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arXiv:2006.15499v3 [math.AG] 9 Nov 2020
ON pq-FOLD REGULAR COVERS
OF THE PROJECTIVE LINE
SEBASTI ´
AN REYES-CAROCCA
Abstract. Let pand qbe odd prime numbers. In this paper we study non-abelian
pq-fold regular covers of the projective line, determine algebraic models for some
special cases and provide a general isogeny decomposition of the corresponding Ja-
cobian varieties. We also give a classification and description of the one-dimensional
families of compact Riemann surfaces as before.
1. Introduction and statement of the results
Compact Riemann surfaces (or, equivalently, smooth complex projective algebraic
curves) and their automorphism groups have been extensively studied since the nine-
teenth century. Foundational results concerning that are:
(1) if the genus of the compact Riemann surface is greater than one then its
automorphism group is finite (see [21] and [42], and also [14]), and
(2) each finite group acts as a group of automorphisms of some compact Riemann
surface of a suitable genus greater than one (see [17] and also [26]).
A general problem that arises naturally with regard to this is to determine necessary
and sufficient conditions under which a given group acts as a group of automorphisms
of a compact Riemann surface satisfying some prescribed conditions. This problem
was successfully studied for cyclic groups by Harvey in [18] and soon after for abelian
groups by Maclachlan in [29]. The same problem for dihedral groups, among other
aspects, was completely solved by Bujalance, Cirre, Gamboa and Gromadzki in [7].
See also [49].
This article is devoted to study those compact Riemann surfaces that are branched
pq-fold regular covers of the projective line, where pand qare prime numbers. Since
the abelian and dihedral cases have been already classified, we shall consider compact
Riemann surfaces endowed with a non-abelian group of automorphisms isomorphic
to the semidirect product of two cyclic groups of odd prime order, in such a way that
the corresponding orbit space is isomorphic to the projective line.
Let pand qbe odd primes such that pdivides q−1 and let rbe a primitive p-th
root of unity in the field of qelements. Throughout the article the unique non-abelian
2010 Mathematics Subject Classification. 30F10, 14H37, 30F35, 14H40.
Key words and phrases. Compact Riemann surfaces, group actions, automorphisms, Jacobians.
Partially supported by Fondecyt Grants 11180024, 1190991 and Redes Grant 2017-170071.
2 SEBASTI ´
AN REYES-CAROCCA
group of order pq will be denote by
Gp,q := ha, b :aq=bp= 1, bab−1=ari∼
=Cq⋊Cp.
The first result of this paper establishes a simple necessary and sufficient condition
for Gp,q to act on a compact Riemann surface of genus greater than one. In order to
state it, we need to bring in the concept of signature. The tuple
(γ;k1,...,kl)∈Zl+1 where γ>0 and ki>2
is called the signature of the action of a group Gon a compact Riemann surface Sif
the genus of the orbit space S/G is γand the branched regular covering map
S→S/G
ramifies over exactly lvalues y1,...,yland the fiber over yiconsists of points with
G-isotropy group of order kifor each i∈ {1,...,l}.Note that kidivides |G|.
Theorem 1. Let n, m >0be integers such that n+m>3. There exists a compact
Riemann surface Sof genus greater than one endowed with a group of automorphisms
isomorphic to Gp,q acting on it with signature
sn,m := (0; p, n
. . ., p, q, m
...,q)
if and only if n>2.In this case, the genus of Sis g= 1 −pq +nq(p−1
2) + mp(q−1
2).
Let Mgdenote the moduli space of compact Riemann surfaces of genus g>2.
For each pair of integers n, m as in Theorem 1, the set of compact Riemann surfaces
admitting a group of automorphisms isomorphic to Gp,q acting on them with signature
sn,m form a family of complex dimension n+m−3 in the singular locus of Mg(see
Subsection §2.4 for a precise definition of family).
Notation. We shall denote the above introduced family by Cn,m.
The zero-dimensional case. A compact Riemann surface is called quasiplatonic if it
has a group of automorphisms such the signature of the action is of the form
(0; k1, k2, k3).
Observe that among the compact Riemann surfaces of Theorem 1, the quasiplatonic
ones split into two cases: those with signature
s2,1= (0; p, p, q) and s3,0= (0; p, p, p).
These compact Riemann surfaces form two zero-dimensional families; namely, there
are, up to isomorphism, finitely many of them. Streit and Wolfart in [44] succeeded
in describing these Riemann surfaces in full detail. More precisely, they determined
algebraic models, the number of isomorphism classes, their automorphism groups and
their minimal fields of definition, among other aspects.
Given a compact Riemann surface Swith a group of automorphisms G, a natural
question that arises is to decide whether or not Sadmits more automorphisms. This
ON pq-FOLD REGULAR COVERS OF THE PROJECTIVE LINE 3
is a challenging problem and its answer depends not only on the signature sof the
action of G, but also on the geometry of a fundamental domain for the surface S.
Singerman in [43] determined all those possible signatures sfor which the Riemann
surface might have more automorphisms (see Subsection §2.2). A direct consequence
of his results is that if the complex dimension of the family Cn,m is greater than one,
then Gp,q is the full automorphism group of the surfaces lying in the interior of Cn,m.
The one-dimensional case will be considered later in this paper.
We recall the obvious fact that Gp,q has a unique normal subgroup Nof order
q, and qpairwise conjugate subgroups of order p; let Hbe one of them. All along
the paper Nand Hwill be used to denote these subgroups. For each S∈Cn,m we
consider the regular covering maps
S→X:= S/N and S→Y:= S/H
given, respectively, by the action of Nand Hon S. As an application of the Riemann-
Hurwitz formula, we see that the genera of Xand Yare
gX=p−1
2(n−2) and gY=q−1
2(m−2) + p−1
2
q−1
pn.
Although the literature still shows few general results in this direction, there is
a great interest in providing explicit descriptions of compact Riemann surfaces as
algebraic curves. See, for instance, [5], [6], [8], [12], [30], [34], [35], [45] and [46].
The following result provides an algebraic description for each compact Riemann
surface Slying in the family C2,m for m>1.This description is given in terms of a
singular plane algebraic curve whose projective desingularization is isomorphic to S.
Theorem 2. Let n, m >0be integers such that n+m>3.
(1) Let Sbelong to the interior of the family Cn,m .Then Sis cyclic ˆq-gonal for
some odd prime ˆqif and only if n= 2 and ˆq=q.
(2) An affine singular algebraic model for S∈C2,m in C2is
yq= Πp−1
i=0 (x−ωi)riΠm
k=2Πp−1
i=0 (x−λkωi)ri
where ωis a primitive p-th root of unity and λ2,...,λmare non-zero complex
numbers such that λp
k6= 1 for each kand λk/∈ {λjωi:i∈Zp}for all j6=k.
(3) In the previous model, the group of automorphisms of Sisomorphic to Gp,q is
generated by
A(x, y) = (x, ξy)and B(x, y) = (ωx, ϕ(x)yr)
where ξis a primitive q-th root of unity,
ϕ(x) = ωme[(x−ωp−1)Πm
k=2(x−λkωp−1)]e(1−r)
and e∈Zis chosen to satisfy 1 + r+···+rp−1=eq.
4 SEBASTI ´
AN REYES-CAROCCA
The one-dimensional case. Observe that, by Theorem 1, there exist three complex
one-dimensional families of compact Riemann surfaces that are non-abelian pq-fold
branched regular covers of the projective line. More precisely, these families are:
family genus signature
C2,2(p−1)(q−1) s2,2= (0; p, p, q, q)
C4,01 + q(p−2) s4,0= (0; p, p, p, p)
C3,11 + pq −1
2(p+ 3q)s3,1= (0; p, p, p, q)
The following three theorems provide a description of these families. More precisely,
we give upper bounds for the number of equisymmetric strata they consists of (see
Subsection §2.4 for a precise definition of equisymmetric stratum), determine their
automorphism groups and show that the complement of the interior in the closure of
each of them is non-empty.
Theorem 3.
(1) The family C2,2consists of at most q(p−1) equisymmetric strata.
(2) If Sbelongs to the interior of C2,2then the automorphism group of Sis iso-
morphic to Gp,q .
(3) The closure of the family C2,2contains a quasiplatonic Riemann surface with
a group of automorphisms of order 2pq isomorphic to
Cq⋊C2pacting with signature (0; q, 2p, 2p).
Theorem 4.
(1) The family C4,0consists of at most q(p−1)(p2−3p+ 3) equisymmetric strata.
(2) If Sbelongs to the interior of C4,0then either the automorphism group of S
is isomorphic to Gp,q or to Cq⋊C2pacting on it with signature (0; 2,2, p, p).
(3) The closure of the family C4,0contains a quasiplatonic Riemann surface with
a group of automorphisms of order 4pq isomorphic to
(Cq⋊C2p)⋊C2acting with signature (0; 2,2p, 2p).
(4) The compact Riemann surface X=S/N is hyperelliptic of genus p−1,and
an affine singular algebraic model for it in C2is
y2= (xp−1)(xp−µp)
where µis a non-zero complex number such that µp6= 1.
Theorem 5.
(1) The family C3,1consists of at most q(p−1)(p−2) equisymmetric strata.
ON pq-FOLD REGULAR COVERS OF THE PROJECTIVE LINE 5
(2) If Sbelongs to the interior of C3,1then the automorphism group of Sis iso-
morphic to Gp,q .
(3) The closure of the family C3,1contains a quasiplatonic Riemann surface with
a group of automorphisms of order 2pq isomorphic to
(Cq⋊Cp)×C2acting with signature (0; p, 2q, 2q).
Remarks.
(1) If Sbelongs to the interior of Cn,m then Sis non-hyperelliptic.
(2) The family C4,0with p= 3 was studied in [24]; see also [23].
(3) The algebraic description of S∈C2,m extends the ones in [44]; see also [9].
Jacobian varieties. Let Sbe a compact Riemann surface of genus g. We denote by
H1(S, C)∗and H1(S, Z)
the dual of the complex vector space of dimension gof its 1-forms and its first integral
homology group, respectively. The Jacobian variety of S, defined as the quotient
JS =H1(S, C)∗/H1(S, Z),
is an irreducible principally polarized abelian variety of dimension g. The importance
of the Jacobian variety of a compact Riemann surface lies, partially, in the classical
Torelli’s theorem, which ensures that, up to isomorphism, the Riemann surface is
uniquely determined by its Jacobian variety. Namely,
S1∼
=S2if and only if JS1∼
=JS2.
Is is known that the action of a group Gon Sinduces a isogeny decomposition
JS ∼Be1
1× · · · × Bes
s
in terms of abelian subvarieties B1,...,Bsof JS in such a way that
Ai=Bei
ifor i= 1,...,s
are pairwise non-G-isogenous.
It is worth emphasizing that whereas this decomposition only depends on the alge-
braic structure of the group, the dimension of the factors Bidoes depend on the way
the group acts. More precisely, it depends on the signature and, in addition, on the
equisymmetric stratum to which Sbelongs. An example of a family exhibiting in an
explicit manner this dependence can be found in [25, Remark 2(6)].
Theorem 6. Let Sbe a compact Riemann surface lying in the family Cn,m.Then the
Jacobian variety JS of Sdecomposes, up to isogeny, as the product
JS ∼B1×Bp
2
where B1and B2are abelian subvarieties of JS of dimension
p−1
2(n−2) and q−1
2(m−2) + p−1
2
q−1
pn.
6 SEBASTI ´
AN REYES-CAROCCA
We point out that in the previous theorem, the dimension of the factors B1and B2
only depends on the signature of the action.
Let Kbe a group of automorphisms of a compact Riemann surface Sand let
S→R=S/K (1.1)
be the associated regular covering map. Assume the genus of Rto be γ>1.Then
(1.1) induces a homomorphism between the associated Jacobian varieties
JR →JS
whose image is an abelian subvariety of JS of dimension γwhich is isogenous to JR.
Keeping the same notations as before, we observe that
gX= dim(JX) = dim(B1) and gY= dim(JY ) = dim(B2).
As a matter of fact, the following result holds.
Theorem 7. Let Sbe a compact Riemann surface lying in the family Cn,m.With the
notations of Theorem 6, there exist isogenies
JX →B1and JY →B2
and, in particular, JS is isogenous to the product of Jacobians of quotients of S
JS ∼JX ×JY p
This article is organized as follows. Section §2will be devoted to briefly review the
basic background: Fuchsian groups and group actions on compact Riemann surfaces
and Jacobian varieties. The proof of the theorems are given in the remaining sections.
2. Preliminaries
2.1. Fuchsian groups. AFuchsian group is a discrete group of automorphisms of
H={z∈C: Im(z)>0}.
If ∆ is a Fuchsian group and the orbit space H/∆ given by the action of ∆ on H
is compact, then the algebraic structure of ∆ is determined by its signature:
s(∆) = (γ;k1,...,kl),(2.1)
where the genus of H/∆ is γand k1,...,klare the branch indices in the universal
canonical projection H→H/∆.
If ∆ is a Fuchsian group of signature (2.1) then ∆ has a canonical presentation in
terms of generators α1,...,αγ,β1,...,βγ, x1,...,xland relations
xk1
1=···=xkl
l= Πγ
i=1[αi, βi]Πl
i=1xi= 1,(2.2)
where the brackets stands for the commutator, and the hyperbolic area of each fun-
damental region of ∆ is given by
µ(∆) = 2π[2γ−2 + Σl
i=1(1 −1
ki)].
ON pq-FOLD REGULAR COVERS OF THE PROJECTIVE LINE 7
Let ∆′be a group of automorphisms of H.If ∆ is a subgroup of ∆′of finite index,
then ∆′is a Fuchsian group and their hyperbolic areas are related by the Riemann-
Hurwitz formula
µ(∆) = [∆′: ∆] ·µ(∆′).
The Teichm¨uller space of ∆ is a complex analytic manifold homeomorphic to the
complex ball of dimension 3γ−3 + l. See, for instance, [43] for further details.
2.2. Group actions on Riemann surfaces. Let Sbe a compact Riemann surface
of genus g>2 and let Aut(S) denote its automorphism group. A finite group G
acts on Sif there is a group monomorphism G→Aut(S).The orbit space S/G of
the action of Gon Sis naturally endowed with a compact Riemann surface structure
such that the canonical projection S→S/G is holomorphic.
By the classical uniformization theorem, there is a unique, up to conjugation, Fuch-
sian group Γ of signature (g;−) such that S∼
=H/Γ.Moreover, Gacts on Sif and
only if there is a Fuchsian group ∆ together with a group epimorphism
θ: ∆ →Gsuch that ker(θ) = Γ.
The action is said to be represented by the surface-kernel epimorphism θ; hence-
forth, we write ske for short. It is said that Gacts on Swith signature s(∆).Note
that this definition agrees with the one given in the introduction.
Assume that G′is a finite group such that G6G′.Then the action of Gon S
represented by the ske θis said to extend to an action of G′on Sif:
(1) there is a Fuchsian group ∆′containing ∆,
(2) the Teichm¨uller spaces of ∆ and ∆′have the same dimension, and
(3) there exists a ske
Θ : ∆′→G′in such a way that Θ|∆=θand ker(θ) = ker(Θ).
Maximal actions are those that cannot be extended in the previous sense. A com-
plete list of pairs of signatures of Fuchsian groups ∆ and ∆′for which it may be
possible to have an extension as before was provided by Singerman in [43].
2.3. Equivalence of actions. Let Hom+(S) denote the group of orientation pre-
serving self-homeomorphisms of S. Two actions ψi:G→Aut(S) of Gon Swith
i= 1,2 are topologically equivalent if there exist ω∈Aut(G) and f∈Hom+(S) such
that
ψ2(g) = fψ1(ω(g))f−1for all g∈G. (2.3)
Each fsatisfying (2.3) yields an automorphism f∗of ∆ where H/∆∼
=S/G. If B
is the subgroup of Aut(∆) consisting of them, then Aut(G)×Bacts on the set of
skes defining actions of Gon Swith signature s(∆) by
((ω, f ∗), θ)7→ ω◦θ◦(f∗)−1.
8 SEBASTI ´
AN REYES-CAROCCA
Two skes θ1, θ2: ∆ →Gdefine topologically equivalent actions if and only if they
belong to the same (Aut(G)×B)-orbit (see [3], [19] and [28]). If the genus of S/G is
zero then Bis generated by the braid transformations Φi, for 1 6i < l, defined by
xi7→ xi+1, xi+1 7→ x−1
i+1xixi+1 and xj7→ xjwhen j6=i, i + 1.
2.4. Stratification of the moduli space and families. We denote the Teichm¨uller
space of a Fuchsian group of signature (g;−) by Tg. It is well-known that the moduli
space Mgarises as the quotient space
π:Tg→Mg:= Tg/Modg
given by the action of the mapping class group Modgof genus gon Tg.Observe that
Mgis endowed with the quotient topology induced by π. Moreover, Mgis endowed
with a structure of complex analytic space of dimension 3g−3,and for g>4 its
singular locus agrees with the branch locus of πand correspond to set of points
representing compact Riemann surfaces with non-trivial automorphisms. See, for
instance, [4, Section 2] for further details.
Following [4, Theorem 2.1], the singular locus of Mgadmits an equisymmetric
stratification
Sing(Mg) = ∪G,θ ¯
MG,θ
g
where each equisymmetric stratum MG,θ
g, if nonempty, corresponds to one topological
class of maximal actions (see also [19]). More precisely:
(1) MG,θ
gconsists of surfaces of genus gwith automorphism group isomorphic to
Gsuch that the action is topologically equivalent to θ,
(2) the closure ¯
MG,θ
gof MG,θ
gis a closed irreducible algebraic subvariety of Mg
and consists of surfaces of genus gwith a group of automorphisms isomorphic
to Gsuch that the action is topologically equivalent to θ, and
(3) if MG,θ
gis nonempty then it is a smooth, connected, locally closed algebraic
subvariety of Mgwhich is Zariski dense in ¯
MG,θ
g.
The aforementioned stratification was the key ingredient in finding all those values
of gfor which the singular locus of Mgis connected; see, for instance, [1] and [2].
As mentioned in the introduction, we shall employ the following terminology.
Definition. Let Gbe a group and let sbe a signature. The subset of Mgof all those
compact Riemann Ssurfaces of genus gwith a group of automorphisms isomorphic
to Gacting with signature swill be called a closed family or simply a family.
Note that the interior of the family consists of those Riemann surfaces whose au-
tomorphism group is isomorphic to Gand is formed by finitely many equisymmetric
strata which are in correspondence with the pairwise non-equivalent topological ac-
tions of G. In addition, the closure of the family is formed by those surfaces whose
automorphism group contains G. If the signature of the action of Gon Sis (2.1)
then the dimension of the family is 3γ−3 + l.
ON pq-FOLD REGULAR COVERS OF THE PROJECTIVE LINE 9
2.5. Decomposition of Jacobians. It is well-known that if Gacts on a compact
Riemann surface Sthen this action induces a Q-algebra homomorphism
Φ : Q[G]→EndQ(JS) = End(JS)⊗ZQ,
from the rational group algebra of Gto the rational endomorphism algebra of J S.
For each α∈Q[G] we define the abelian subvariety
Aα:= Im(α) = Φ(nα)(S)⊂JS
where nis some positive integer chosen in such a way that nα ∈Z[G].
Let W1,...,Wsbe the rational irreducible representations of G. For each Wjwe
denote by Vja complex irreducible representation of Gassociated to it. As proved in
[27, Theorem 2.2] (see also [11, Section 5]) the decomposition 1 = e1+···+es,where
ej∈Q[G] is a central idempotent computed explicitly from Wj, yields an isogeny
JS ∼Ae1× · · · × Aes
which is G-equivariant. Moreover, there are idempotents fj1,...,fjnjsuch that
ej=fj1+···+fj njwhere nj=dj/sj
is the quotient of the degree djand the Schur index sjof Vj. If Bj=Afj1then we
have the isogeny decomposition
JS ∼Bn1
1× · · · × Bns
s
called the group algebra decomposition of JS with respect to G. See also [36].
Assume that (2.1) is the signature of the action of Gon Sand that this action is
represented by θ: ∆ →G, with ∆ as in (2.2). If W1is the trivial representation,
then dim(B1) = γand n1= 1.If 2 6j6sthen, following [38, Theorem 5.12]
dim(Bj) = cj[dj(γ−1) + 1
2Σl
i=1(dj−dhθ(xi)i
j)] (2.4)
where cjis the degree of the extension Q≤Ljwith Ljdenoting a minimal field of
definition for Vj.
The problem of decomposing Jacobian varieties is old and goes back to Wirtinger
[47] and Schottky and Jung [40]. For recent works concerning that for special classes
of groups we refer to the articles [10], [13], [15], [20], [22], [31], [32], [33], [37] and [39].
3. Proof of Theorem 1
Let n, m >0 be integers such that n+m>3 and let ∆ be a Fuchsian group of
signature sn,m = (0; p, n
. . ., p, q, m
. . ., q) with canonical presentation
hx1,...,xn, y1, . . . , ym:xp
1=···=xp
n=yq
1=···=yq
m= Πn
i=1xiΠm
j=1yj= 1i.
Assume the existence of a compact Riemann surface Swith a group of automor-
phisms isomorphic to Gp,q acting on it with signature sn,m.Then there exists a ske
θ: ∆ →Gp,q such that S∼
=H/ker(θ).
10 SEBASTI ´
AN REYES-CAROCCA
Observe that there is no homomorphism ∆ →Gp,q provided that n= 1,since
otherwise the image of x1y1···ymwould not be trivial. In addition, every homomor-
phism ∆ →Gp,q with n= 0 is non-surjective, since bdoes not belong to its image.
Hence, if there exists a compact Riemann surface Sas before then n>2.
Conversely, for each integer n>2 we shall construct a ske θ: ∆ →Gp,q explicitly.
For the sake of simplicity, we shall identify θwith the tuple
θ= (θ(x1),...,θ(xn), θ(y1),...,θ(ym))
and write (x, y, t
. . ., x, y) to denote the 2t-uple with xin the i-th entry for iodd, and
yin the i-th entry for ieven.
Assume m= 0.If nis even then consider
(b, b−1,t
. . ., b, b−1, ab, (ab)−1)
where t=n
2−1.If nis odd and n>5 then consider
(b, b−1,t
. . ., b, b−1, ab, (ab)−1, b2, b−1, b−1)
where t=n−5
2.If n= 3 consider (ab, b, (ab2)−1).
Assume m= 1.If nis even then consider
(b, b−1,t
. . ., b, b−1, b, (ab)−1, a)
where t=n
2−1.If nis odd then consider
(b, b−1,t
. . ., b, b−1, b2, b−1,(ab)−1, a)
where t=n−3
2.
Assume m>2.If nand mare even then consider
(b, b−1,t1
. . ., b, b−1, a, a−1,t2
. . ., a, a−1)
where t1=n
2and t2=m
2.If nand mare odd then consider
(b, b−1,t1
. . ., b, b−1,(b2, b−1, b−1), a, a−1,t2
. . ., a, a−1,(a2, a−1, a−1))
where t1=n−3
2and t2=m−3
2.If nis even and mis odd then consider
(b, b−1,t1
. . ., b, b−1, a, a−1,t2
. . ., a, a−1,(a2, a−1, a−1))
where t1=n
2and t2=m−3
2.Finally, if nis odd and mis even then consider
(b, b−1,t1
. . ., b, b−1,(b2, b−1, b−1), a, a−1,t2
. . ., a, a−1)
where t1=n−3
2and t2=m
2.
The value of gis computed as an application of the Riemann-Hurwitz formula to
the branched regular covering map
H/ker(θ)→H/∆
induced by the inclusion ker(θ)⊳∆.
ON pq-FOLD REGULAR COVERS OF THE PROJECTIVE LINE 11
4. Proof of Theorem 2
Proof of statement (1). The sufficient condition is obvious. Assume that Sadmits
a cyclic ˆq-gonal morphism for some odd prime number ˆq. If the automorphism group
of Sis isomorphic to Gp,q then ˆpequals por q. The former case is not possible since the
genus of Yis always positive. Thus, ˆq=qand necessarily n= 2.If the automorphism
group of Shas order strictly greater than pq then, by [43, Theorem 1], the pair (n, m)
might only belong to
{(2,1),(3,0),(2,2),(4,0)}.
However, for the first two cases the automorphism group of Sis isomorphic to either
Gp,q or Cq⋊C2pas proved in Theorems 1 and 2 in [44], and the same fact holds for
the third and fourth ones as it will be proved later in our Theorems 3and 4.
Proof of statement (2). Let Sbe a compact Riemann surface lying in the family
C2,m with m>1.The q-gonal morphism S→X∼
=P1given by the action of N=hai
on Sramifies over pm values, all of them marked with q. If we denote these values by
ui,k for i∈ {0,...,p−1}and k∈ {1,...,m}
and assume that none of them equal ∞then, following [16] (see also [19] and [48]),
the affine singular algebraic curve
yq= Πp−1
i=0 Πm
k=1(x−ui,k )ni,k
is (after normalization) isomorphic to S, for suitable values 1 6ni,k 6q−1 in such
a way that their sum is congruent to 0 modulo q. Note that
P=Gp,q/N ∼
=Cp=hβ:βp= 1i
acts on Xwith signature (0; p, p).In other words, the action of Pon Xhas two fixed
points and morbits of length p. It follows that, up to a M¨obius transformation, we
can assume that the fixed points of βare 0 and ∞and that the orbits of length pare
{ui,1=ωi:i∈Zp}and {ui,k =λkωi:i∈Zp}for 2 6k6m,
where λ2,...,λmare non-zero complex numbers as in the statement of the theorem
(note that the conditions imposed on them guarantee that the orbits are disjoint).
Thus, after replacing ωby an appropriate power of it and after replacing λkby ωukλk
for some ukif necessary, we conclude that Sis isomorphic to the normalization of
yq= Πp−1
i=0 (x−ωi)riΠm
k=2Πp−1
i=0 (x−λkωi)ri(4.1)
Proof of statement (3). Observe that with the previous identification, the action
of Pon Xis then given by z7→ β(z) = ωz and the regular covering map X→X/P
is given by z7→ zp.Thus, after identifying
S/Gp,q ∼
=X/P ∼
=P1,
12 SEBASTI ´
AN REYES-CAROCCA
the branch values of S→S/Gp,q are ∞,0 marked with pand 1, λpmarked with q.
Note that the lift to Sof βhas the form
(x, y)7→ (ωx, ψ(x, y))
where ψ(x, y)q=f(ωx) with f(x) denoting the right side part of (4.1). In fact
ψ(x, y) = ϕ(x)yr
with ϕas in the theorem. Finally, it is a direct computation to verify that
A(x, y) = (x, ξy) and B(x, y) = (ωx, ϕ(x)yr)
satisfy BAB−1=Arand therefore hA, Bi∼
=Gp,q as claimed.
5. Proof of Theorem 3
Proof of statement (1). Consider a Fuchsian group ∆ of signature s2,2
∆ = hx1, x2, y1, y2:xp
1=xp
2=yq
1=yq
2=x1x2y1y2= 1i
and let θ: ∆ →Gp,q be a ske representing an action of Gp,q on S∈C2,2. Then
θ(x1) = al1bn1and θ(x2) = al2bn2
for some l1, l2∈Zqand n1, n2∈Z∗
p.After a suitable conjugation, we can suppose
l2= 0.Moreover, after applying an automorphism of Gof the form a7→ aiand b7→ b
for some i∈Z∗
q,we can assume that θis given by
θ(x1) = al1bn1, θ(x2) = bn2, θ(y1) = al3and θ(y2) = a.
The fact that x1x2y1y2= 1 implies that n2=−n1and l3=−l1−1,and therefore θ
is equivalent to the ske θl,n given by
x17→ albn, x27→ b−n, y17→ a−l−1and y27→ a,
for l∈Zqand n∈Z∗
p.Thus, there are at most q(p−1) pairwise non-equivalent skes.
Proof of statement (2). As a consequence of [43, Theorem 1], the action of Gp,q
on Smight be only extended to an action of a group of order 2pq with signature
(0; 2,2, p, q). We claim that such extension is not possible. Indeed, if G′is a group of
order 2pq such that Gp,q 6G′then, by the Schur-Zassenhaus theorem, we have that
G′=Gp,q ⋊C2where C2=ht:t2= 1i.
As haicontains all the elements of Gp,q of order qand as thas order two, we observe
that tat−1=aǫwhere ǫ=±1. We write tbt−1=ambnfor m∈Zqand n∈Z∗
pand
observe that the fact that
b=t2bt−2=amǫ(ambn)n(5.1)
implies that n=±1.
(1) Assume ǫ= 1.The equality (5.1) shows that m= 0.Observe that if n=−1
then bab−1=arimplies b−1ab =ar; a contradiction. Consequently n= 1 and
G′=Gp,q ×C2.Note the tis the unique involution of the group.
ON pq-FOLD REGULAR COVERS OF THE PROJECTIVE LINE 13
(2) Assume ǫ=−1.If n=−1 then equality (5.1) shows that m= 0,and
bab−1=arimplies b−1a−1b=a−r; a contradiction. It follows that n= 1.We
write b′:= aubwhere 2u≡mmod qand notice that tb′t−1=b′.Thus, we can
assume m= 0 and the involutions of the group are takwhere k∈Zq.
The contradiction is obtained after noticing that if there were a ske from a Fuchsian
group of signature (0; 2,2, p, q) onto G′then the group would contain an element of
order qand two involutions whose product has order p; however, in both cases, this
is not possible.
Proof of statement (3). Let ∆′be a Fuchsian group of signature (0; q, 2p, 2p)
∆′=hz1, z2, z3:zq
1=z2p
2=z2p
3=z1z2z3= 1i
and consider the group G′of order 2pq
G′=ha, c :aq=c2p= 1, cac−1=a−ri∼
=Cq⋊C2p.
Then the map Θ : ∆′→G′given by
z17→ a, z27→ cand z37→ (ac)−1
is a ske and then the orbit space Z=H/ker(Θ) is a compact Riemann surface with
a group of automorphisms isomorphic to G′acting on it with signature (0; q, 2p, 2p).
We now proceed to prove that Z∈C2,2.Define
ˆx1:= z−2
3,ˆx2:= z1z2
3z−1
1,ˆy1:= z1and ˆy2:= z−2
3z−1
1z2
3
and observe that ˆx1,ˆx2have order p, ˆy1,ˆy2have order qand the product of them
equals 1. Note that, by letting b=c2,we have that
Θ(ˆx1) = a1−rb, Θ(ˆx2) = a1+(r−2)r−1b−1,Θ(ˆy1) = aand Θ(ˆy2) = a−r
and therefore
Θ|hˆx1,ˆx2,ˆy1,ˆy2i:hˆx1,ˆx2,ˆy1,ˆy2i∼
=∆→ ha, bi
is equivalent to one of the skes θl,n as before. Consequently, Z∈C2,2.
6. Proof of Theorem 4
Proof of statement (1). Consider a Fuchsian group ∆ of signature s0,4
∆ = hx1, x2, x4, x4:xp
1=xp
2=xp
3=xp
4=x1x2x3x4= 1i
and let θ: ∆ →Gp,q be a ske representing an action of Gp,q on S∈C4,0. Then
θ(xi) = alibnifor i= 1,2,3,4
where li∈Zqand ni∈Z∗
p.After a suitable conjugation, we can suppose l1= 0.
In addition, note that if each liequals zero then θis not surjective. Then, after
considering the braid transformations Φ2or Φ2◦Φ3if necessary, we can suppose
14 SEBASTI ´
AN REYES-CAROCCA
l26= 0.Now, we consider the automorphism of Gp,q given by a7→ ak2and b7→ b
where k2l2≡1 mod q, to assume l2= 1.It follows that θis given by
θ(x1) = bn1, θ(x2) = abn2, θ(x3) = al3bn3and θ(x4) = al4bn4
where, as x1x2x3x4= 1, we have that
n1+n2+n3+n4= 0 and 1 + l3rn2+l4rn2+n3= 0.
Thus, θis equivalent to the ske θn1,n2,n3,l3given by
x17→ bn1, x27→ abn2, x37→ al3bn3and x47→ a−r−n2−n3−l3r−n3b−n1−n2−n3
where ni∈Z∗
psuch that n1+n2+n36= 0 and l3∈Zq.Then, there are at most
q[(p−1)3−(p−1)(p−2)] = q(p−1)(p2−3p+ 3)
pairwise non-equivalent skes.
Proof of statement (2). By [43, Theorem 1], the action of Gp,q on Smight be
extended to an action of a group G′of order 2pq with signature (0; 2,2, p, p) and this
action, in turn, might be extended to only an action of a group of order 4pq with
signature (0; 2,2,2, p).
Note that G′cannot be isomorphic to Gp,q ×C2since it has only one involution.
Claim 1. The action of Gp,q on the surfaces lying in certain strata of C4,0extends
to an action of
G′=ha, b, t :aq=bp=t2= 1, bab−1=ar,(ta)2= [t, b] = 1i∼
=Cq⋊C2p
with signature (0; 2,2, p, p).
Let ∆′be a Fuchsian group of signature (0; 2,2, p, p)
∆′=hz1, z2, z3, z4:z2
1=z2
2=zp
3=zp
4=z1z2z3z4= 1i,
and let Θ : ∆′→G′be the ske defined by
Θ(z1) = taL,Θ(z2) = taL−1,Θ(z3) = abNand Θ(z4) = b−N,
where L∈Zqand N∈Z∗
p.Set
ˆx1:= z3,ˆx2:= z4,ˆx3:= z1z3z1and ˆx4:= z1z4z1
and note that they generate a Fuchsian group isomorphic to ∆.The claim follows
after noticing that the restriction
Θ|hˆx1,ˆx2,ˆx3,ˆx4i: ∆ → ha, bi∼
=Gp,q
is equivalent to some ske θn1,n2,n3,l3as before, with n2=−n1=N.
Claim 2. The action of Gp,q on each Sdoes not extend to any action of a group
of order 4pq with signature (0; 2,2,2, p).
ON pq-FOLD REGULAR COVERS OF THE PROJECTIVE LINE 15
Assume that the action of Gp,q on Sextends to an action of a group G′′ of order
4pq with signature (0; 2,2,2, p).Then G′′ is isomorphic to either
(Cq⋊Cp)⋊C4or (Cq⋊Cp)⋊C2
2.
In the former case, if C4=ht:t4= 1ithen
tat−1=aǫand tbt−1=anb
where n∈Zqand ǫequals 1, −1 or a primitive fourth root of unity in the field of
qelements. If ǫ= 1 then the product is direct, if ǫ=−1 then t2is the unique
involution and if ǫis a primitive fourth root of unity in the field of qelements then
the involutions are t2akwhere k∈Zq.
In the latter case, if C2
2=ht, u :t2=u2= (tu)2= 1ithen
tat =aǫ1, tbt =an1b, uau =aǫ2and ubu =an2b
where ni∈Zqand ǫi=±1.If ǫ1=ǫ2= 1 then the product is direct, if ǫ1=ǫ2=−1
then the involutions are takand uakwhere k∈Zq,and if ǫ1=−ǫ2= 1 or ǫ1=−ǫ2=
−1 then the involutions are takor uakrespectively, where k∈Zq.
The proof of the claim follows after noticing that, in each case, the group G′′ cannot
be generated by three involutions.
Proof of statement (3). Let ∆′′ be a Fuchsian group of signature (0; 2,2p, 2p)
∆′′ =hz1, z2, z3:z2
1=z2p
2=z2p
3=z1z2z3= 1i
and consider the group G′′ = (Cq⋊C2p)⋊C2presented as
ha, c, t :aq=c2p=t2= 1, cac−1=a−r,(ta)2= [t, c] = 1i.
The map Θ : ∆′′ →G′′ given by
z17→ ta, z27→ a−1cand z37→ c−1t
is a ske and therefore V=H/ker(Θ) is a compact Riemann surface with a group of
automorphisms isomorphic to G′′ acting on it with signature (0; 2,2p, 2p).
If we define
ˆx1:= z2
3,ˆx2:= z1z2
2z1,ˆx3:= z1z2
3z1and ˆx4:= z2
2
then they generate a Fuchsian group isomorphic to ∆ and the restriction
Θ|hˆx1,ˆx2,ˆx3,ˆx4i: ∆ → ha, b := c2i∼
=Gp,q
is equivalent to some ske θn1,n2,n3,l3as before. Hence, V∈C4,0.
16 SEBASTI ´
AN REYES-CAROCCA
Proof of statement (4). Consider the subgroups N=haiand K=ha, tiof G′and
the quotient J=K/N ∼
=C2. Note that Kis isomorphic to the dihedral group of
order 2q. The regular covering map S→S/K ramifies over exactly 2pvalues marked
with two and, consequently, the quotient S/K has genus zero. It is clear that Nacts
on Swithout fixed points and therefore the genus of the quotient X=S/N equals
γ=p−1.Note that Xadmits the action of Jand the corresponding two-fold regular
covering map
X→X/J ∼
=S/K ∼
=P1
ramifies over 2p= 2γ+ 2 values, showing that Xis hyperelliptic. We observe that,
as the group P=G′/K ∼
=Cpacts on X/J with signature (0; p, p),we can suppose
that P∼
=hβiwhere β(z) := ωz, and that the branch values of X→X/J are
{ωk:k∈Zp}and {µωk:k∈Zp}
where ωis a primitive p-th root of unity and µis a non-zero complex number such
that µp6= 1.The result follows by arguing as in the proof of Theorem 2.
7. Proof of Theorem 5
The proof is similar to the ones of Theorems 3and 4; so, we avoid some details.
Proof of statement (1). Consider a Fuchsian group ∆ of signature s3,1
∆ = hx1, x2, x3, y1:xp
1=xp
2=xp
3=yq
1=x1x2x3y1= 1i
and let θ: ∆ →Gp,q be a ske representing an action of Gp,q on S∈C3,1. Up to
equivalence, we can suppose that θagrees with the ske θl,n1,n2given by
x17→ a−1−lrn1bn1, x27→ albn2, x37→ b−n1−n2and y17→ a,
for some l∈Zqand ni∈Z∗
psuch that n16=−n2.Thus, there are at most q(p−1)(p−2)
pairwise non-equivalent skes.
Proof of statement (2). It follows from the fact that, by [43, Theorem 1], the
signature (0; p, p, p, q) is maximal.
Proof of statement (3). Let ∆′be a Fuchsian group of signature (0; p, 2p, 2q)
∆′=hz1, z2, z3:zp
1=z2p
2=z2q
3=z1z2z3= 1i
and consider the direct product G′=Gp,q ×C2with C2=ht:t2= 1i.The map
Θ : ∆′→G′given by z17→ a−1b, z27→ b−1tand z37→ ta
is a ske and then W=H/ker(Θ) is a Riemann surface with a group of automorphisms
isomorphic to G′acting on it with signature (0; p, 2p, 2q). To see that W∈C3,1,define
ˆx1:= z1,ˆx2:= z2
2,ˆx3:= z3z1z−1
3and ˆy1:= z2
3
and notice that they generate a Fuchsian group isomorphic to ∆ and that the restric-
tion of Θ to it is equivalent to some ske θl,n1,n2as before.
ON pq-FOLD REGULAR COVERS OF THE PROJECTIVE LINE 17
8. Proof of Theorems 6and 7
Rational irreducible representations. We refer to [41] for basic background con-
cerning representations of groups. Set ωt:= exp(2πi
t) where i2=−1.The group Gp,q
has, up to equivalence, pcomplex irreducible representations of degree 1, given by
χl:a7→ 1 and b7→ ωl
pfor l∈Zp.
They give rise to two non-equivalent rational irreducible representations of the group;
namely, the trivial one W0=χ0and the direct sum of the non-trivial ones
W1=χ1⊕ · · · ⊕ χp−1.
Consider the equivalence relation Ron Z∗
qgiven by
uRv ⇐⇒ u=rnvfor some n∈Zp
where ris primitive p-th root of unity in the field of qelements. Let d=q−1
pand
let {k1,...,kd}be a maximal collection of representatives of R. Then Gp,q has, up to
equivalence, dcomplex irreducible representations of degree p
ψj:a7→ diag(ωkj
q, ωkjr
q, ωkjr2
q,...,ωkjrp−1
q) and b7→
0 1 0 ··· 0
0 0 1 ··· 0
...
0 0 0 ··· 1
1 0 0 ··· 0
for j∈ {1,...,d}.The direct sum of these representations yields a rational irreducible
representation of the group; namely
W2=ψ1⊕ · · · ⊕ ψd.
Observe that the set {W0, W1, W2}is a maximal set of pairwise non-equivalent
rational irreducible representations of Gp,q .In addition, χ1and ψ1have Schur index
one and their character fields have degree p−1 and (q−1)/p over Qrespectively.
Proof of Theorem 6.Let Sbe a compact Riemann surface endowed with a group
of automorphisms isomorphic to Gp,q .Then, as explained in Subsection §2.5, the in-
formation concerning the rational irreducible representations described above, allows
us to obtain that the group algebra decomposition of J S with respect to Gp,q is
JS ∼B0×B1×Bp
2(8.1)
where the factor Blis associated to the representation Wl(and, in turn, W1, W2and
W3are associated to χ0, χ1and ψ1respectively). Note that B0is isogenous to the
Jacobian variety of S/Gp,q .
We now assume that S∈Cn,m and that the action of Gp,q on Sis represented by
the ske θ: ∆ →Gp,q,with ∆ presented as in the proof of Theorem 1. Observe that,
independently of jand independently of the choice of the ske θ, we have that
hθ(yj)i=haiand hθ(xj)i ∼chbi
18 SEBASTI ´
AN REYES-CAROCCA
where ∼cstands for conjugation. It follows that the dimension of the fixed subspaces
of χ1and ψ1under the action of the corresponding isotropy groups hθ(xj)iand hθ(yj)i
do not depend on θ, and are given by
dim(χhθ(xj)i
1) = 0 and dim(χhθ(yj)i
1) = 1
and
dim(ψhθ(xj)i
1) = 1 and dim(ψhθ(yj)i
1) = 0
for each j. We now apply the equation (2.4) to conclude that the dimension of the
factors B1and B2in (8.1) are given by
dim(B1) = (p−1)[−1 + 1
2(n(1 −0) + m(1 −1))] = (n−2)( p−1
2)
dim(B2) = q−1
p[−p+1
2(n(p−1) + m(p−0))] = q−1
p[p(m
2−1) + n(p−1
2)],
as claimed. Clearly B0= 0 since S/Gp,q ∼
=P1.
Proof of Theorem 7.Let Sbe a compact Riemann surface lying in the family Cn,m
and consider the group algebra decomposition of JS with respect to Gp,q
JS ∼B1×Bp
2
obtained in Theorem 6. Following [11, Proposition 5.2], if Kis a subgroup of Gp,q
then
J(S/K)∼BnK
1
1×BnK
2
2(8.2)
where nK
1and nK
2are the dimension of the vector subspaces of χ1and ψ1fixed under
the action of K. Now, if we consider (8.2) with K=Nand K=Hwe obtain
JX =J(S/N)∼B1
1×B0
2and JY =J(S/H)∼B0
1×B1
2
respectively, and the result follows.
Acknowledgements. The author is grateful to the referee for valuable comments
and suggestions.
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Departamento de Matem´
atica y Estad
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ıstica, Universidad de La Frontera, Avenida
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