Generalized superelliptic Riemann surfaces

Abstract

A conformal automorphism τ\tau, of order n≥2n \geq 2, of a closed Riemann surface \X, of genus g≥2g \geq 2, which is central in {\rm Aut}(\X) and such that \X/\langle \tau\rangle has genus zero, is called a superelliptic automorphism of level n. If n=2, then τ\tau is called hyperelliptic involution and it is known to be unique. In this paper, for the case n≥3n \geq 3, we investigate the uniqueness of the cyclic group ⟨τ⟩\langle \tau \rangle.
Let τ1\tau_{1} and τ2\tau_{2} be two superelliptic automorphisms of level n of \X. If n≥3n \geq 3 is odd, then ⟨τ1⟩=⟨τ2⟩\langle \tau_{1} \rangle =\langle \tau_{2} \rangle.
In the case that n≥2n \geq 2 is even, then the same uniqueness result holds, up to some explicit exceptional cases.

Full text

GENERALIZED SUPERELLIPTIC RIEMANN SURFACES
RUB´
EN A. HIDALGO, SA´
UL QUISPE, AND TONY SHASKA
Abstract. A conformal automorphism τ, of order n≥2, of a closed Riemann
surface X, of genus g≥2, which is central in Aut(X) and such that X/hτihas
genus zero, is called a superelliptic automorphism of level n. If n= 2, then τ
is called hyperelliptic involution and it is known to be unique. In this paper,
for the case n≥3, we investigate the uniqueness of the cyclic group hτi. Let
τ1and τ2be two superelliptic automorphisms of level nof X. If n≥3 is odd,
then hτ1i=hτ2i. In the case that n≥2 is even, then the same uniqueness
result holds, up to some explicit exceptional cases.
1. Introduction
Let Xbe a closed Riemann surface of genus g≥2 and let G= Aut (X) be its
group of conformal automorphisms. It is well known that Aut(X) is finite [23] of
order at most 84(g−1) [16]. In this paper, we concider certain cyclic subgroups of
Aut(X) which behaves similar to the hyperelliptic involution.
If τ∈Ghas order n≥2 and X/hτihas genus zero, then it is called n-gonal.
In this case, we also say that hτi∼
=Cnis a n-gonal group and that Xis a cyclic
n-gonal Riemann surface. A 2-gonal automorphism, also called a hyperelliptic
involution, is known to be unique in G; in particular, it is central in G. If n≥3
is a prime integer and s≥3 is the number of fixed points of τ, then hτiis known
to be the unique n-Sylow subgroup of Xif either 2n < s [7] or n≥5s−7 [12].
For a general n≥3, under the assumption that the fixed points of each non-trivial
power of τis also fixed by τ, the uniqueness of hτiis also true under the assumption
that g > (n−1)2[18] (as a consequence of results in [1]). In this last case, the
computation of Ghas been done in [22].
Let Nbe the normalizer of hτiin G. If n= 2 (the hyperelliptic case), then τis
central in N=G. For n≥3, τdoes not need to be central in Nand, if it is central
in N, it might be that N6=G. It follows from the results in [27], that generically
τis central in G.
If τis central in G(respectively, central in N), then we called it a superelliptic
automorphism of level n (respectively, generalized superelliptic automor-
phism of level n); we also say that H=hτiis a superelliptic group of level
n(respectively, generalized superelliptic group of level n), and that Xis a
superelliptic curve of level n (respectively, generalized superelliptic curve
of level n).
In this paper, we are interested in the uniqueness of superelliptic groups. Our
main result is the following.
2010 Mathematics Subject Classification. 14H37; 14H45; 30F10.
Key words and phrases. generalized superelliptic curves, cyclic gonal curves, automorphisms,
Riemann surfaces.
The first two authors were partially supported by Projects Fondecyt 1230001 and 1220261.
1

2 RUB´
EN A. HIDALGO, SA´
UL QUISPE, AND TONY SHASKA
Theorem 1. Let Xbe a Riemann surface, admitting two superelliptic automor-
phisms τand η, both of level n, such that hτi 6=hηi. Then n= 2d≥4and it can
be represented by a cyclic n-gonal curve of the form
(1) X:y2d=x2x2−1l1x2−a2
1l2QL
j=3 x2−a2
j2b
lj,
where (i) l1, l2,2b
l3,...,2b
lL∈ {1,...,2d−1}, (ii) l1is odd and (iii) either (1) or
(2) below holds for l2.
(1) If l2= 2b
l2, then gcd d, l1,b
l2,...,b
lL= 1.
(2) If l2odd, then l1+l2= 2dand gcd d, l1, l2,b
l3,...,b
lL= 1.
In these cases, τ(x, y) = (x, ω2dy)and η(x, y) = (−x, ω2dy)and K=hτ , ηi∼
=
C2d×C2.
Remark 1. The superelliptic Riemann surfaces, described by the cyclic 2d-gonal
curves in Theorem 1, will be called exceptionals. Note that the Riemann surfaces
defined by equations as in the previous theorem are not all of them necessarilly
superelliptic; the theorem only asserts that the exceptional ones are some of them.
For example, the cyclic 2d-gonal curve
y2d=x2(x2−1)l1,
with l1=d−1 and d≥2 even, admits the extra automorphism
(2) α(x, y) = x(x2−1)d/2
yd,yl1
(x2−1)(l2
1−1)/2d,
which does not commute with τ.
Corollary 1. Let Xbe a Riemann surface admittiing a superelliptic group Hof
level n. Then His the unique superelliptic group of level nof Xif either: (1) n≥3
is odd or (2) n= 2, or (iii) n≥4is even and X/H has no cone point of order
n/2.
In order to prove Theorem 1, we first find necessary and sufficient conditions for
an-gonal automorphism to be generalized superelliptic.
Let us consider a pair (X, τ ), where τis a n-gonal automorphism of X, and
H=hτi∼
=Cn. An algebraic model for (X, τ ) can be constructed as follows. Let us
consider a Galois branched covering π:X → b
C, whose deck covering group is H=
hτi, and let p1, . . . , ps∈b
Cbe its branch values. Then there are integers l1, . . . , ls∈
{1, . . . , n −1}satisfying that l1+···+lsis a multiple of nand gcd(n, l1, . . . , ls) = 1,
such that Xcan be described by an affine irreducible algebraic curve (which might
have singularities) of the following form (called a cyclic n-gonal curve)
(3) yn=Qs
j=1(x−pj)lj,
If one of the branch values is ∞, say ps=∞, then we need to delete the factor
(x−ps)lsfrom the above equation. In this algebraic model, τand πare given by
τ(x, y) = (x, ωny), where ωn=e2πi/n , and π(x, y) = x.
Theorem 2. Let Xbe a cyclic n-gonal Riemann surface, described by the cyclic
n-gonal curve Eq. (3), and Nbe the normalizer of H=hτ(x, y) = (x, ωny)iin
Aut (X). Let θ:N→N=N/H the canonical projection homomorphism. Then τ
is a generalized superelliptic automorphism of level nif and only if for all pjand
piin the same θ(N)-orbits it holds that lj=li.

GENERALIZED SUPERELLIPTIC RIEMANN SURFACES 3
Corollary 2. Let Xbe a cyclic n-gonal Riemann surface, described by the cyclic
n-gonal curve Eq. (3). If lj=l, for every j, where gcd(n, l) = 1, then τ(x, y) =
(x, ωny)∈Aut (X)is a generalized superelliptic automorphism of level n.
Remark 2. The condition impossed to the integers ljin the above corollary is
equivalent to say that: (i) every fixed point of a non-trivial power of τis also a
fixed points of τ, and (ii) the rotation number of τat each of its fixed points is the
same.
Notations. We denote by Cnthe cyclic group of order n, by Dnthe dihedral
group of order 2n, by Anthe alternating group, and by Snthe symmetric group.
2. Preliminaries
2.1. The finite groups of M¨obius transformations. Up to PSL2(C)-conjugation,
the finite subgroups of the group PSL2(C) of M¨obius transformations are given by
(see, for instance, [3])
(4)
Cm:= a(x) = ωmx, Dm:= Da(x) = ωmx, b(x) = 1
xE, A4:= Da(x) = −x, b(x) = i−x
i+xE,
S4:= Da(x) = ix, b(x) = i−x
i+xE, A5:= Da(x) = ω5x, b(x) = (1−ω4
5)x+(ω4
5−ω5)
(ω5−ω3
5)x+(ω2
5−ω3
5)E,
where ωmis a primitive m-th root of unity. For each of the above finite groups A,
a Galois branched covering fA:b
C→b
C, with deck group A, is given as follows
fCm(x) = xm; branching: (m, m).
fDm(x) = xm+x−m; branching: (2,2, m).
fA4(x) = (x4−2i√3x2+ 1)3
−12i√3x2(x4−1)2; branching: (2,3,3).
fS4(x) = (x8+ 14x4+ 1)3
108x4(x4−1)4; branching: (2,3,4).
fA5(x) = (−x20 + 228x15 −494x10 −228x5−1)3
1728x5(x10 + 11x5−1)5; branching: (2,3,5),
see [14]. In the above, the branching corresponds to the tuple of branch orders of
the cone points of the orbifold b
C/A.
2.2. Fuchsian groups. AFuchsian group is a discrete subgroup Kof PSL2(R),
the group orientation-preserving isometries of the hyperbolic plane H. It is called
co-compact if the quotient orbifold H/K is compact; its signature is the tuple
(g;n1, . . . , ns), where gis the genus of the quotient orbifold H/K,sis the num-
ber of its cone points they having branch orders n1, . . . , ns. The group Khas a
presentation as follows:
K=ha1, b1, . . . , ag, bg, c1, . . . , cs:
cn1
1=··· =cns
s= 1, c1···cs[a1, b1]···[ag, bg] = 1i,
(5)
where [a, b] = aba−1b−1. The hyperbolic area of the orbifold H/K is equal to
(6) µ(K)=2π2g−2 + Ps
j=1 1−1
nj.
If a co-compact Fuchsian group Γ has no torsion, then X=H/Γ is a closed
Riemann surface of genus g≥2 and its signature is (g;−). Conversely, by the
uniformization theorem, every closed Riemann surface of genus g≥2 can be repre-
sented as above. By Riemann’s existence theorem, a finite group Gacts faithfully

4 RUB´
EN A. HIDALGO, SA´
UL QUISPE, AND TONY SHASKA
as a group of conformal automorphisms of Xif and only if there is a co-compact
Fuchsian group Kand a surjective homomorphism θ:K→Gwhose kernel is Γ.
2.3. Cyclic n-gonal Riemann surfaces. Let Xbe a cyclic n-gonal Riemann
surface of genus g≥2, τ∈Aut (X) be a n-gonal automorphism and π:X → b
C
be a Galois branched cover whose deck group is the n-gonal group H=hτi∼
=Cn.
Let p1, . . . , ps∈b
Cbe the branch values of πand let us denote by nj≥2 (which is
a divisor of n) the branch order of πat pj.
Let Kbe a Fuchsian group such that (up to biholomorphisms) H2/K =X/hτi.
Then Khas signature (0; n1, . . . , ns) and a presentation
(7) K=hc1, . . . , cs:cn1
1=··· =cns
s= 1, c1···cs= 1i.
The branched Galois covering πis determined by a surjective homomorphism
ρ:K→Cn=hτiwith a torsion-free kernel Γ such that X=H2/Γ. (The
homomorphism ρis uniquely determined up to post-composition by automorphisms
of Cnand pre-composition by an automorphism of K.) Let ρ(cj) = τlj, where cj
is as in Eq. (5), for l1, . . . , ls∈ {1, . . . , n −1}.
As a consequence of Harvey’s criterion [11],
(a) n= lcm(n1, . . . , nj−1, nj+1, . . . , ns) for all j;
(b) if nis even, then #{j∈ {1, . . . , s}:n/njis odd}is even.
The equality c1·· ·cs= 1 is equivalent to have l1+···+ls≡0 mod(n), and the
condition for Γ = ker(ρ) to be torsion free is equivalent to have gcd(n, lj) = n/nj,
for j= 1, . . . , s. The surjectivity of ρis equivalent to have gcd(n, l1, . . . , ls) = 1,
which in our case is equivalent to condition (a). Condition (b) is equivalent to
say that for neven the number of ljbeing odd is even, which trivially holds.
Summarizing all the above,
(1) l1, . . . , ls∈ {1, . . . , n −1},
(2) l1+· ·· +ls≡0 mod(n),
(3) nj=n/ gcd(n, lj), for all j,
(4) gcd(n, l1, . . . , ls) = 1.
As a consequence, the Riemann surface Xcan be described by the curve
(8) X:yn=Qs
j=1(x−pj)lj,
where
(1) l1, . . . , ls∈ {1, . . . , n −1}, gcd(n, lj) = n/nj,
(2) l1+· ·· +ls≡0 mod(n),
(3) gcd(n, l1, . . . , ls) = 1.
In the above, if one of the branched values is infinity, say ps=∞, then we delete
the factor (x−ps)lsin the above equation.
In such an algebraic model, τ(x, y)=(x, ωny), where ωn=e2πi/n, and π(x, y ) =
x. The branch order of πat pjis nj=n/ gcd(n, lj) and, by the Riemann-Hurwitz
formula, the genus gof Xis given by
(9) g= 1 + 1
2(s−2)n−Ps
j=1 gcd(n, lj).
3. Proof of Theorem 2
Let X,τ∈G= Aut (X) and πbe as in Eq. (3).
Let Nbe the normalizer of H=hτiin G. There is a short exact sequence
(10) 1→H=hτi → Nθ
→N=N/H →1,

GENERALIZED SUPERELLIPTIC RIEMANN SURFACES 5
where θ(η)◦π=π◦η, for every η∈N.
The reduced group of automorphisms N=N/H < PSL2(C) is a finite group
keeping invariant the set {p1, . . . , ps}.
The following describes the form of those elements of N.
Lemma 1. Let η∈Nand l∈ {1, . . . , n −1}(necessarily relatively prime to n)
such that ητ η−1=τl. If b=θ(η), then η(x, y) = (b(x), ylQ(x)), where Q(x)is a
suitable rational map.
Proof. Let us note that η(x, y) = (b(x), R(x, y )), where R(x, y) is a suitable ra-
tional map. As η(τ(x, y )) = η(x, ωny) = (b(x), R(x, ωny)) and τl(η(x, y)) =
τl(b(x), R(x, y)) = (b(x), ωl
nR(x, y)), the condition ητη−1=τlholds if and only
if R(x, ωny) = ωl
nR(x, y), that is, R(x, y) = Q(x)yl, for a suitable rational map
Q(x)∈C(x). 
In particular, the above lemma asserts that the form of those η∈Ncommuting
with τhave the form η(x, y)=(b(x), Q(x)y).
Let us observe that, if t∈PSL2(C), then replacing πby t◦πonly change the
set of branch points {p1, . . . , ps}by {t(p1), . . . , t(ps)}but keeps invariant the set of
exponents l1, . . . , ls.
Let η∈Nand assume θ(η) has order m≥2. As there is a suitable t∈PSL2(C)
so that tθ(η)t−1(x) = ωmx, we may assume (by post-composing πwith t) that
θ(η)(x) = ωmx. So the cyclic n-gonal curve Eq. (8) can be written as
(11) yn=xsQL
j=1(x−qj)lj,1(x−ωmqj)lj,2···(x−ωm−1
mqj)lj,m ,
where
{s, l1,1, . . . , l1,m, l2,1, . . . , l2,m , . . . , lL,1, . . . , lL,m}={l1, . . . , lr}.
In this model, τ(x, y)=(x, ωny) and, by Lemma 1,η(x, y)=(ωmx, Q(x)yl), for
a suitable rational map Q(x)∈C(x). (ηcommutes with τif and only if l= 1).
If R(x) denotes the right side of Eq. (11), then Q(x)nyln =R(ωmx) on X, where
(12)
R(ωmx) = xsQL
j=1
ωqj
m(x−qj)lj,1(x−ωmqj)lj,2···(x−ωm−1
mqj)lj,m
(x−qj)lj,1−lj,2(x−ωmqj)lj,2−lj,3···(x−ωm−1
mqj)lj,m−lj,1
=ωPL
j=1(s+lj,1+···+lj,m )
myn
QL
j=1(x−qj)lj,1−lj,2(x−ωmqj)lj,2−lj,3···(x−ωm−1
mqj)lj,m−lj,1,
and qj=s+lj,1+· ·· +lj,m, that is,
(13) Q(x)nyln =ωPL
j=1(s+lj,1+···+lj,m )
myn
QL
j=1(x−qj)lj,1−lj,2(x−ωmqj)lj,2−lj,3···(x−ωm−1
mqj)lj,m−lj,1.
In particular,
(14) Q(x)ny(l−1)n=ωPL
j=1(s+lj,1+···+lj,m )
m
QL
j=1(x−qj)lj,1−lj,2(x−ωmqj)lj,2−lj,3···(x−ωm−1
mqj)lj,m−lj,1.
Let us assume ηcommutes with τ, that is, l= 1. We proceed to prove that the
exponents lj,i are the same for every i= 1, . . . , m. As θ(η)m= 1, it follows that
ηm∈ hτi, from which we must have that
(15) Qm−1
j=0 Q(ωj
mx)n= 1.

6 RUB´
EN A. HIDALGO, SA´
UL QUISPE, AND TONY SHASKA
Claim 1. Equation Eq. (15)asserts that Q(x)is either a nm-root of unity or it
has the form
Q(x) = λQA
u=1
x−αu
x−ωqu
mαu,
where λnm = 1 and qu∈ {1, . . . , m −1}.
Proof. If we write
Q(x) = λQA
u=1(x−αu)
QB
v=1(x−βv),
then Qm−1
j=0 Q(ωj
mx) = λmQm−1
j=0 ω(A−B)j
mQA
u=1(x−ωm−j
mαu)
QB
v=1(x−ωm−j
mβv)=
=λmω(A−B)m(m−1)/2
mQA
u=1(xm−αm
u)
QB
v=1(xm−βm
v).
Equation Eq. (15) asserts that
A=B, QB
v=1(xm−βm
v) = λnm QA
u=1(xm−αm
u).
So, λnm = 1 and, up to a permutation of indices, we may assume αm
u=βm
u, for
u= 1, . . . , A.
By Claim 1, either lj,i −lj,i+1 = 0 or ωi−1
mqjmust be either a zero or a pole
of order nof the left side of Eq. (14), that is, each lj,i −lj,i+1 ∈ {0,±n}. As
lj,i ∈ {1, . . . , n −1}, it follows that lj,1=··· =lj,m .
In the other direction, let us assume that lj,1=··· =lj,m =lj, for every
j= 1, . . . , L. In this case, Xhas equation
(16) yn=xsQL
j=1(xm−qm
j)lj.
A lifting of θ(η) under π(x, y) = xis of the form bη(x, y) = (ωmx, ωs/n
my). This
asserts that η=bητk, for some k∈ {0, . . . , n −1}, i.e., η(x, y)=(ωmx, ωk
nωs/n
my),
that is l= 1.
3.1. A consequence. The above permits us to observe that, if τis a generalized
superelliptic automorphism of level n, then Xcan be represented by a cyclic n-gonal
curve of the form
(17) X:yn=xl0(xm−1)l1QL
j=2(xm−am
j)lj,
where the following Harvey’s conditions are satisfied:
(1) l0= 0, m(l1+· · · +lL)≡0 mod(n) and gcd(n, l1, . . . , lL) = 1; or
(2) l06= 0 and gcd(n, l0, l1, . . . , lL) = 1,
where τ(x, y)=(x, ωny).
4. Proof of Theorem 1and Corollary 1
Let us assume Xadmits two superelliptic automorphisms τand η, both of level
n, that is, each one being central in G= Aut (X), Let H=hτiand the reduced
group G=G/H. We proceed to investigate when hτi 6=H.
As the case n= 2 corresponds to the hyperelliptic situation, and the hyperelliptic
involution is unique, we only need to restrict to the case when n≥3.
4.1. Proof of Theorem 1.

GENERALIZED SUPERELLIPTIC RIEMANN SURFACES 7
4.1.1. First case. As for the Platonic groups and the dihedral groups of order not
divisible by 4 there is no non-trivial central element, this uniqueness property fol-
lows.
Proposition 1. If Gis either trivial, a dihedral group of order not divisible by 4
or A4or S4or A5, then H=hηi.
Proof. Assume, by the contrary, that there is a superelliptic automorphism ηof
level nwith η6∈ H. Then ηinduces a non-trivial central element of the reduced
group G, a contradiction. 
4.1.2. Second case. Let us assume η /∈H=hτi. As a consequence of the above
case, Gis either a non-trivial cyclic group or a dihedral group of order 4m.
Let us consider, as before, the canonical quotient homomorphism θ:G→G,
and let π:X → b
Cbe a Galois branched cover with deck group H. As τis central,
K=hτ, η i< G is an abelian group and K=K/H =hθ(η)i∼
=Cm, where n=md
and m≥2. Since θ(η) has order m,ηm∈Hand it has order d. So, replacing τby
a suitable power (still being a generator of H) we may assume that ηm=τm. Now,
as noted in Section 3.1, we may assume Xto be represented by a cyclic n-gonal
curve of the form
(18) X:yn=xl0(xm−1)l1QL
j=2(xm−am
j)lj,
where the following Harvey’s conditions are satisfied:
(1) l0= 0, m(l1+· · · +lL)≡0 mod (n) and gcd(n, l1, . . . , lL) = 1; or
(2) l06= 0 and gcd(n, l0, l1, . . . , lL) = 1.
In this algebraic model, τ(x, y ) = (x, ωny), π(x, y) = xand θ(η)(x) = ωmx,
where ωt=e2πi/t. In this way, η(x, y) = (ωmx, ωl0/n
my). As we are assuming
ηm=τmand ηhas order n, we may assume the following
(19) if l06=0: η(x, y) = (ωmx, ωny) and l0=m,
if l0=0: η(x, y)=(ωmx, y) and n=m.
i): Case l0=m.In this case, η(x, y)=(ωmx, ωny) and we are in case (2) above.
The η-invariant algebra C[x, y]hηiis generated by the monomials u=xm, v =yn
and those of the form xayb, where a∈ {0,1, . . . , m−1}and b∈ {0,1, . . . , n−1}(the
case a=b= 0 not considered) satisfy that a+b/d ≡0 mod (m). In particular,
b=dr for r∈ {0,1,...,[(n−1)/d]}so that a+r≡0 mod (m). As 0 ≤a+r≤
(m−1) + [(n−1)/d]≤(m−1) + [(md −1)/d]<2m, it follows that a+r∈ {0, m}.
As the case a+r= 0 asserts that a=b= 0, which is not considered, we must have
a+r=m, from which we see that the other generators are given by t1, . . . , tm−1,
where tj=xm−jydj . As consequence of invariant theory, the quotient curve X/hηi
corresponds to the algebraic curve
(20) Y:













tm
1=um−1v,
tm
2=um−2v2,
.
.
.
tm
m−1=uvm−1,
v=u(u−1)l1QL
j=2(u−am
j)lj.
The curve Yadmits the automorphisms T1, . . . , Tm−1, where Tjis just amplifica-
tion of the tj-coordinate by ωmand acts as the identity on all the other coordinates.

8 RUB´
EN A. HIDALGO, SA´
UL QUISPE, AND TONY SHASKA
The group generated by all of these automorphisms is
(21) U=hT1, . . . , Tm−1i∼
=Cm−1
m.
The Galois branched cover map πU:Y → b
C: (u, v, t1, . . . , tm−1)7→ uhas U
as its deck group. Let us observe that the values 0, am
1, . . . , am
Lbelongs to the
branch set of πU. Since Y=X/hηihas genus zero and the finite abelian groups of
automorphisms of the Riemann sphere are either the trivial group, a cyclic group
or V4=C2
2, the group Uis either one of these three types. As m≥2, the group U
cannot be the trivial group nor it can be isomorphic to the Klein group V4=C2
2. It
follows that Uis a cyclic group; so m= 2 and, in particular, n= 2d, where d≥2,
and
(22) X:y2d=x2(x2−1)l1QL
j=2(x2−a2
j)lj.
Harvey’s condition (a) is equivalent to have gcd(2d, 2, l1, . . . , lL) = 1, which is
satisfied if some of the exponents ljis odd. Without loss of generality, we may
assume that l1is odd. In this case the curve Yis given by
(23) Y:t2
1=uv,
v=u(u−1)l1QL
j=2(u−a2
j)lj,
which is isomorphic to the curve
(24) w2= (u−1)l1QL
j=2(u−a2
j)lj.
As this curve must have genus zero, and l1is odd, the number of indices j∈
{2, . . . , L}for which ljis odd must be at most one.
(i) If l1is the only odd exponent and lj= 2b
lj, for j= 2, . . . , L, then it must
hold gcd(2d, 2, l1,2b
l2,...,2b
lL) = 1, equivalently, gcd(d, l1,b
l2,...,b
lL) = 1.
(ii) If there are exactly two of the exponents being odd, then we may assume,
without loss of generality, that l1and l2are the only odd exponents. In
this case, l1+l2≡0 mod (2d), that is, l1+l2= 2d. If we write lj= 2b
lj,
for j= 3, . . . , L, then we must have gcd(2d, 2, l1, l2,2b
l3,...,2b
lL) = 1, which
is equivalent to gcd(d, l1, l2,b
l3,...,b
lL) = 1.
ii): Case l0= 0.In this case, m=n,η(x, y) = (ωnx, y ) and we are in case
(1) above. The η-invariants algebra C[x, y]hηiis generated by the monomials u=
xn, v =y. As consequence of invariant theory, the quotient curve X/hηicorresponds
to the algebraic curve
(25) Y:nvn= (u−1)l1QL
j=2(u−an
j)lj.
As Ymust have genus zero and n≥3, we should have either case (1) or (2) below.
(26) (1) X:yn= (xn−1)l1.
(2) X:yn= (xn−1)l1(xn−an
2)l2, l1+l2≡0 mod (n).
Note that, for situation (1) above, we may assume l1= 1 (this is the classical
Fermat curve of degree n). As the group of automorphisms of classical Fermat curve
of degree nis C2
noS3, we may see that τis not central; that is, it is not a generalized
superelliptic Riemann surface of level n. In case (2), Harvey’s conditions holds

GENERALIZED SUPERELLIPTIC RIEMANN SURFACES 9
exactly when gcd(n, l1, l2) = 1. As l1+l2≡0 mod (n) and l1, l2∈ {1, . . . , n −1},
we have that l1+l2=n. If we write l2=n−l1, then
(27) xn−1
xn−an
2l1=yn
(xn−an
2)n,
and by writing l1=n−l2we also have that
(28) xn−an
2
xn−1l2=yn
(xn−1)n.
Then the M¨obius transformation M(x) = a/x induces the automorphism
(29) α(x, y) = ωna2
x,−al2
2(xn−1)(xn−an
2)
xny,
which does not commute with η(x, y)=(ωnx, y) since n≥3, a contradiction.
4.2. Proof of Corollary 1.Let Xbe a a cyclic n-gonal Riemann surface admitting
superelliptic automorphisms τ, η, both of level n, such that hτi 6=hηi. In this case,
Theorem 1asserts that n= 2d, where d≥2. The quotient orbifold X/K has
signature 0; 2,2d, 2d
gcd(2d,l1),2d
gcd(2d,l2),d
gcd(d,b
l3),..., d
gcd(d,b
lL),
if 1 + l1+l2+ 2 b
l3+· ·· +b
lL≡0 mod (d), otherwise, it has signature
0; 2,2d, 2d
gcd(2d,l1),2d
gcd(2d,l2),d
gcd(d,b
l3),..., d
gcd(d,b
lL),d
gcd(d,1+l1+l2+b
l3+···+b
lL).
The genus of Xis, in the first case, equal to
2d(1 + L)−gcd(2d, l1)−gcd(2d, l2)−2PL
j=3 gcd d, b
lj,
and, in the second case, equal to
d(3 + 2L)−gcd(2d, l1)−gcd(2d, l2)−2PL
j=3 gcd d, b
lj−gcd d, 1 + l1+l2+ 2 PL
j=3 b
lj.
It follows from the above that the quotient orbifold X/H has a cone point of
order n/2. This is not possible in under the hypothesis of Corollary 1.
5. Field of moduli of superelliptic curves
5.1. Field of definition. As a consequence of the Riemann-Roch theorem, every
closed Riemann surface Xcan be described as a complex projective irreducible
algebraic curve, say defined as the common zeros of the homogeneous polynomials
P1, . . . , Pr. If σ∈Gal(C), the group of field automorphisms of C, then Xσwill
denote the curve defined as the common zeros of the polynomials Pσ
1, . . . , P σ
r, where
Pσ
jis obtained from Pjby applying σto its coefficients. The new algebraic curve
Xσis again a closed Riemann surface of the same genus. Let us observe that, if
σ, τ ∈Gal(C), then Xστ = (Xσ)τ(we multiply the permutations from left to right).
A subfield Lof Cis called a field of definition of Xif there is a curve Y, defined over
L, which is isomorphic to Xover C. Weil’s descent theorem [28] provides sufficient
conditions for a given subfield of Cto be a field of definition of X. These conditions
holds if Xhas no non-trivial automorphisms (a generic situation for g≥3).

10 RUB´
EN A. HIDALGO, SA´
UL QUISPE, AND TONY SHASKA
5.2. Field of moduli. If GXis the subgroup of Gal(C) consisting of those σso
that Xσis isomorphic to X, then the fixed field MXof GXis called the field of
moduli of X. The notion of the field of moduli was originally introduced by Shimura
[26] for the case of abelian varieties and later extended to more general algebraic
varieties by Koizumi [17]. In that same paper, Koizumi observed the following two
facts:
(i) MXis the intersection of all the fields of definition of X, and
(ii) Xhas a field of definition being a finite extension of MX.
5.3. Field of moduli versus field of definition. There are examples for which
the filed of moduli si not a field of definition [8,26]. In [13] the following sufficient
condition for a curve to be definable over its field of moduli was obtained.
Theorem 3. Let Xbe a curve of genus g≥2admitting a subgroup H < Aut (X)so
that X/H has genus zero. If His unique in Aut (X)and the reduced group Aut (X)
is different from trivial or cyclic, then Xis definable over its field of moduli.
If Xis a hyperelliptic curve and His the cyclic group generated by the hyper-
elliptic involution, then the above result is due to Huggins [15].
Another sufficient condition of a curve Xto be definable over its field of moduli,
which in particular contains the case of quasiplatonic curves, was provided in [2].
We say that Xhas odd signature if X/Aut (X) has genus zero and in its signature
one of the cone orders appears an odd number of times.
Theorem 4. Let Xbe a curve of genus g≥2. If Xhas odd signature, then it can
be defined over its field of moduli.
5.4. Minimal fields of definition of superelliptic curves. As exceptional su-
perelliptic curves of level nhave odd signature, Theorem 4asserts that they are
definable over their field of moduli. At the level of the non-exceptional ones, as a
consequence of the uniqueness of the superelliptic group of level n(Corollary 1),
we may apply Theorem 3and Theorem 4to obtain that conditions for them to be
definable over their fields of moduli.
Theorem 5. Let Xbe a superelliptic curve of level n≥2and let H∼
=Cnbe a
superelliptic group of it. If G= Aut (X), then Xis definable over its field of moduli
in any of the following situations.
(1) Xis exceptional.
(2) Xis non-exceptional with either (i) G/H different from trivial or cyclic or
(ii) G/H either trivial or cyclic and Xhas odd signature.
As a consequence, the only cases were the superelliptic curves cannot be defined
over their field of moduli are those non-exceptional superelliptic curves with reduced
group Aut (X)/H being either trivial or cyclic and with Xhaving not an odd
signature.
6. Appendix A: Algebraic equations for generalized superelliptic
curves
Let Xbe a generalized superelliptic curve of level nand τ∈G= Aut (X) be a
generalized superelliptic automorphism of level n(so, it is central in its normalizer
N). We proceed to describe explicit algebraic equations for Xand also explicit

GENERALIZED SUPERELLIPTIC RIEMANN SURFACES 11
generators for N, by making a subtle modification of the classical method done by
Horiuchi in [14] for the hyperelliptic situation.
Let π:X → b
Cbe a Galois branched cover with deck group H=hτiand let
Bπ={p1, . . . , ps} ⊂ b
Cbe its set of branch values. Let θ:N→Nbe the surjective
homomorphism satisfying θ(η)◦π=π◦η, for every η∈N. Recall that Nis one
of the finite subgroups of PSL2(C) (as described in Section 2.1) keeping the set Bπ
invariant.
Let us consider the Galois branched cover f=fN:b
C→b
Cwith Nas its deck
group (as described in Section 2.1). Let P(x), Q(x)∈C[x] be relatively prime
polynomials such that f(x) = P(x)
Q(x).
The collection Bπis N-invariant and, by Theorem 2, if for t∈Nit holds
that t(pi) = pj, then li=lj. In particular, we may consider the partition Bπ=
Bcrit
π∪B∗
π, where Bcrit
πconsists of those branch values with non-trivial N-stabilizer.
For simplicity, we assume ∞/∈ B∗
π(but it might happen that ∞∈Bcrit
π).
6.1. Horiuchi’s general process.
6.1.1. Computing algebraic models. There is at most T≤3 disjoint N-orbits of the
points in Bcrit
π.
If N∼
=Cm, then T≤2; each such orbit has cardinality one.
If N∼
=Dm, then T≤3; at most one orbit of cardinality 2 and at most two
others, each of cardinality m.
If N∼
=A4, then T≤3; at most one orbit of cardinality 6 and at most two
others, each of cardinality 4.
If N∼
=S4, then T≤3; at most one orbit of cardinality 8, one of cardinality 6
and another of cardinality 12.
If N∼
=A5, then T≤3; at most one orbit of cardinality 20, one of cardinality 30
and another of cardinality 12.
Let us denote these orbits (eliminating ∞from its orbit if it is a branch value of
π) by
(30) Ocrit
u={qu,1, . . . , qu,su}, u = 1, . . . , T ,
where s=s1+··· +sTis the cardinality of Bcrit
πif ∞/∈ Bcrit
π(otherwise, this
cardinality is s+ 1).
Similarly, let the disjoint N-orbits of the points in B∗
πbe given by
(31) O∗
k={pk,1, . . . , pk,|N|}, k = 1, . . . , L,
(so, L|N|is the cardinality of B∗
π).
As, for k= 1, . . . , L,
Q|N|
j=1(x−pk,j ) = P(x)−f(pk,1)Q(x),
our curve can be written as
(32) X:yn=QT
u=1 Ru(x)b
luQL
k=1 (P(x)−f(pk,1)Q(x))e
lk,
where
(1) Ru(x) = Qsu
j=1(x−qu,j )
(2) b
lu∈ {0,1, . . . , n −1}and e
lk∈ {1, . . . , n −1}.
(3) gcd(n, b
l1, . . . b
lT,e
l1,...,e
lL) = 1 (where we eliminate a zero if appears).

12 RUB´
EN A. HIDALGO, SA´
UL QUISPE, AND TONY SHASKA
(4) if ∞/∈ Bcrit
π, then
PT
u=1 sub
lu+PL
k=1 |N|e
lk≡0 mod n.
(5) If ∞∈Ocr it
v, then
(1 + sv)b
lv+PT
u=1,u6=vsub
lu+PL
k=1 |N|e
lk≡0 mod n.
6.1.2. Computing the elements of N.Let η∈Nand b=θ(η). As τcommutes
with η, by Lemma 1,η(x, y)=(b(x), F (x)y), where F(x)∈C(x). Below, we sketch
how to compute such F(x).
Lemma 2. Let O={a1, . . . , ar}a full N-orbit (in our case, this is one of the
Ocrit
uor O∗
k). If b∈N, then the following hold.
(1) If ∞/∈ O, then
Qr
j=1(b(x)−aj)=(b0(x))r/2Qr
j=1 b0(aj)1/2Qr
j=1(x−aj).
(2) If ar=∞and b(∞) = ∞, then
Qr−1
j=1(b(x)−aj) = (b0(x))(r−1)/2Qr−1
j=1 b0(aj)1/2Qr−1
j=1(x−aj).
(3) If ar=∞,b(ar−1) = ∞and b(∞) = as, where s6=r−1, and b(at) = ar−1,
then Qr−1
j=1(b(x)−aj) =
(ar−1−as)1/2(ar−1−at)1/2
x−ar−1(b0(x))(r−1)/2Qr−2
j=1 b0(aj)1/2Qr−1
j=1(x−aj).
(4) If ar=∞,b(ar−1) = ∞and b(∞) = ar−1, then
Qr−1
j=1(b(x)−aj) = −(b0(x))r/2Qr−2
j=1 b0(aj)1/2Qr−1
j=1(x−aj).
Proof. The equalites are consequence of the fact that, for a, b(a)∈C,
b(x)−b(a) = b0(x)1/2b0(a)1/2(x−a).

If, in the above lemma, we replace Oby Ocrit
u, then we obtain an equality
Ru(b(x)) = Qsu
j=1(b(x)−qu,j ) = Qu(x)Qsu
j=1(x−qu,j ) = Qu(x)Ru(x).
Similarly, if we replace Oby O∗
kand set
Sk(x) := P(b(x)) −f(pk,1)Q(b(x)) = Q|N|
j=1(x−pk,j ),
then we obtain an equality
Sk(b(x)) = Q|N|
j=1(b(x)−pk,j ) = Lk(x)Q|N|
j=1(x−pk,j ) = Lk(x)Sk(x).
It can be checked, by plugging directly into the equation for X, that
(33) F(x)n=QT
u=1 Qu(x)b
luQL
k=1 Lk(x)e
lk.

GENERALIZED SUPERELLIPTIC RIEMANN SURFACES 13
6.2. Explicit computations. Below, for each of the possibilities for N, we pro-
ceed to explicitly describe the above procedure. In the following, if lu>0, then we
set nu= gcd(n, lu).
Theorem 6. Let Xbe a generalized superelliptic curve of level n,τ∈G= Aut (X)
be a generalized superelliptic automorphism of order nand Nbe the normalizer of
H=hτiin G. Then, up to isomorphisms, X,τand Nare described as indicated
in the above cases.
¯
NEquation Genus
CmEq. (34)Eq. (35)
DmEq. (36)Eq. (37)
A4Eq. (38)Eq. (39)
S4Eq. (40)Eq. (41)
A5Eq. (42)Eq. (43)
Table 1. default
Proof. We will consider all cases one by one.
Case N ∼
=Cm: In this case, N=a(x) = ωmxand the curve Xhas the form
(34) X:yn=xl0(xm−1)l1Qr
j=2(xm−am
j)lj,
where (i) a2, . . . , ar∈C− {0,1}, am
i6=am
jand (ii) gcd(n, l0, l1, . . . , lr) = 1. If
α(x, y)=(ωmx, ωl0/n
my),
then
N=hτ, α :τn= 1, αm=τl0, τ α =ατi.
The signature of X/H is









0; n
n1,m
. . ., n
n1,..., n
nr,m
. . ., n
nr,if l0= 0, m Pr
j=1 lj≡0 mod (n),
0; n
n0,n
n1,m
. . ., n
n1,..., n
nr,m
. . ., n
nr,if l06= 0, l0+mPr
j=1 lj≡0 mod (n),
0; n
n0,n
nr+1 ,n
n1,m
. . ., n
n1,..., n
nr,m
. . ., n
nr,if l06= 0, l0+mPr
j=1 lj6≡ 0 mod (n),
where (in the last situation) lr+1 ∈ {1, . . . , n −1}is the class of −(l0+mPr
j=1 lj)
module n. The signature of X/N is









0; m, m, n
n1,n
n2,..., n
nr,if l0= 0, mPr
j=1 lj≡0 mod (n),
0; m, mn
n0,n
n1,n
n2,..., n
nr,if l06= 0, l0+mPr
j=1 lj≡0 mod (n),
0; mn
n0,mn
nr+1 ,n
n1,n
n2,..., n
nr,if l06= 0, l0+mPr
j=1 lj6≡ 0 mod (n),
The genus of Xis
(35)









1 + 1
2(rm −2)n−mPr
j=1 nj,if l0= 0, m Pr
j=1 lj≡0 mod (n),
1 + 1
2(rm −1)n−mPr
j=1 nj,if l06= 0, l0+mPr
j=1 lj≡0 mod (n),
1 + 1
2rmn −mPr
j=1 nj,if l06= 0, l0+mPr
j=1 lj6≡ 0 mod (n).

14 RUB´
EN A. HIDALGO, SA´
UL QUISPE, AND TONY SHASKA
Case N ∼
=Dm: In this case, Dm:= Da(x) = ωmx, b(x) = 1
xEand the curve Xhas
the form
(36)
X:yn=xl0(xm−1)lr+1 (xm+ 1)lr+2 Qr
j=1(x2m−(am
j+a−m
j)xm+ 1)lj,
where (i) a±m
i6=a±m
j6= 0,±1, (ii) 2l0+m(lr+1 +lr+2)+2m(l1+· ·· +lr)≡0
mod (n) and (iii) gcd(n, l0, l1, . . . , lr+2) = 1. If αand βare as follows
α(x, y) = (ωmx, ωl0/n
my), β(x, y) = 1
x,(−1)lr+1/n
x(2l0+m(lr+1+lr+2 +2(l1+···+lr)))/n y,
then
N=hτ, α, β :τn= 1, αm=τl0, β2=τlr+1 , τ α =ατ, τβ =βτi.
The signature of X/H is
0; n
n0,n
n0,n
nr+1 ,m
. . ., n
nr+1 ,n
nr+2 ,m
. . ., n
nr+2 ,n
n1,2m
. . ., n
n1,..., n
nr,2m
. . ., n
nr,
the signature of X/N is0; mn
n0,2n
nr+1 ,2n
nr+2 ,n
n1,n
n2,..., n
nr,
and the genus of Xis
(37) g= 1 + 1
22m(r+ 1)n−2n0−mnr+1 +nr+2 + 2 Pr
j=1 nj.
Case N ∼
=A4: In this case, A4:= Da(x) = −x, b(x) = i−x
i+xEand Xhas the form
(38)
X:yn=R1(x)lr+1 R2(x)lr+2 R3(x)lr+3 Qr
j=1 R1(x)3+ 12ibj√3R3(x)2lj,
where
R1(x) = x4−2i√3x2+ 1,
R2(x) = x4+ 2i√3x2+ 1,
R3(x) = x(x4−1),
f(x) = R1(x)3
−12i√3R3(x)2,
such that (i) bj6=bi∈C\ {0,1}, (ii) 4(lr+1 +lr+2) + 6lr+3 + 12(l1+· ·· +lr)≡0
mod (n), and (iii) gcd(n, l1, . . . , lr+3) = 1. If
α(x, y)=(−x, (−1)lr+3 /ny), β(x, y)=(b(x), F (x)y),
where
F(x) = 2(lr+1+lr+2 )/n (1−I√3)lr+1/n (1+I√3)lr+2/n (8i)lr+3 /n(−64)(l1+···+lr)/n
(x+i)(4(lr+1+lr+2 )+6lr+3+12(l1+···+lr))/n ,
then
N=hτ, α, β :τn= 1, α2=τlr+3 , β3=τlr+1 +lr+2 +lr+3+l1+···+lr,
(αβ)3=τlr+1 +lr+2+3lr+3 +l1+···+lr, τ α =ατ, τ β =βτ i.
The signature of X/H is
0; n
nr+1 ,4
. . ., n
nr+1 ,n
nr+2 ,4
. . ., n
nr+2 ,n
nr+3 ,6
. . ., n
nr+3 ,n
n1,12
. . ., n
n1,..., n
nr,12
. . ., n
nr,
the signature of X/N is
0; 3n
nr+1 ,3n
nr+2 ,2n
nr+3 ,n
n1,n
n2,..., n
nr,

GENERALIZED SUPERELLIPTIC RIEMANN SURFACES 15
and the genus of Xis
(39) g= 1 −n+ 2nr+1 + 2nr+2 + 3nr+2 −6Pr
j=1 nj.
Case N ∼
=S4: In this case, S4:= Da(x) = ix, b(x) = i−x
i+xEand Xhas the form
(40) X:yn=R1(x)lr+1 R2(x)lr+2 R3(x)lr+3 Qr
j=1(R1(x)3−108bjR3(x)4)lj,
where
R1(x) = x8+ 14x4+ 1,
R2(x) = x12 −33x8−33x4+ 1,
R3(x) = x(x4−1),
f(x) = R1(x)3
108R3(x)4,
such that (i) bj6=bi∈C\ {0,1}, (ii) 8lr+1 + 12lr+2 + 6lr+3 + 24(l1+·· · +lr)≡0
mod (n) and (iii) gcd(n, l1, . . . , lr+3) = 1. If
α(x, y) = (ix, ilr+3 /ny), β(x, y)=(b(x), F (x)y),
F(x) = 16lr+1/n ·(−64)lr+2 /n·(8i)lr+3 /n·4096(l1+···+lr))/n
(x+i)(8lr+1+12lr+2 +6lr+3+24(l1+···+lr))/n ,
then
N=hτ, α, β :τn= 1, α4=τlr+3 , β3=τlr+1 +lr+2 +lr+3+l1+···+lr,
(αβ)2=τ , τ α =ατ , τ β =βτ i.
The signature of X/H is
0; n
nr+1 ,8
. . ., n
nr+1 ,n
nr+2 ,12
. . ., n
nr+2 ,n
nr+3 ,6
. . ., n
nr+3 ,n
n1,24
. . ., n
n1,..., n
nr,24
. . ., n
nr,
the signature of X/N is
0; 3n
nr+1 ,2n
nr+2 ,4n
nr+3 ,n
n1,n
n2,..., n
nr,
and the genus of Xis
(41) g= 1 + 12(1 + r)n−4nr+1 −6nr+2 −3nr+3 −12 Pr
j=1 nj.
Case N ∼
=A5: In this case,
A5:= Da(x) = ω5x, b(x) = (1 −ω4
5)x+ (ω4
5−ω5)
(ω5−ω3
5)x+ (ω2
5−ω3
5)E
and Xhas the form
(42) X:yn=R1(x)lr+1 R2(x)lr+2 R3(x)lr+3 Qr
j=1(R1(x)3−1728bjR3(x)5)lj,
where
R1(x) = −x20 + 228x15 −494x10 −228x5−1,
R2(x) = x30 + 522x25 −10005x20 −10005x10 −522x5+ 1,
R3(x) = x(x10 + 11x5−1),
f(x) = R1(x)3
1728R3(x)5,
such that
(i) bj6=bi∈C\ {0,1},
(ii) 20lr+1 + 30lr+2 + 12lr+3 + 60(l1+· ·· +lr)≡0 mod (n),
(iii) gcd(n, l1, . . . , lr+3) = 1.

16 RUB´
EN A. HIDALGO, SA´
UL QUISPE, AND TONY SHASKA
N=hτ, α, β :α5=τlr+3 , β3=τl,(αβ)3=τti,
such that α(x, y)=(a(x), ωlr+3 /n
5y) and β(x, y)=(b(x), F (x)y), where F(x) is a
rational map satisfying
F(b2(x)) ·F(b(x)) ·F(x) = ωl
n,
for a suitable l∈ {0, . . . , n −1}, and
F(x)n=Tlr+1+3(l1+···+lr)
1(x)Tlr+2
2(x)Tlr+3
3(x),
where Tj(x) = Rj(b(x))
Rj(x), for j= 1,2,3.
The signature of X/H is
0; n
nr+1 ,20
. . ., n
nr+1 ,n
nr+2 ,30
. . ., n
nr+2 ,n
nr+3 ,12
. . ., n
nr+3 ,n
n1,60
. . ., n
n1,..., n
nr,60
. . ., n
nr,
and the signature of X/N is
0; 3n
nr+1 ,2n
nr+2 ,5n
nr+3 ,n
n1,n
n2,..., n
nr,
and the genus of Xis
(43) g= 1 + 30(r+ 1)n−10nr+1 −15nr+2 −6nr+3 −30 Pr
j=1 nj.

7. Appendix B: Computing cyclic n-gonal curves
Consider the collection Fgof all the tuples (n, s;n1, . . . , ns) satisfying the fol-
lowing Harvey’s conditions:
(1) n≥2, s≥3, 2 ≤n1≤n2≤ · ·· ≤ ns≤n;
(2) njis a divisor of n, for each j= 1, . . . , s;
(3) lcm (n1, . . . , nj−1, nj+1 , . . . , ns) = n, for every j= 1, . . . , s;
(4) if nis even, then #{j∈ {1, . . . , s}:n/njis odd}is even;
(5) 2(g−1) = ns−2−Ps
j=1 n−1
j.
For each tuple (n, s;n1, . . . , ns)∈ Fgwe consider the collection Fg(n, s;n1, . . . , ns)
of tuples (l1, . . . , ls) so that
(1) l1, . . . , ls∈ {1, . . . , n −1};
(2) l1+· ·· +ls≡0 mod (n);
(3) gcd(n, lj) = n/nj, for each j= 1, . . . , s.
Now, for each such tuple (l1, . . . , ls)∈ Fg(n, s;n1, . . . , ns) we may consider the
epimorphism
(44) θ:∆=hc1, . . . , cs:cn1
1=··· =cns
s=c1···cs= 1i → Cn=hτi:cj7→ τlj.
Our assumptions ensure that the kernel Γ = ker(θ) is a torsion free normal co-
compact Fuchsian subgroup of ∆ with X=H/Γ a closed Riemann surface of genus
gadmitting a cyclic group H∼
=Cnas a group of conformal automorphisms with
quotient orbifold X/H =H/∆; a genus zero orbifold with exactly scone points of
respective orders n1, . . . , ns. The surface Xcorresponds to a cyclic n-gonal curve
(45) C(n, s;l1, . . . , ls;p1, . . . , ps) : yn=Qs
j=1(x−pj)lj,
for pairwise different values p1, . . . , ps∈C, and Hgenerated by τ(x, y)=(x, ωny).
Different tuples (l1, . . . , ls),(l0
1, . . . , l0
s)∈ Fg(n, s;n1, . . . , ns) might provide iso-
morphic pairs (X, H) and (X0, H0) (i.e., there is an isomorphism between the Rie-
mann surfaces conjugating the cyclic groups). In general this is a difficult problem

GENERALIZED SUPERELLIPTIC RIEMANN SURFACES 17
to determine if different tuples define isomorphic pairs. But, in the non-exceptional
fully generalized superelliptic situation (see Theorem 1) the uniqueness of the su-
perelliptic cyclic group of level npermits us to see that (X, H ) and (X0, H0) are
isomorphic pairs if and only if the corresponding curves C(n, s;l1, . . . , ls;p1, . . . , ps)
and C(n, s;l0
1, . . . , l0
s;p0
1, . . . , p0
s) are isomorphic, this last being equivalent to the
existence of M¨obius transformation t∈PSL2(C), a permutation η∈Ssand an
element u∈ {1, . . . , n −1}with gcd(u, n) = 1, such that
(a) l0
j≡ulη(j)mod (n), for j= 1, . . . , s,
(b) p0
η(j)=t(pj), for j= 1, . . . , s.
The above (together with Theorem 2) may be used to construct all the possible
(generalized) superelliptic curves of lower genus in a similar fashion as done in [20]
for the superelliptic case.
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Departamento de Matem´
atica y Estad
´
ıstica, Universidad de La Frontera, Temuco,
Chile.
Email address:ruben.hidalgo@ufrontera.cl, saul.quispe@ufrontera.cl
Department of Mathematics, Oakland University, Rochester, MI, 48386.
Email address:shaska@oakland.edu