequence, which tends to ∞. As a result, (4) provides

limn→∞

(p2n)

(α1 − α2)

α3

= limn→∞

(p2n) lim

n→∞

(rnpn

)[limn→∞

(snrn

)− lim

n→∞

(qnpn

)]=

limn→∞

(m2nkn) =∞,

so that limn→∞

(p2n) =∞ and lim

n→∞|pn| =∞. Now by (2) one has

an = pn

(τn−1 + qn

pn

τn

)(5)

and there follows the divergence of {an}∞n=1, whenever {deg(µan)}∞n=1 ⊆ N is notbounded.

If {mn}∞n=1 ⊆ N and {kn}∞n=1 ⊆ N are bounded, there is no loss of generality inassuming that ∀mn = m and ∀kn = k, after passing to an infinite subsequence. Then(4) reads as

p2n =

m2k(rnpn

)(snrn− qn

pn

)and implies the convergence of {p2

n}∞n=1 ⊂ N. After passing to a subsequence, one has

limn→∞

pn = limn→∞

|pn| or limn→∞

pn = − limn→∞

|pn|.

13

Making use of (5), one concludes the convergence of {an}∞n=1.We claim that if {an}∞n=1 converges to a∞ ∈ C then the sequences {pn}∞n=1, {qn}∞n=1,

{rn}∞n=1, {sn}∞n=1 ⊂ Z are constant. Indeed, by an = rnτn−1 + sn one expressesrn = Im(an)

Im(τn−1)and infers the convergence of {rn}∞n=1. Then sn = an − rτn−1 is to

converge, as well. Similarly, the convergence of {τn}∞n=1 and {an}∞n=1 implies theconvergence of{

pn =Im(anτn)

Im(τn−1)

}∞n=1

and {qn = anτn − pnτn−1}∞n=1 .

The limit transition in

(rnτn−1 + sn)τn = anτn = pnτn−1 + qn

provides

rτ 2∞ + (s− p)τ∞ − q = 0 or τ∞ =

p− s±√D

2rwith D = (p− s)2 + 4qr ∈ Z.

IfD ≥ 0 then τ∞ ∈ R is an absurd. ThereforeD = −d for d = −(p−s)2−4qr ∈ N andτ∞ ∈ Q(

√−d). Thus, E∞ = C/(Z + τ∞Z) has complex multiplication by Q(

√−d).

2 Ball quotient surfaces with infinitely many non-birational co-abelian Galois covers

In order to study the effect of a principal isogeny

µa : (A2, D(2) = µ−1a D(1)) −→ (A1, D(1))

on the corresponding automorphism groups, one needs to describe the elliptic curvesDj on an abelian surface A = E × E. The identical inclusion Id : Dj → A is amorphism of abelian varieties and lifts to an affine-linear map Ij : Dj = C→ C2 = Aof the corresponding universal covers. If Dj and A have a common origin oDj = oA,then Ij is a linear map and Ij : Dj → Ij(Dj) is an isomorphism onto a line Ij(Dj) ⊂C2 = A through the origin (0, 0) ∈ C2. Let Ij(1) = (aj, bj) ∈ C2 \ {(0, 0)}, in orderto have Ij(t) = (ajt, bjt) for ∀t ∈ C = Dj and

Ij(Dj) = {(ajt, bjt) ∈ C2 | t ∈ C} = {(u, v) ∈ C2 | bju− ajv = 0}.

The complete pre-image ofDj with respect to the universal covering UA : A = C2 → Ais

U−1A (Dj) = {(ajt+ λ1, bjt+ λ2) | t ∈ C, λ1, λ2 ∈ π1(E)},

14

so that Dj can be represented as

Dj = U−1A (Dj)/π1(A) = {(ajt+ π1(E), bjt+ π1(E)) | t ∈ C}.

From now on, we denote such Dj by E(aj, bj).

Lemma 9. Let µa : (A2, D(2)) → (A1, D(1)) be a principal isogeny of abelian ballquotient models (Ai, D(i)) with Ai = Ei × Ei, Ei = C/Λi, Λi = Z + τiZ, τi ∈F(PSL(2,Z)), D(2) = µ−1

a D(1) and

Aut(Ai, D(i))Rj := {h ∈ Aut(Ai, D(i)) | L(h) ∈ GL(2, Rj)}.

Thenψa : Aut(A2, D(2))R1 −→ Aut(A1, D(1))R2 ,

ψa(τ(U,V )L(h)) = τµa(U,V )L(h) for ∀h ∈ τ(U,V )L(h) ∈ Aut(A2, D(2))R1

is a group epimorphism with kernel ker(ψa) = (ker(µa),+).In particular, for any subgroup H1 ≤ Aut(A1, D(1))R2 the complete preimage

H2 = ψ−1a (H1) is a subgroup of Aut(A2, D(2))R1, containing (ker(µa),+) as a normal

subgroup with quotient H2/ ker(µa) ' H1 and there is a commutative diagram

(A2, D(2)) (A1, D(1))

(Aj/Hj, D(j)/Hj)

-µa

QQQQQQQs

ζH2

?

ζH1 , (6)

relating the Hj-Galois covers ζHj for j ∈ {1, 2}.

Proof. First of all, any image of ψa : Aut(A2, D(2))R1 → Aut(A1)R2 leaves invariantD1. More precisely, for any irreducible component D(2)ij = E2(ai, bi) + (Pij, Qij) ⊂µ−1a D(1)i of D(2) over D(1)i = E1(ai, bi) + (Pi, Qi), µa(Pij, Qij) = (Pi, Qi) and anyh = τ(U,V )L(h) ∈ Aut(A2, D(2))R1 , one has

E2(ak, bk) + (Pkl, Qkl) = D(2)kl = hD(2)ij = L(h)E2(ai, bi) + L(h)(Pij, Qij) + (U, V ).

That amounts toL(h)E2(ai, bi) = E2(ak, bk) and

L(h)(Pij, Qij) + (U, V ) ∈ E2(ak, bk) + (Pkl, Qkl).

Acting by µa, one obtains

L(h)E1(ai, bi) = L(h)µaE2(ai, bi) = µaL(h)E2(ai, bi) = µaE2(ak, bk) = E1(ak, bk)

15

andτµa(U,V )L(h)(Pi, Qi) = L(h)µa(Pij, Qij) + µa(U, V ) =

= µaL(h)(Pij, Qij) + µa(U, V ) ∈ µaE2(ak, bk) + µa(Pkl, Qkl) = E1(ak, bk) + (Pk, Qk).

Therefore

ψa(τ(U,V )L(h))D(1)i = L(h)E1(aibi) + τµa(U,V )L(h)(Pi, Qi) =

= E1(ak, bk) + (Pk, Qk) = D(1)k

and ψaAut(A2, D(2))R1 acts on D(1).According to

ψa(τ(U1,V1)L(h1)τ(U2,V2)L(h2)) = ψa(τ(U1,V1)+L(h1)(U2,V2)L(h1)L(h2)) =

= τµa(U1,V1)+L(h1)µa(U2,V2)L(h1)L(h2) = (τµa(U1,V1)L(h1))(τµa(U2,V2)L(h2)) =

= ψa(τ(U1,V1)L(h1))ψa(τ(U2,V2)L(h2)) for ∀hj = τ(Uj ,Vj)L(hj) ∈ Aut(A2, D(2))R1 ,

the map ψa is a group homomorphism. For arbitrary h1 = τ(U1,V1)L(h1) from thegroup Aut(A1, D(1))R2 and (U2, V2) ∈ µ−1

a (U1, V1), we claim that h2 = τ(U2,V2)L(h1) ∈Aut(A2, D(2))R1 . Due to ψa(h2) = h1, the homomorphism ψa turns to be surjective.Indeed, if

L(h1)E1(ai, bi) + L(h1)(Pi, Qi) + (U1, V1) = h1D(1)i = D(1)k = E1(ak, bk) + (Pk, Qk)

then

L(h1)E1(ai, bi) = E1(ak, lk) and L(h1)(Pi, Qi) + (U1, V1) ∈ E1(Ak, bk) + (Pk, Qk).

For an arbitrary irreducible component D(2)ij = E2(ai, bi) + (Pij, Qij) of theisogeny pull-back µ−1

a D(1)i ⊂ D(2) there hold

L(h1)E2(ai, bi) = E2(ak, bk) and

L(h1)(Pij, Qij) + (U2, V2) ∈ E2(ak, bk) + (Pkl, Qkl) = D(2)kl ⊂ µ−1a D(1)k ⊂ D(2),

according to

µa(L(h1)(Pij, Qij) + (U2, V2)) = L(h1)µa(Pij, Qij) + µa(U2, V2) =

= L(h1)(Pi, Qi) + (U1, V1) ∈ E1(ak, bk) + (Pk, Qk) = D(1)k.

Thus, h2 = τ(U2,V2)L(h1) acts on D(2) and belongs to Aut(A2, D(2))R1 . The kernel ofthe group epimorphism ψa is

ker(ψa) = {h = τ(U,V )L(h) ∈ Aut(A2, D(2))R1 | ψa(h) = τµa(U,V )L(h) = IdA1} =

= {h = τ(U,V ) ∈ Aut(A2, D(2))R1 | µa(U, V ) = oA1} = (ker(µa),+).

16

For an arbitrary biholomorphism group G, acting on a manifold X and a normalsubgroup K of G, the quotient group G/K acts on the quotient variety X/K andgives rise to a commutative diagram

X X/K

X/G = (X/K)/(G/K)

-ζK

QQQQQQQs

ζG

?

ζG/K

with H-Galois covers ζH , H ∈ {G,K,G/K}. For an arbitrary subgroup H1 ≤Aut(A1, D(1))R2 and its complete preimage H2 = ψ−1

a (H1) ≤ Aut(A2, D(2))R1 , con-taining (ker(µa),+) as a normal subgroup with quotient group H2/ ker(µa) ' H1, onehas a commutative diagram

(A2, D(2)) (A1, D(1))

(Aj/Hj, D(j)/Hj)

-µa

QQQQQQQs

ζH2

?

ζ′H1 (7)

with H1-Galois cover ζ ′H1, induced by the H1-action

(h2 ker(µa))(µa(P2, Q2)) = µah2(P2, Q2)

for ∀h1 = h2 ker(µa) ∈ H2/ ker(µa), ∀(P2, Q2) ∈ A2. There remains to be checkedthat (7) coincides with the original H1-action on (A1, D(1)). Indeed, for any h1 =τµa(U,V )L(h2) ∈ H1 and (P1, Q1) = µa(P2, Q2) with h2 = τ(U,V )L(h2) ∈ H2, (P2, Q2) ∈A2, one identifies h1 with the coset h2 ker(µa) and observes that (7) reads as

h1(P1, Q1) = (h2 ker(µa))(µa(P2, Q2)) = µah2(P2, Q2) =

= µaL(h2)(P2, Q2) + µa(U, V ) = L(h2)(P1, Q1) + µa(U, V )

and coincides with the original H1-action.

Corollary 10. In the notations from Lemma (9), let((B/Γj)′ , T (j)

)be the blow-ups

of (Aj, D(j)) at D(j)sing and(B/ΓHj , T (j)/Hj

)be their Hj-Galois quotients. Then(

B/ΓH2 , T (2)/H2

)=(B/ΓH1 , T (1)/H1

)coincide and fit in a commutative diagram(

(B/Γ2)′ , T (2)) (

(B/Γ1)′ , T (1))

(B/ΓHj , T (j)/Hj

)-

µ′a

HHHHH

HHjζ′H2 ?

ζ′H1 , (8)

17

where ζ ′Hj are Hj-Galois covers and µ′a is an unramified ker(µa)-Galois cover.

Proof. By Holzapfel’s Proposition 4 from [1], there is a commutative diagram

(A2, D(2))((B/Γ2)′ , T (2)

)

(A1, D(1))((B/Γ1)′ , T (1)

)?

µa

�ξ2

?

µ′a

�ξ1

, (9)

where ξj are the blow-ups of Aj at D(j)sing, T (j) are the proper transforms of D(j)under ξj and µ′a is a finite unramified morphism. We claim that µ′a is ker(µa)-Galoiscover. To this end, note that if D(1) =

∑i

D(1)i and µ−1a D(1)i =

∑j

D(1)i,j then

D(2) = µ−1a D(1) =

∑i

∑j

D(1)i,j with D(1)ij ∩D(1)i,k = ∅ for j 6= k. Then

D(2)sing =∑i1<i2

∑j1,j2

D(1)i1,j1 ∩Di2,j2 = µ−1a D(1)sing

and the exceptional divisor of ξ2 is ker(µa)-invariant. The (ker(µa),+)-action on A2

is by translations, so that extends by the identity over the irreducible componentsξ−1

2 (P,Q) ' P1, (P,Q) ∈ D(2)sing of the exceptional divisor ξ−12 D(2)sing of ξ2. Thus,

the morphism µ′a : (B/Γ2)′ → (B/Γ1)′ is ker(µa)-Galois cover. In general, if H is afinite automorphism group of an abelian surface A and ξ : A′ → A is the blow-up ofA at a finite H-invariant subset S of A then the H-action extends to A′ and providesa commutative diagram

A A′

A/H A′/H?

ζH

�ξ

?

ζ′H

�ξH

with H-Galois covers ζH , ζ ′H and a birational morphism ξH , shrinking the irreduciblecomponents of ξ−1Dsing/H to points. This is due to the H-equivariance of the ex-ceptional divisor ξ−1(Dsing) of ξ. The group H < Aut(A) consists of affine-lineartransformations with respect to global holomorphic coordinates (u, v) on the uni-versal cover A = C2 of A. At any point po = (uo, vo) + π1(A) ∈ A = A/π1(A),(uo, vo) ∈ C2 consider the local holomorphic coordinates u−uo, v−vo, centered at po.The action of an arbitrary h ∈ H in these local coordinates is given by the linear partL(h) ∈ GL(2, R), R = End(E), A = E × E. The projectivization of L(h) (namely,

18

the coset of L(h) modulo the center of GL(2, R)) belongs to PGL(2,C) and extendsh : A→ A with h(po) = qo to

L(h) : ξ−1(po) ' P1 −→ P1 ' ξ−1(qo).

In the case under consideration, one obtains commutative diagrams (9) and

(Aj, D(j))((B/Γj)′ , T (j)

)

(Aj/Hj, D(j)/Hj)(B/ΓHj , T (j)/Hj

)?

ζHj

�ξj

?

ζ′Hj

�ξHj

with Hj-Galois covers ζHj , ζ ′Hj . Towards the commutativity of (8), note that itcoincides with (6) on (B/Γj)′ \

[ξ−1j D(j)sing

]and B/ΓHj \

[ξHj(D(j)sing/Hj

)]for

1 ≤ j ≤ 2. For an arbitrary (P2, Q2) ∈ D(2)sing one has

ξ−12 (P2, Q2) ξ−1

1 µa(P2, Q2)

ξ−12 (P2, Q2)/H2 = ξ−1

1 µa(P1, Q1)/H1

HHHHHHH

HHHHj

ζ′H2

-µ′a

?

ζ′H1 ,

as far as (ker(µa),+) consists of automorphisms with trivial linear part I2 (i.e., trans-lations) and any h2 ∈ H2 corresponds to ψa(h2) ∈ H1 with the same linear partL(h2) = Lψa(h2). That justifies the commutativity of (8).

We say that a principal isogeny µa : A2 → A1 is non-trivial if

deg(µa) = | ker(µa)| > 1.

Combining Lemma 2, Lemma 1 and Corollary 10, one obtains the following

Corollary 11. Let µan : (An, D(n)) → (An−1, D(n − 1)) be an infinite sequence ofnon-trivial principal isogenies of abelian ball quotient models (An, D(n)) with An =En × En, En = C/(Z + τnZ) and different τn ∈ F(PSL(2,Z)). Then any subgroupH1 of Aut(A1, D(1)) with linear part L(H1) ≤ GL(2,Z) admits an infinite sequenceof extensions

. . . Hnψ−1an...a2

(H1) Hn−1 = ψ−1an−1...a2

(H1) . . .-ψan+1 -

ψan -ψan−1

19

. . . H2 = ψ−1a2

(H1) H1 1-ψa3 -

ψa2 -ψa1

,

such that(B/ΓHn , T (n)/Hn

)≡(B/ΓH1 , T (1)/H1

)coincide for ∀n ∈ N,

ζ ′Hn :((B/Γn)′ , T (n)

)−→

(B/ΓH1 , T (1)/H1

)are finite Galois covers of strictly increasing degrees

deg(ζ ′Hn) = |Hn| = |H1|n∏i=2

| ker(µai)|

and the sequence{(

(B/Γn)′ , T (n))}∞

n=1contains infinitely many mutually non-bira-

tional co-abelian ball quotient surfaces.

3 Infinite series of birational ball quotient modelsLemma 12. Let µa : (A,D(2) = µ−1

a D(1)) → (A,D(1)) with A = E × E, R =End(E), a ∈ R be a principal isogeny of abelian ball quotient models (A,D(j)),1 ≤ j ≤ 2,

ψa : Aut(A,D(2)) −→ Aut(A,D(1)),

ψa(τ(U,V )L(h)) = τµa(U,V )L(h)

be the induced epimorphism of the automorphism groups and H be a subgroup ofAut(A,D(1)).

Then µa is H-equivariant, i.e., µah = hµa for ∀h = τ(U,V )L(h) ∈ H if and only ifµa(U, V ) = (U, V ). If so, then H is a subgroup of ψ−1

a (H),

ψ−1a (H) = ker(µa) hH

is a semi-direct product of H and its normal subgroup (ker(µa),+).Moreover, there is a commutative diagram

(A,D(2)) (A,D(1))

(A/H,D(2)/H) (A/H,D(1)/H)?

ζH

-µa

?

ζH

-µa,H

(10)

with H-Galois cover ζH and a finite, not necessarily Galois, eventually ramified coverµA,H .

20

Proof. If h = τ(U,V )L(h) ∈ H then µah = µaτ(U,V )L(h) = τµa(U,V )aL(h) and hµa =τ(U,V )aL(h), so that µa is H-equivariant exactly when µa(U, V ) = (U, V ) for thetranslation parts (U, V ) of all h ∈ H. If µa(U, V ) = (U, V ) for ∀h = τ(U,V )L(h) ∈ Hthen

ψa(h) = ψa(τ(U,V )L(h)) = τµa(U,V )L(h) = τ(U,V )L(h) = h

for ∀h ∈ H, so that H is contained in ψ−1a (H) and ψ−1

a |H = IdH . In such a way,ψa : ψ−1

a (H) → H is an epimorphism with kernel ker(ψa) = (ker(µa),+), whichrestricts to the identity of H and ψ−1

a (H) = ker(µa) hH is a semi-direct product ofker(µa) and H.

Note that H is a subgroup of Aut(A,D(2)), as far as H is a subgroup of ψ−1a (H)

and ψ−1a (H) is a subgroup of Aut(A,D(2)). Therefore ζH : A→ A/H restricts to an

H-Galois cover ζH : D(2) → D(2)/H. The H-equivariance µah = hµa of µa impliesthat the values of µa on an H-orbit constitute an H-orbit and there is a correctlydefined morphism

µa,H : A/H −→ A/H,

µa,HζH(P,Q) = ζHµa(P,Q),

closing the commutative diagram (10). Since µa and ζH are finite and surjective, µa,His a finite cover.

Definition 13. If Q(√−d) is an imaginary quadratic number field with integers ring

O−d then SU2,1(O−d) is called the full Picard modular group over O−d.We refer to the subgroups Γ of SU2,1(O−d) as Picard modular groups.

The following proposition establishes the existence of Aut(A,D(1))-equivariantprincipal isogenies µa : (A,D(2) = µ−1

a D(1)) → (A,D(1)) for the abelian minimalmodels (A,D(1)) of Picard modular torsion free compactifications

((B/Γ)′ , T (1)

).

Proposition 14. If (A,D(1)) is an abelian minimal model of a Picard modular tor-sion free toroidal compactification

((B/Γ)′ , T (1)

)of a ball quotient B/Γ then there

exist infinitely many integers z ∈ Z \ {0}, such that

µz : (A,D(2) = µ−1z D(1)) −→ (A,D(1))

are Aut(A,D(1))-equivariant principal isogenies.

Proof. Let Γ be a subgroup of SU2,1(O−d) for the integers ring O−d of an imaginaryquadratic number field Q(

√−d). According to [3], the morphism ξ : B/Γ → A

induces an epimorphism ξ∗ : Γ → π1(A) of the corresponding fundamental groups,so that π1(A) = π1(E) × π1(E) is also a Picard modular group over O−d. In otherwords, E has complex multiplication by Q(

√−d). That amounts to a fundamental

group π1(E) = c(Z + τZ) for some τ ∈ Q(√−d), c ∈ C∗ and an endomorphism ring

21

End(E) = R−d,f = Z + fO−d with conductor f ∈ N. Up to an isomorphism of Eand, therefore, of A, one can assume that c = 1.

The Γ-cusps of B/Γ are the Γ-orbits of the Γ-rational boundary points p ∈ ∂ΓB.Note that p ∈ ∂ΓB are defined over Q(

√−d), as far as they are the fixed points

of the parabolic γ ∈ Γ < SU2,1(O−d). Therefore the Γ-cusps and the irreduciblecomponents T (1)i of the toroidal compactifying divisor T (1) = (B/Γ)′ \ (B/Γ) aredefined over Q(

√−d). Since (A,D(1)) is a birational model of

((B/Γ)′ , T (1)

)and

T (1) intersects the exceptional divisor E(ξ) of ξ in a finite set of points, D(1)i aredefined over Q(

√−d). According to [3], one can represent

Di = {(u+ π1(E), v + π1(E)) | biu− aiv ∈ aiπ1(E) + biπ1(E)}

with slope vector (ai, bi) = I(1), where I : Di = C → C2 = A is the unique liftingof Id : Di → A with I(0) = (0, 0). (The morphisms of abelian varieties are affinelinear transformations and the condition I(0) = (0, 0) is equivalent to the linearityof I. ) As a result, the singular locus D(1)sing =

∑1≤i<j≤h

D(1)i ∩ D(1)j is defined

over Q(√−d) and lifts to a discrete subset of Q(

√−d) × Q(

√−d) ⊂ C2 = A. Due

to Q * Q(τ) ⊆ Q(√−d), one has Q(τ) = Q(

√−d), so that [Q(τ) : Q] = 2 and

Q(√−d) = Q(τ) = Q + τQ. Thus, any singular point (Po, Qo) ∈ D(1)sing is of the

form(Po, Qo) =

(p1

q1

+r1

s1

τ + π1(E),p2

q2

+r2

s2

τ + π1(E)

)for some pi, qi, ri, si ∈ Z, qi 6= 0, si 6= 0. If mo is the least common multiple ofq1, s1, q2, s2, then mo(Po, Qo) = (π1(E), π1(E)) = oA or (Po, Qo) ∈ (A,+) is of finiteorder ro ∈ N, dividing mo. If m is the least common multiple of the orders ro of all(Po, Qo) ∈ D(1)sing, then D(1)sing ⊆ Am−tor.

By Lemma 12, µz : (A,D(2) = µ−1z D(1)) → (A,D(1)) with z ∈ Z \ {0} is an

Aut(A,D(1))-equivariant principal isogeny if and only if the translation parts (U, V )of all h = τ(U,V )L(h) ∈ Aut(A,D(1)) are (z − 1)-torsion. If one moves the origin oAof A at a point of D(1)sing, then for any h ∈ Aut(A,D(1)) there holds

(U, V ) = τ(U,V )(oA) = τ(U,V )L(h)(oA) = h(oA) ∈ D(1)sing,

due to Aut(A,D(1))-invariance of D(1)sing =∑

1≤i<j≤hD(1)i ∩ D(1)j. Therefore the

translation parts (U, V ) of all h = τ(U,V )L(h) ∈ Aut(A,D(1)) are m-torsion and forany z ∈ mZ + 1 the principal isogeny µz : (A,D(2) = µ−1

z D(1))n → (A,D(1)) isAut(A,D(1))-equivariant.

Lemma 15. Let (A,D(1)) be an abelian ball quotient model, H be a subgroup ofAut(A,D(1)),

µa : (A,D(2) = µ−1a D(1)) −→ (A,D(1))

22

be an H-equivariant principal isogeny and

ψa : Aut(A,D(2)) −→ Aut(A,D(1),

ψa(τ(U,V )L(h)) = τµa(U,V )L(h)

be the induced epimorphism of the automorphism groups. Then for any point (P,Q) ∈A the stabilizers Stabψ−1

a (H)(P,Q) = StabH(P,Q) in ψ−1a (H) and H coincide.

Proof. For any point (P,Q) ∈ A and any subgroup G of Aut(A,D(2)) one hasStabG(P,Q) = τ(P,Q)StabG(oA)τ(−P,−Q). Namely,

τ(P,Q)StabG(oA)τ(−P,−Q) ⊆ StabG(P,Q)

by τ(P,Q)StabG(oA)τ(−P,−Q)(P,Q) = τ(P,Q)StabG(oA)(oA) = τ(P,Q)(oA) = (P,Q) and

StabG(P,Q) ⊆ τ(P,Q)StabG(oA)τ(−P,−Q)

is equivalent to τ(−P,−Q)StabG(P,Q)τ(P,Q) ⊆ StabG(oA), according to

τ(−P,−Q)StabG(P,Q)τ(P,Q)(oA) = τ(−P,−Q)StabG(P,Q)(P,Q) = τ(−P,−Q)(P,Q) = oA.

If A = E ×E and R = End(E) then StabG(oA) = StabG∩GL(2,R)(oA). More precisely,for any g = τ(U,V )L(g) ∈ StabG(oA) there holds

(U, V ) = (U, V ) + oA = τ(U,V )L(g)(oA) = g(oA) = oA,

so that StabG(oA) ⊂ GL(2, R) and StabG(oA) ⊆ StabG∩GL(2,R)(oA). We claim thatψ−1a (H)∩GL(2, R) = H ∩GL(2, R). Indeed, by Lemma 12, the H-equivariance of µa

implies that H is a subgroup of ψ−1a (H) and ψ−1

a (H) = ker(µa) hH is a semi-directproduct of ker(µa)/ψ

−1a (H) andH. ThereforeH∩GL(2, R) ⊆ ψ−1

a (H)∩GL(2, R). Onthe other hand, any element of ψ−1

a (H) is of the form τ(Po,Qo)h for uniquely determinedτ(Po,Qo) ∈ ker(µa) and h = τ(U,V )L(h) ∈ H. Thus, τ(Po,Qo)h = τ(Po,Qo)+(U,V )L(h) ∈GL(2, R) if and only if (U, V ) = −(Po, Qo) ∈ ker(µa). By Lemma 12, one has

(U, V ) = µa(U, V ) = −µa(Po, Qo) = oA,

whereas τ(Po,Qo)h = h = L(h) ∈ H∩GL(2, R) and ψ−1a (H)∩GL(2, R) ⊆ H∩GL(2, R).

As a result,Stabψ−1

a (H)(P,Q) = τ(P,Q)Stabψ−1a (H)(oA)τ(−P,−Q) =

= τ(P,Q)Stabψ−1a (H)∩GL(2,R)(oA)τ(−P,−Q) = τ(P,Q)StabH∩GL(2,R)(oA)τ(−P,−Q) =

= τ(P,Q)StabH(oA)τ(−P,−Q) = StabH(P,Q).

23

According to Lemma 9, there is a commutative diagram

(A,D(2)) (A,D(1))

(A/ψ−1

a (H), D(2)/ψ−1a (H)

)= (A/H,D(1)/H)

HHHH

HHHHHH

HHHj

ζψ−1a (H)

-µa

?

ζH .

Combining with (10), one obtains

(A,D(2))

(A/H,D(2)/H)(A/ψ−1

a (H), D(2)/ψ−1a (H)

)= (A/H,D(1)/H)

�����

������

������

ζH

?

ζψ−1a (H)

-µa,H

. (11)

Lemma 16. Let µa : (A,D(2))→ (A,D(1)) be a principal isogeny of a split abeliansurface A = E × E with E = C/(Z + τZ), τ ∈ F(PSL(2,Z),

ψa : Aut(A,D(2)) −→ Aut(A,D(1)),

ψa(τ(U,V )L(h)) = τµa(U,V )L(h)

be the induced epimorphism of the automorphism groups, H ≤ Aut(A,D(1)), (P,Q) ∈A and L(P,Q) = LStabH(P,Q) be the linear part of the stabilizer StabH(P,Q) of(P,Q) in H. Then L(P,Q) acts on ker(µa) and the fibre of µa,H over ζψ−1

a (H)(P,Q)admits an isomorphism of sets

Φ(P,Q) : µ−1a,Hζψ−1

a (H)(P,Q) = ∪(Po,Qo)∈ker(µa)ζh(τ(Po,Qo)(P,Q)

)−→ ζL(P,Q) ker(µa),

Φ(P,Q)ζH(τ(Po,Qo)(P,Q)

)= ζL(P,Q)(Po, Qo).

In particular, if µa is unramified over the point (P,Q) then µa,H is unramified overζψ−1

a (H)(P,Q).When the fundamental group π1(E) = O−d coincides with the integers ring of an

imaginary quadratic number field Q(√−d), the finite morphism µa,H is unramified

over ζψ−1a (H) if and only if L(P,Q) is a subgroup of the principal congruence group

GL(2,O−d)(aO−d) = ker[GL(2,O−d) −→ GL(2,O−d/aO−d].

24

Proof. An arbitrary linear transformation g′ ∈ GL(2, R) commutes with the scalarmatrix µa = aI2 ∈ GL(2,C) and acts on ker(µa), according to

µag(Po, Qo) = gµa(Po, Qo) = goA = oA

for ∀(Po, Qo) ∈ ker(µa).The fibre of µa,H will be described along the commutative diagram (11). Namely,

µa,Hζψ−1a (H)(P,Q) = ζHζ

−1H µ−1

a,Hζψ−1a (H)(P,Q) =

= ζ(µa,HζH)−1ζψ−1a (H)(P,Q) = ζHζ

−1

ψ−1a (H)

ζψ−1a (H)(P,Q) = ζHOrbψ−1

a (H)(P,Q),

according to ζHζ−1H (x) = x for ∀x = ζψ−1

a (H)(P,Q). Let

H/StabH(P,Q) = ∪mi=1hiStabH(P,Q) (12)

with h1 = IdA be the coset decomposition of H with respect to the stabilizerStabH(P,Q) of (P,Q). By Lemma 15, one has Stabψ−1

a (H)(P,Q) = StabH(P,Q).According to Lemma 12, ψ−1

a (H) = ker(µa)H = ker(µa)hH is a semi-direct productof its normal subgroup ker(µa) and the subgroup H. Therefore

ψ−1a (H)/Stabψ−1

a (H)(P,Q) = ker(µa)H/StabH(P,Q) = (13)

= ∪(Po,Qo)∈ker(µa) ∪mi=1 τ(Po,Qo)hiStabH(P,Q)

is the coset decomposition of ψ−1a (H) with respect to Stabψ−1

a (H)(P,Q), as far as anyelement of ψ−1

a (H)/Stabψ−1a (H)(P,Q) has a representative τ(Po,Qo)hiStabH(P,Q) with

(Po, Qo) ∈ ker(µa) and the index

[ψ−1a (H) : Stabψ−1

a (H)(P,Q)] = [ker(µa)H : StabH(P,Q)] =

= [ker(µa)H : H][H : StabH(P,Q)] = [ker(µa) : H ∩ ker(µa)]m = | ker(µa)|m.The orbit Orbψ−1

a (H)(P,Q) is isomorphic as a set to

ψ−1a (H)/Stabψ−1

a (H)(P,Q) = ∪(Po,Qo)∈ker(µa) ∪mi=1 τ(Po,Qo)hiStabH(P,Q) '

' ∪(Po,Qo)∈ker(µa) ∪mi=1 τ(Po,Qo)hiStabH(P,Q)(P,Q) =

= ∪(Po,Qo)∈ker(µa) ∪mi=1 τ(Po,Qo)hi(P,Q),

so that the fibre

µa,Hζψ−1a (H)(P,Q) = ζHOrbψ−1

a (H)(P,Q) = ∪(Po,Qo)∈ker(µa) ∪mi=1 ζH(τ(Po,Qo)hi(P,Q).

Since ψ−1a (H) = ker(µa) h H = H i ker(µa) is a semi-direct product of the normal

subgroup ker(µa) and the subgroup H, any τ(Po,Qo)hi has unique representative in theform hiτ(P ′o,Q

′o) for some (Po, Qo) ∈ ker(µa) and

µ−1a,Hζψ−1

a (H)(P,Q) = ∪(Po,Qo)∈ker(µa) ∪mi=1 ζH(hiτ′(Po,Q′o)

(P,Q)) =

25

= ∪(Po,Qo)∈ker(µa)ζH(τ(Po,Qo)(P,Q)).

The mapΦ(P,Q) : µ−1

a,Hζψ−1a (H)(P,Q) −→ ζL(P,Q) ker(µa),

Φ(P,Q)ζH(τ(Po,Qo)(P,Q)) = ζL(P,Q)(Po, Qo)

is correctly defined as far as for ζH(τ(Po,Qo)(P,Q) = ζh(τ(P1,Q1)(P,Q) with (Po, Qo),(P1, Q1) ∈ ker(µa) there exists h ∈ H with hτ(Po,Qo)(P,Q) = τ(P1,Q1)(P,Q). Sinceker(µa) is a normal subgroup of ψ−1

a (H), one has hτ(Po,Qo)h−1 = τ(P ′o,Q

′o) for some

(P ′o, Q′o) ∈ ker(µa) and

τ(P1,Q1)(P,Q) = τ(P ′o,Q′o)h(P,Q).

We claim that

τ−(P1,Q1)+(P ′o,Q′o)h ∈ Stabψ−1

a (H)(P,Q) = StabH(P,Q)

implies h ∈ StabH(P,Q). In the notations from (12), let h = hiho for some ho ∈StabH(P,Q). Then τ−(P1,Q1)+(P ′o,Q

′o)hi ∈ StabH(P,Q) forces (P ′o, Q

′o) = (P1, Q1), ac-

cording to the coset decomposition (13) of ψ−1a (H) modulo the common stabilizer

Stabψ−1a (H)(P,Q) = StabH(P,Q). As a result, h = hiho ∈ StabH(P,Q) and

τ(P1,Q1) = τ(P ′o,Q′o) = hτ(Po,Qo)h

−1 = τL(h)(Po,Qo).

Thus, ζL(P,Q)(P1, Q1) = ζL(P,Q)(L(h)(Po, Qo)) = ζL(P,Q)(Po, Qo) and the map Φ(P,Q)

is bijective, as far as for any ζL(P,Q)(Po, Qo) with (Po, Qo) ∈ ker(µa) there is apoint ζH(τ(Po,Qo)(P,Q)) from the fibre µ−1

a,Hζψ−1a (H)(P,Q) of µa,H over ζψ−1

a (H)(P,Q),so that Φ(P,Q)ζH(τ(Po,Qo)(P,Q)) = ζL(P,Q)(Po, Qo). If ζL(P,Q)(Po, Qo) = ζL(P,Q)(P1, Q1)for some (Po, Qo), (P1, Q1) ∈ ker(µa) then (P1, Q1) = L(h)(Po, Qo) for some h ∈StabH(P,Q) and τ(P1,Q1) = hτ(Po,Qo)h

−1. Consequently,

ζH(τ(P1,Q1)(P,Q)) = ζH(hτ(Po,Qo)h−1(P,Q)) = ζH(τ(Po,Qo)(P,Q)),

due to ζHh(U, V ) = ζH(U, V ) for ∀(U, V ) ∈ A and h−1(P,Q) = (P,Q) for h ∈StabH(P,Q). That justifies the bijectiveness of Φ(P,Q).

For a generic point (P,Q) ∈ A with StabH(P,Q) = {IdA} one has L(P,Q) =LStabH(P,Q) = {IdA = I2}, whereas ζL(P,Q) = Idker(µa). Thus, µ−1

a,Hζψ−1a (H)(P,Q)

is isomorphic to the fibre ker(µa) of µa : A → A and µa,H is unramified overζψ−1

a (H)(P,Q).Suppose that π1(E) = O−d coincides with integers ring O−d of an imaginary

quadratic number field Q(√−d), d ∈ N. Recall that O−d = Z + ω−dZ with

ω−d =

{√−d for −d 6≡ 1(mod4)

1+√−d

2for −d ≡ 1(mod4),

26

and observe that

a−1O−d/O−d = (a−1 +O−d)Z + (a−1ω−d +O−d)Z

is generated by a−1 + O−d and a−1ω−d as a Z-module. Note that O−d ⊂ a−1O−dis equivalent to aO−d ⊂ O−d and implies that a ∈ O−d. The finite cover µa,H isunramified over the point ζψ−1

a (H)(P,Q) exactly when the fibre µ−1a,Hζψ−1

a (H)(P,Q) 'ζL(P,Q) ker(µa) is isomorphic to ker(µa). Equivalently, any L(P,Q)-orbit on ker(µa)consists of a single point or (g− I2)(Po, Qo) = oA for all g ∈ L(P,Q) and all Po, Qo ∈Ea−tor = a−1O−d/O−d. Note that it is sufficient to have (g − I2)(Po, Qo) = oA for allg ∈ L(P,Q) and Po, Qo among the generators of Ea−tor. In particular,

(g − I2)(a−1 +O−d,O−d) = ((g11 − 1)a−1 +O−d, g12a−1 +O−d) = (O−d,O−d) = oA

and

(g − I2)(O−d, a−1 +O−d) = (g21a−1 +O−d, (g22 − 1)a−1 +O−d) = (O−d,O−d) = oA

imply that (g − I2)ij ∈ aO−d for ∀1 ≤ i, j ≤ 2 and g ∈ GL(2,O−d)(aO−d). Straight-forwardly, for any g ∈ GL(2,O−d)(aO−d) one has

(g − I2)(a−1 +O−d,O−d) = oA, (g − I2)(O−d, a−1 +O−d) = oA,

(g − I2)(a−1ω−d +O−d,O−d) = oA, (g − I2)(O−d, a−1ω−d +O−d) = oA,

whereas g(Po, Qo) = (Po, Qo) for all (Po, Qo) ∈ ker(µa).

Corollary 17. Let (A = E × E,D(1)) be an abelian minimal model of a torsionfree ball quotient compactification

((B/Γ1)′ , T (1)

), a ∈ R = End(E) and H be a

subgroup of Aut((B/Γ1)′ , T (1)

), whose translation part TH = {τ(h) | h ∈ H} is

contained in A(a−1)−tor. Then there is an infinite sequence{(

B/ΓH,n, T (n)/H)}∞

n=1

of birational models of(B/ΓH,1 = (B/Γ1)′ /H, T (1)/H

)with finite, not necessarily

Galois, eventually ramified coverings

µ′a,H,n :(B/ΓH,n, T (n)/H

)−→

(B/ΓH,n−1, T (n− 1)/H

).

Proof. It suffices to establish the existence of a finite, not necessarily Galois, even-tually ramified covering

µ′a,H,2 = µ′a,H :(B/ΓH,2, T (2)/H

)−→

(B/ΓH,1, T (1)/H

).

27

More precisely, the commutative square

((B/Γ2) , T (2))((B/Γ1)′ , T (1)

)

(A,D(2)) (A,D(1))?

ξ2

-µ′a

?

ξ1

-µa

with ker(µa)-covers µa, µ′a and blow-ups ξ1, ξ2 arises from Holzapfel’s Proposition 4.1from [1]. The compatibility of the blow-downs of the rational curves with the action ofH ≤ Aut(A,D(j)) = Aut

((B/Γj)′ , T (j)

)is expressed by the commutative diagrams

(A,D(j))((B/Γj)′ , T (j)

)

(A/H,D(j)/H)(B/ΓH,j, T (j)/H

)?

ζH

�ξj

?

ζ′H,j

�ξj,H

for j = 1 and j = 2.Recall from Corollary 10 that µ′a is a ker(µa)-Galois cover. The H-equivariance of

µa implies the H-equivariance of µ′a and the presence of a commutative diagram((B/Γ)′ , T (2)

) ((B/Γ1)′ , T (1)

)

(B/ΓH,2, T (2)/H

) (B/ΓH,1, T (1)/H

)?

ζ′H,2

-µ′a

?

ζ′H,1

-µ′a,H

with finite, not necessarily Galois, eventually ramified cover µ′a,H .There remains to be justified the commutativity of(

B/ΓH,2, T (2)/H) (

B/ΓH,1, T (1)/H)

(A/H,D(2)/H) (A/H,D(1)/H)?

ξ2,H

-µ′a,H

?

ξ1,H

-µa,H

. (14)

The finite covers µa,H and µ′a,H coincide on

A/H \(D(2)sing/H

)= B/ΓH,2 \

(ξ−1

2 D(2)sing/H),

28

as far as ξ2,H |B/ΓH,2\(ξ−12 D(2)sing/H) = IdB/ΓH,2\(ξ−1

2 D(2)sing/H). Recall that ξ1,H shrinks

the smooth rational irreducible components of ξ−11 D(1)sing/H to D(1)sing/H and µa,H

transforms D(2)sing/H into D(1)sing/H, as far as µaD(2)sing = D(1)sing. By its verydefinition,

µ′a,H : B/ΓH,2 −→ B/ΓH,1,

µ′a,Hζ′H,2(p) = ζ ′H,1µ

′a,H(p) for ∀p ∈ (B/Γ2)′ .

Therefore

µ′a,H(ξ−1

2 D(2)sing/H)

= µ′a,Hζ′H,2ξ

−12 D(2)sing = ζ ′H,1µ

′ξ−12 D(2)sing =

= ζ ′H,1ξ−11 D(1)sing = ξ−1

1 D(1)sing/H

and there is a commutative diagram

ξ−12 D(2)sing/H ξ−1

1 D(1)sing/H

D(2)sing/H D(1)sing/H

?

ξ2,H

-µ′a,H

?

ξ1,H

-µa,H

,

which yields (14).

The next proposition counts the number of the irreducible components of anisogeny pull-back of an elliptic curve E(aj, bj) ⊂ A = E × E. To this end, con-sider the morphism

ϕ(aj ,bj) : A = E × E −→ D′j = C/(ajπ1(E) + bjπ1(E)),

ϕ(aj ,bj)(u+ π1(E), v + π1(E)) = bju− ajv + ajπ1(E) + bjπ1(E)

of A onto the elliptic curve D′j = C/(ajπ1(E) + bjπ1(E)). If aj 6= 0 then bju− ajv =ajλ1 + bjλ2 for some u, v ∈ C and λ1, λ2 ∈ π1(E) is equivalent to∣∣∣∣ u = ajt+ λ2

v = bjt− λ1for t =

u− λ2

aj∈ C.

Thus, the central fibre

ϕ(aj ,bj)(oD′j) = {(u+ π1(E), v + π1(E)) | bju− ajv ∈ ajπ1(E) + bjπ1(E)} =

{(ajt+ λ2 + π1(E), bjt− λ1 + π1(E)) = (ajt+ π1(E), bjt+ π1(E)) | t ∈ C} = Dj

coincides with Dj and ϕ(aj ,bj) foliates A by elliptic curves, parallel to Dj.

29

Lemma 18. Let µa : A2 = E2 × E2 → E1 × E1 = A1 be a principal isogeny of splitabelian surfaces, Dj = E1(aj, bj) + (Pj, Qj) be an elliptic curve on A1 and

ϕi : Ai −→ Ei(aj, bj)′

ϕi(u+ π1(Ei), v + π1(Ei)) = bju− ajv + ajπ1(Ei) + bjπ1(Ei)

be the foliations of Ai by elliptic curves, parallel to Ei(aj, bj), 1 ≤ i ≤ 2. Then µainduces an isogeny νa : E2(aj, bj)

′ → E1(aj, bj)′, fitting in a commutative diagram

A2 A1

E2(aj, bj)′ E1(aj, bj)

?

ϕ2

-µa

?

ϕ1

-νa

(15)

and the isogeny pull-back

µ−1a (Dj) =

∑Sij∈ν−1

a ϕ1(Pj ,Qj)

ϕ−12 (Sij). (16)

In particular, µ−1a (Dj) has

kj = [π1(E1(aj, bj)′) : aπ1(E2(aj, bj)

′)] = [ajπ1(E1)+bjπ1(E1) : a(ajπ1(E2)+bjπ1(E2))]

irreducible components.

Proof. By the very definition of ϕi, the fundamental group of Ei(aj, bj)′ is

π1(Ei(aj, bj)) = ajπ1(Ei) + bjπ1(Ei).

Note that µa : A2 → A1 is a principal isogeny if and only if

aπ1(A2) = a(π1(E2)× π1(E2)) = aπ1(E2)× aπ1(E2) ⊆ π1(E1)× π1(E1) = π1(A1),

which is equivalent toaπ1(E2) ⊆ π1(E1).

Therefore

aπ1(E2(aj, bj)′) = a(ajπ1(E2) + bjπ1(E2)) ⊆ ajπ1(E1) + bjπ1(E1) = π1(E1(aj, bj)

′)

and νa : E2(aj, bj)′ → E1(aj, bj)

′ is an isogeny of elliptic curves. Straightforwardly,

ϕ1πa(u+ π1(E2), v + π1(E2)) = ϕ1(au+ π1(E1), av + π1(E1)) =

= bj(au)− aj(av) + ajπ1(E1) + bjπ1(E1) = νa(bju− ajv + ajπ1(E2) + bjπ1(E2) =

30

= νaϕ2(u+ π1(E2), v + π1(E2))

for ∀(u+ π1(E2), v + π1(E2)) ∈ A2, so that νa closes the diagram (15)We claim that

ϕ2µ−1a (Dj) = ν−1

a ϕ1(Pj, Qi). (17)

Indeed, for any

(P,Q) = (aju+ pij + π1(E2), bju+ qij + π1(E2)) ∈ E2(aj, bj) + (Pij, Qij) ⊂

⊂ µ−1a (E1(aj, bj) + (Pj, Qj))

with µa(Pij, Qij) = µa(pij+π1(E2), qij+π1(E2)) = (pj+π1(E2), qj+π1(E1)) = (Pj, Qj),one has

ϕ2(P,Q) = bj(aju+ pij)− aj(bju+ qij) + ajπ1(E2) + bjπ1(E2) =

= bjpij − ajqij + ajπ1(E2) + bjπ1(E2)

withνaϕ2(P,Q) = bj(apij)− aj(aqij) + ajπ1(E1) + bjπ1(E1) =

= bjpj − ajqj + ajπ1(E1) + bjπ1(E1) = ϕ1(Pj, Qj),

so that ϕ2µ−1a (Dj) ⊆ ν−1

a ϕ1(Pj, Qj). For the opposite inclusion ν−1a ϕ1(Pj, Qj) ⊆

ϕ2µ−1a (Dj), note that ϕ2 : A2 → E2(aj, bj)

′ is surjective and pick up a point S =ϕ2(P,Q) ∈ ν−1

a ϕ1(Pj, Qi) with some (P,Q) ∈ A2. By the commutativity of (15),

ϕ1µa(P,Q) = νaϕ2(P,Q) ∈ ϕ1(Pj, Qj).

For the linear map ϕ1, that implies µa(P,Q)−(Pj, Qj) ∈ ker(ϕ1). Note that ker(ϕ1) =E1(aj, bj). Indeed, if (P ′, Q′) = (aju+ π1(E1), bju+ π1(E1)) ∈ E1(aj, bj) then

ϕ1(P ′, Q′) = bj(aju)−aj(bju)+ajπ1(E1)+bjπ1(E1) = ajπ1(E1)+bjπ1(E1) = oE1(aj ,bj)′

or (P ′, Q′) ∈ ker(ϕ1). Conversely, if (P ′, Q′) = (u + π1(E1), v + π1(E1)) ∈ ker(ϕ1)then

oE1(aj ,bj)′ = ϕ1(P ′, Q′) = bju− ajv + ajπ1(E1) + bjπ1(E1) = ajπ1(E1) + bjπ1(E1).

In other words,

bju− ajv = ajλ1 + bjλ2 for some λ1, λ2 ∈ π1(E1).

In the case of aj 6= 0 choose t := u−λ2aj

to express u = ajt + λ2, v = bjt − λ1 andobserve that

(P ′, Q′) = (ajt+ λ2 + π1(E1), bjt− λ1 + π1(E1)) =

= (ajt+ π1(E1), bjt+ π1(E1)) ∈ E1(aj, bj).

31

Similarly, for bj 6= 0 one chooses t := v+λ1bj

to have v = bjt − λ1, u = aju + λ2

and (P ′, Q′) ∈ E1(aj, bj). That justifies ker(ϕ1) = E1(aj, bj). Now, µa(P,Q) −(Pj, Qj) ∈ ker(ϕ1) = E1(aj, bj) is equivalent to µa(P,Q) ∈ E1(aj, bj) + (Pj, Qj) = Dj

and (P,Q) ∈ µ−1a (Dj). As a result, S = ϕ2(P,Q) ∈ ϕ2µ

−1a (Dj) for ∀S ∈ ν−1

a ϕ1(Pj, Qj)and ν−1

a ϕ1(Pj, Qj) ⊆ ϕ2µ−1a (Dj). That concludes the verification of (17) and reduces

(16) toµ−1a (Dj) =

∑Sij∈ϕ2µ

−1a (Dj)

ϕ−12 (Sij).

For the last equality recall that µ−1a (Dj) =

∑i

E2(aj, bj) + (Pij, Qij) with (Pij, Qij) ∈

µ−1a (Pj, Qj) consists of elliptic curves, parallel to E2(aj, bj). Therefore ϕ2 with kernel

ker(ϕ2) = E2(aj, bj) shrinks any irreducible component Dij = E2(aj, bj) + (Pij, Qij)of µ−1

a (Dj) to the point Sij := ϕ2(Pij, Qij). The homomorphism ϕ2 : A2 → E2(aj, bj)′

has fibres ϕ−12 (Sij) = (Pij, Qij) + ker(ϕ2) = (Pij, Qij) + E2(aj, bj) = Dij ⊂ µ−1

a (Dj).Therefore,

mu−1a (Dj) =

∑(Pij ,Qij)∈µ−1

a (Pj ,Qj)

E(aj, bj) + (Pij, Qij) =

=∑

(Pij ,Qij)∈µ−1a (Pj ,Qj)

ϕ−12 ϕ2(Pij, Qij) =

∑Sij=ϕ2(Pij ,Qij)∈ϕ2µ

−1a (Pj ,Qj)

ϕ−12 (Sij).

In particular, the number of the irreducible components of µ−1a (Dj) equals the car-

dinality of ϕ2µ−1a (Pj, Qj) = ν−1

a ϕ1(Pj, Qj). Bearing in mind that the isogeny νa :E2(aj, bj)

′ → E1(aj, bj)′ is a homomorphism of the additive groups, one has

|ν−1a ϕ1(Pj, Qj)| = |ν−1

a oE1(aj ,bj)′ | =∣∣∣∣1aπ1(E1(aj, bj)

′)/π1(E2(aj, bj)′)

∣∣∣∣ =

= [π1(E1(aj, bj)′) : aπ1(E2(aj, bj)

′)] =

[ajπ1(E1) + bjπ1(E1) : a(ajπ1(E2) + bjπ1(E2))] = kj.

The following two corollaries are immediate consequences of Lemma 18:

Corollary 19. Let (A1 = E1 × E1, D(1)) with D(1) =h∑j=1

E1(aj, bj) + (Pj, Qj) be an

abelian minimal model of a torsion free toroidal compactification((B/Γ1)′ , T (1)

), H1

be a subgroup of Aut(A1, D(1)) with linear part L(H1) ⊆ GL(2,Z) and

µcn : (An, D(n)) −→ (An−1, D(n− 1))

be an infinite sequence of principal isogenies of abelian minimal models (An, D(n))of torsion free ball quotient compactification, supported by An = En × En, En =

32

C/(Z + τnZ) with different τn ∈ F(PSL(2,Z)). Then(B/ΓH1 , T (1)/H1

)admits an

infinite sequence of co-abelian, finite Galois covers((B/Γn)′ , T (n)

)with

h∑j=1

(n∏i=2

[ajπ1(Ei−1) + bjπ1(Ei−1) : ci(ajπ1(Ei) + bjπ1(Ei))]

)irreducible components and

|D(1)sing|n∏i=2

[π1(Ei−1) : ciπ1(Ei)]2

singular points.

Corollary 20. Let (A = E × E,D(1)) with D(1) =h∑j=1

E(aj, bj) + (Pj, Qj) be an

abelian minimal model of a torsion free toroidal compactification((B/Γ1)′ , T (1)

),

c ∈ R = End(E) and H be a subgroup of Aut(A,D(1)) with

TH = {τ(h) | h ∈ H ⊆ A(a−1)−tor}.

Then(B/ΓH , T (1)/H

)admits an infinite isogeny series

µ′c :((B/Γn)′ , T (n)

)−→

((B/Γn−1)′ , T (n− 1)

)of co-abelian, finite H-Galois covers

((B/Γn)′ , T (n)

)with

n∑j=1

[ajπ1(E) + bjπ1(E) : c(ajπ1(E) + bjπ1(E)]n−1

irreducible components and

|D(1)sing|[π1(E) : cπ1(E)]2(n−1)

singular points.

References[1] Holzapfel R.-P., Complex Hyperbolic Surfaces of Abelian Type, Serdica Math.

Jour. 30 (2004) 207–238.

[2] Kasparian A., Galois groups of co-abelian ball-quotient covers, Preprint.

[3] Kasparian A., Elliptic configurations, Preprint.

[4] Uludag M., Covering relations between ball-quotient orbifolds, Math. Annalen308 (2004), 503-523.

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