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ON FAMILIES OF RIEMANN SURFACES WITH AUTOMORPHISMS
MILAGROS IZQUIERDO, SEBASTI ´
AN REYES-CAROCCA, AND ANITA M. ROJAS
Dedicated to our friend Antonio F. Costa on the occasion of his 60th birthday
Abstract. In this article we determine the maximal possible order of the automorphism group of the
form ag +b, where aand bare integers, of a complex three and four-dimensional family of compact
Riemann surfaces of genus g, appearing for all genus. In addition, we construct and describe explicit
complex three and four-dimensional families possessing these maximal numbers of automorphisms.
1. Introduction
The classification of groups of automorphisms of compact Riemann surfaces is a classical subject of
study which has attracted broad interest ever since Schwarz and Hurwitz proved that the automorphism
group of a compact Riemann surface of genus g>2 is finite and its order is at most 84g−84.
Riemann surfaces of genus gwith a group of automorphisms of order of the form
ag +bwhere a, b are integers
can be found in the literature plentiful supply. The most classical example concerning that is the class
of the Riemann surfaces which possess exactly 84g−84 automorphisms; they are regular covers of the
projective line ramified over three values, marked with 2, 3 and 7. Another remarkable example is the
cyclic case. Wiman in [41] showed that the largest cyclic group of automorphisms of a Riemann surface
of genus g>2 has order at most 4g+ 2.Furthermore, the Riemann surface given by
y2=x2g+1 −1
shows that, for each value of g, this upper bound is attained; see also [18]. Kulkarni in [25] proved
that, for gsufficiently large, the aforementioned curve is the unique Riemann surface of genus gwith
an automorphism of order 4g+ 2.
Riemann surfaces with 4gautomorphisms have been classified in [9]; the Jacobian varieties of these
surfaces were studied in [32]. Riemann surfaces with 8(g+ 3) automorphisms were considered in [1] and
[28]. Under the assumption that g−1 is a prime number, the case ag −ahas been completely classified
in [5], [22], [23] and [33]. Recently, the case 3g−3 in which g−1 is assumed to be the square of a prime
number was classified in [11].
It is worth mentioning that the order of the automorphism group of a Riemann surface of genus g
need not be of the form ag +b. See, for instance, [16].
This paper is aimed to address the problem of determining the maximal possible order of the au-
tomorphism group of the form ag +b, where aand bare integers, of a family of Riemann surfaces of
genus g, appearing for all genus. To review known facts and to state the results of this paper, inspired
by Accola’s notation introduced in [1], we shall bring in the following definition.
2010 Mathematics Subject Classification. 30F10, 14H37, 14H30, 14H40.
Key words and phrases. Riemann surfaces, Fuchsian groups, Group actions, Jacobian varieties.
The first and second authors were partially supported by Redes Grant 2017-170071. The second author was partially
supported by Fondecyt Grants 11180024, 1190991. The third author was partially supported by Fondecyt Grant 1180073.
2 MILAGROS IZQUIERDO, SEBASTI ´
AN REYES-CAROCCA, AND ANITA M. ROJAS
Definition. For each d>0 and A⊆N− {1}we define Nd(g, A) to be the unique integer of the form
ag +bwhere a, b ∈Z, if exists, which satisfies:
(1) for each g∈Athere is a complex d-dimensional family of Riemann surfaces of genus gwith a
group of automorphisms of order Nd(g, A),and
(2) for at least one g∈A, there is no a complex d-dimensional family of Riemann surfaces of genus
gwith strictly more than Nd(g, A) automorphisms.
If A=N− {1}then we simply write Nd(g) instead of Nd(g, A).
In the sixties, Accola [1] and Maclachlan [28] considered the zero-dimensional case; namely, they
dealt with the problem of determining the largest order of the automorphism group of compact Riemann
surfaces appearing for all genus. Independently, they proved that
N0(g)=8g+ 8
by considering the Riemann surface given by the curve y2=x2g+2 −1.Later, the uniqueness problem
was addressed by Kulkarni in [25]. Concretely, he succeeded in proving that for g6≡ 3 mod 4 sufficiently
large, the aforementioned curve is the unique Riemann surface of genus gwith 8g+ 8 automorphisms.
The one-dimensional case was studied in [15]. For each g>2,there is a complex one-dimensional
family of Riemann surfaces of genus gwith a group of automorphisms isomorphic to Dg+1 ×C2and
N1(g)=4g+ 4.
The uniqueness problem was also studied by noticing that for g≡3 mod 4,there exists another one-
dimensional family with the same number of automorphisms. Besides, the two-dimensional case was
addressed in [33], where a classification of compact Riemann surfaces of genus gendowed with a maximal
non-large group of automorphisms was studied. By means of this classification, it was noticed that
N2(g)=4g−4
due to the existence, for all g>2,of a complex two-dimensional family of Riemann surfaces of genus g
with dihedral action. In addition, it was proved that if g−1 is a prime number then the aforementioned
family is the unique complex two-dimensional family with this number of automorphisms.
This article is devoted to extend the previous results by dealing with the complex three and four-
dimensional cases. Concretely, we first prove that the equality
N3(g)=2g−2
holds. We then observe that this case as well as the zero, one and two-dimensional cases are very much
in contrast with the four-dimensional situation. Indeed, we prove that if B={g:g>3}then
N4(g, B) does not exist.
In proving the non-existence of N4(g, B), we obtain the following facts, which are interesting in their
own right. If A1and A2consist of those values of g>3 that are odd and even respectively, then
N4(g, A1) = g−1 and N4(g, A2) = g.
The strategy to prove the results is to find upper bounds for the number of automorphisms and
then to construct in a very explicit manner complex three and four-dimensional families attaining these
bounds. After that, we study in detail these families; concretely:
(1) we address the uniqueness problem by providing conditions under which they turn into unique,
(2) we describe the families themselves as subsets of the moduli space, and
(3) we provide an isogeny decomposition of the corresponding families of Jacobian varieties.
ON FAMILIES OF RIEMANN SURFACES WITH AUTOMORPHISMS 3
Some computations will be done with the help of SageMath [38].
Section §2is devoted to review the basic preliminaries. The three-dimensional case will be considered
in Sections §3and §4. The four-dimensional case will be considered in Sections §5,§6and §7.
2. Preliminaries
2.1. Fuchsian groups and group actions. Let ∆ be a Fuchsian group, namely, a discrete group
of automorphisms of the upper half-plane H.If the orbit space H∆given by the action of ∆ on His
compact, then the algebraic structure of ∆ is determined by its signature; namely, the tuple
σ(∆) = (h;m1, . . . , ml),(2.1)
where hdenotes the genus of the quotient surface H∆and m1, . . . , mlthe branch indices in the associated
universal projection H→H∆.If l= 0 then it is said that ∆ is a surface Fuchsian group.
If ∆ is a Fuchsian group of signature (2.1) then ∆ has a canonical presentation with generators
α1, . . . , αh,β1, . . . , βh, x1, . . . , xland relations
xm1
1=· · · =xml
l= Πh
i=1[αi, βi]Πl
j=1xj= 1,(2.2)
where [u, v] stands for the commutator uvu−1v−1.The Teichm¨uller space of ∆ is a complex analytic
manifold homeomorphic to the complex ball of dimension 3h−3 + l.
Let ∆2be a group of automorphisms of H.If ∆ is a subgroup of ∆2of finite index then ∆2is also
Fuchsian. Moreover, if the signature of ∆2is (h2;n1, . . . , ns) then
2h−2+Σl
i=1(1 −1
mi) = [∆2: ∆](2h2−2+Σs
i=1(1 −1
ni)).
This equality is called the Riemann-Hurwitz formula. We refer to [19], [37] and [40] for more details.
Let Sbe a compact Riemann surface and let Aut(S) denote its automorphism group. A finite group
Gacts on Sif there is a group monomorphism G→Aut(S).The orbit space SGis endowed with a
Riemann surface structure such that the canonical projection S→SGis holomorphic.
By uniformization theorem, there is a surface Fuchsian group Γ such that Sand HΓare isomorphic.
Moreover, Riemann’s existence theorem ensures that Gacts on S∼
=HΓif and only if there is a Fuchsian
group ∆ containing Γ together with a group epimorphism
θ: ∆ →Gsuch that ker(θ)=Γ.
Note that SG∼
=H∆.It is said that Gacts on Swith signature σ(∆) and that this action is represented
by the surface-kernel epimorphism θ. For the sake of simplicity, we usually identify θwith the tuple of
its images or generating vector: (see, for example, [7] and [37])
θ= (θ(α1), . . . , θ(αh), θ(β1), . . . , θ(βh), θ(x1), . . . , θ(xl)).
2.2. Equivalence of actions. Let S1and S2be two compact Riemann surfaces of the same genus.
Two actions ψi:G→Aut(Si), for i= 1,2,are topologically equivalent if there exist ω∈Aut(G) and
an orientation-preserving homeomorphism f:S1→S2such that
ψ2(g) = fψ1(ω(g))f−1for all g∈G. (2.3)
Observe that if we write (Si)G∼
=H∆ithen each homeomorphism fsatisfying (2.3) induces an isomor-
phism f∗: ∆1→∆2. Thus, in order to describe the topological equivalence classes of a given group,
one may assume in the above that ∆1= ∆2.We shall write ∆ instead of ∆iand Sinstead of Si.
We denote the subgroup of Aut(∆) consisting of those f∗by B. It is known that θ1, θ2: ∆ →G
define topologically equivalent actions if and only if there are ω∈Aut(G) and f∗∈Bsuch that
4 MILAGROS IZQUIERDO, SEBASTI ´
AN REYES-CAROCCA, AND ANITA M. ROJAS
θ2=ω◦θ1◦f∗(see [7] and [19]). We recall for later use that, with the notations (2.2), if the genus h
of SGis zero, then Bis generated by the braid transformations Φidefined by:
Φi:xi7→ xi+1, xi+1 7→ x−1
i+1xixi+1 and xj7→ xjwhen j6=i, i + 1 (2.4)
for each i∈ {1, . . . , l −1}.Meanwhile, if h= 1, then, in addition to (2.4), Bcontains
A1,n :α17→ α1, β17→ β1αn
1, xj→xj, A2,n :α17→ α1βn
1, β17→ β1, xj→xj
where n∈Z,and the transformations
C1,i :α17→ x1α1, β17→ β1, xi7→ y1xiy−1
1, xj7→ xjfor each j6=i
C2,i :α17→ α1, β17→ x2β1, xi7→ y2xiy−1
2, xj7→ xjfor each j6=i
for i∈ {1, . . . , l}, where x1=β−1
1wz, y1=zβ−1
1w, x2=wzα1, y2=zα1w,w= Πk<i xkand z= Πk>ixk.
See, for example, [3], [7] and [19].
Let Bgdenote the locus of orbifold-singular points of the moduli space Mgof Riemann surfaces of
genus g. It was proved in [8] (see also [19]) that Bgadmits an equisymmetric stratification,
Bg=∪G,θ ¯
MG,θ
g
where the non-empty equisymmetric strata are in bijective correspondence with the topological classes
of actions that are maximal (in the sense of [39]). Concretely:
(1) the equisymmetric stratum MG,θ
gconsists of those Riemann surfaces Sof genus gwith (full)
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 Mgand consists of
those Riemann surfaces Sof genus gwith a group of automorphisms isomorphic to Gsuch that
the action is topologically equivalent to θ, and
(3) if the equisymmetric stratum MG,θ
gis non-empty then it is a smooth, connected, locally closed
algebraic subvariety of Mgwhich is Zariski dense in ¯
MG,θ
g.
Definition. Let Gbe a group and let σbe a signature. The subset of Mgconsisting of all those Riemann
surfaces of genus gendowed with a group of automorphisms isomorphic to Gacting with signature σ
will be called a closed family or simply a family.
We recall that:
(1) the complex dimension of the family agrees with the complex dimension of the Teichm¨uller
space associated to a Fuchsian group of signature σ,
(2) the interior of a family, if non-empty, consists of those Riemann surfaces whose (full) automor-
phism group is isomorphic to Gand is formed by finitely many equisymmetric strata which are
in correspondence with the pairwise non-equivalent topological actions of G, and
(3) the complement of the interior (with respect to the family) is formed by those Riemann surfaces
that have strictly more automorphisms than G.
Definition. A family is called equisymmetric if its interior consists of exactly one equisymmetric stratum.
2.3. Jacobian and Prym varieties. Let Sbe a compact Riemann surface of genus g>2.We denote
by H1(S, C)∗the dual of the g-dimensional complex vector space of 1-forms on S, and by H1(S, Z) the
first integral homology group of S. We recall that the Jacobian variety of S, defined by
JS =H1(S, C)∗/H1(S, Z),
is an irreducible principally polarized abelian variety of dimension g. The relevance of the Jacobian
variety lies, partially, in Torelli’s theorem, which establishes that two Riemann surfaces are isomorphic if
and only if the corresponding Jacobian varieties are isomorphic as principally polarized abelian varieties.
ON FAMILIES OF RIEMANN SURFACES WITH AUTOMORPHISMS 5
If H6Aut(S) then the associated regular covering map π:S→SHinduces a homomorphism
π∗:JSH→JS
between the associated Jacobians. The image of π∗is an abelian subvariety of JS isogenous to JSH.
Thereby, the classical Poincar´e’s Reducibility theorem implies that there exists an abelian subvariety
of JS, henceforth denoted by Prym(S→SH) and called the Prym variety associated to π, such that
JS ∼JSH×Prym(S→SH),
where ∼stands for isogeny. See [6] for more details.
If Gacts on a compact Riemann surface Sthen this action induces a Q-algebra homomorphism
ρ:Q[G]→EndQ(JS) = End(J S)⊗ZQ,
from the rational group algebra of Gto the rational endomorphism algebra of JS. For α∈Q[G], set
Aα:= Im(α) = ρ(nα)(JS)⊆J S
where nis a suitable positive integer chosen in such a way that nα ∈Z[G].
Let W1, . . . , Wrbe the rational irreducible representations of Gand for each Wldenote by Vla
complex irreducible representation of Gassociated to it. Following [26], the decomposition of 1 as the
sum e1+· · ·+erin Q[G],where elis central idempotent associated to Wl,yields a G-equivariant isogeny
JS ∼Ae1× · · · × Aer.
Moreover, for each lthere are idempotents fl1, . . . , flnlsuch that el=fl1+· · · +flnlwhere nl=dVl/sVl
is the quotient of the degree dVlof Vland its Schur index sVl. These idempotents provide nlpairwise
isogenous subvarieties of J S; let Blbe one of them, for each l. Thus, the following isogeny is obtained
JS ∼Bn1
1× · · · × Bnr
r(2.5)
and is called the group algebra decomposition of JS with respect to G. See [13] and also [35].
If W1denotes the trivial representation then n1= 1 and B1∼JSG.
If H6Gthen we denote by dH
Vlthe dimension of the vector subspace VH
lof Vlof the elements fixed
under H. Following [13], the group algebra decomposition (2.5) induces the following isogenies.
(1) The Jacobian variety JSHof the quotient SHdecomposes as
JSH∼BnH
1
1× · · · × BnH
r
rwhere nH
l=dH
Vl/sVl.(2.6)
(2) Let H16H2be subgroups of G. The Prym variety associated to SH1→SH2decomposes as
Prym(SH1→SH2)∼BnH1,H2
1
1× · · · × BnH1,H2
r
rwhere nH1,H2
l=nH1
l−nH2
l.(2.7)
The previous induced isogenies have been useful to provide decomposition of Jacobian varieties J S
whose factors are isogenous to Jacobians of quotients of Sand Pryms of intermediate coverings; see,
for example, [10], [11] and [34].
Assume that (γ;m1, . . . , ml) is the signature of the action of Gon Sand that this action is represented
by the surface-kernel epimorphism θ: ∆ →G, with ∆ canonically presented as in (2.2). As proved in
[37, Theorem 5.12], the dimension of Biin (2.5) for i>2 is given by
dim Bi=kVi[dVi(γ−1) + 1
2Σl
k=1(dVi−dhθ(xk)i
Vi)] (2.8)
where kViis the degree of the extension Q≤LViwith LVidenoting a minimal field of definition for Vi.
Note that the dimension of B1equals γ.
The decomposition of Jacobian varieties with group actions goes back to old works of Wirtinger,
Schottky and Jung. For decompositions of Jacobians with respect to special groups, we refer to the
articles [2], [12], [17], [20], [21], [27], [29], [30] and [36].
6 MILAGROS IZQUIERDO, SEBASTI ´
AN REYES-CAROCCA, AND ANITA M. ROJAS
Notation. We denote the cyclic group of order nby Cnand the dihedral group of order 2nby Dn.
3. The three-dimensional case
Theorem 1. N3(g)=2g−2.
The proof of the theorem will follow directly from Lemmata 3.1 and 3.2 stated and proved below.
Lemma 3.1. Let g>2be an integer. There are no complex three-dimensional families of compact
Riemann surfaces of genus gwith strictly more than 2(g−1) automorphisms.
Proof. Assume the existence of a complex three-dimensional family of Riemann surfaces Sof genus g
with a group of automorphisms Gof order strictly greater that 2(g−1).If the signature of the action
of Gon Sis (h;m1, . . . , ml) then, by the Riemann-Hurwitz formula, we have that
2(g−1) >2(g−1)[2h−2+Σl
j=1(1 −1
mj)],
or, equivalently, Σl
j=1 1
mj>2h+l−3.As the dimension 3h−3 + lof the family is assumed to be 3,
Σl
j=1 1
mj>1 + l
3where l∈ {0,3,6}.(3.1)
If l= 0 then (3.1) turns into 0 >1.Besides, if l= 3 or l= 6 then (3.1) turns into Σ3
j=1 1
mj>2 and
Σ6
j=1 1
mj>3 respectively. In both cases this contradicts the fact that each mjis at least 2.
Lemma 3.2. Let g>2be an integer. There is a complex three-dimensional family of compact Riemann
surfaces Sof genus gwith a group of automorphisms Gisomorphic to the dihedral group of order 2(g−1)
such that the signature of the action of Gon Sis (0; 2,6
. . ., 2).
Proof. Let ∆ be a Fuchsian group of signature (0;2,6
. . ., 2) with canonical presentation
∆ = hx1, . . . , x6:x2
1=· · · =x2
6=x1· · · x6= 1i,
and consider the dihedral group Dg−1=hr, s :rg−1=s2= (sr)2= 1i.Note that if g>3 then
∆→Dg−1given by x1, . . . , x47→ sand x5, x67→ sr (3.2)
is a surface-kernel epimorphism of signature (0; 2,6
. . ., 2). If g= 2 then the group is C2=hsiand the
surface-kernel epimorphism can be chosen to be xj7→ sfor each 1 6j66.In addition, for each g>2,
the equality
2(g−1) = 2(g−1)[0 −2 + 6(1 −1
2)]
shows that the Riemann-Hurwitz formula is satisfied for a 2(g−1)-fold regular covering map from a
Riemann surface of genus gonto the projective line with six branch values marked with 2.
Thus, the existence of the desired family follows from Riemann’s existence theorem.
Notation. From now on, we shall denote the family of all those surfaces Sof genus g>2 with a
group of automorphisms Gisomorphic to the dihedral group of order 2(g−1) such that the signature
of the action of Gon Sis (0; 2,6
. . ., 2) by Fg.
4. The family Fg
Proposition 1. Let g>3be an integer. If g−1is a prime number then Fgis the unique complex
three-dimensional family of compact Riemann surfaces of genus gwith 2(g−1) automorphisms.
Proof. Set q=g−1.Let Fbe a complex three-dimensional family of Riemann surfaces of genus gwith
a group of automorphisms Gof order 2q. By considering the Riemann-Hurwitz formula and by arguing
similarly as done in the proof of Lemma 3.1, one sees that the unique solution of
1=2h−2+Σl
j=1(1 −1
mj)
ON FAMILIES OF RIEMANN SURFACES WITH AUTOMORPHISMS 7
is h= 0, l = 6 and mj= 2 for each 1 6j66. Thus, the signature of the action of Gon each S∈ F
is necessarily equal to (0; 2,6
. . ., 2).If we now assume qto be prime then Gis isomorphic to either the
dihedral group or the cyclic group. We claim that the latter case is impossible. In fact, otherwise there
would exist a surface-kernel epimorphism ∆ →C2qwhere ∆ is a Fuchsian group of signature (0; 2,6
. . ., 2).
This, in turn, would imply that C2qcan be generated by involutions; a contradiction. It follows that G
is isomorphic to the dihedral group and therefore Fagrees with the family Fgas desired.
Proposition 2. Let g>4be an integer. If g−1is a prime number then Fgis equisymmetric.
Proof. Set q=g−1 and assume qto be prime. Let θ: ∆ →Dq=hr, s :rq=s2= (sr)2= 1ibe a
surface-kernel epimorphism representing an action of Gon S∈ Fg.What we need to prove is that θ
is equivalent to the surface-kernel epimorphism (3.2) of Lemma 3.2. To accomplish this task we shall
introduce some notation. We write
srnj=θ(xj) where j= 1,...,6 and nj∈ {0, . . . , q −1},
and if nj6= 0 then we shall denote by mjits inverse in the field of qelements. Also, we denote by φα,β
the automorphism of Dqgiven by (r, s)7→ (rα, srβ) for 1 6α6q−1 and 0 6β6q−1.
Claim 1. Let j∈ {1,...,5}fixed. If nj= 1 and nk= 0 for all k < j then, up to equivalence, we
can assume that nj+1 = 0 or nj+1 = 1.
Assume nj+1 6= 0.Then the transformation φmj+1,0◦Φjinduces the correspondence
(s, j−1
. . . , s, sr, srnj+1 )→(s, j−1
. . . , s, sr, srf(nj+1 )) where f(u)=2−1
u.
The claim follows by noting that the rule u7→ f(u) fixes 1 and has an orbit of length is q−1.
Claim 2. Up to equivalence, we can assume n1=n2= 0.
Note that if n1=n2then it is enough to consider φ1,−n1to obtain the claim. Thus, we shall assume
that n16=n2.If α:= (n2−n1)−1and β:= n1(n1−n2)−1(where the inverses are taken in the field of
qelements) then the automorphism φα,β ensures that, up to equivalence, n1= 0 and n2= 1.Now:
(a) if n3= 0 then Φ2shows that we can assume n1=n2= 0,and
(b) if n36= 0 then, by Claim 1, we can assume n3= 1.We now apply Φ2◦Φ1◦φ−1,1to obtain that,
up to equivalence, n1=n2= 0.
The proof of the claim is done.
We proceed by studying two cases separately, according to n3= 0 or n36= 0.
Type 1. Assume that n3= 0.
(a) If n4= 0 then necessarily n5and n6are equal and different from zero. We consider φm5,0to
obtain that θis equivalent to (3.2).
(b) If n46= 0 then, we consider φm4,0to assume n4= 1.Now, by Claim 1, we can ensure that
n5= 0 or n6= 1; thus, θis equivalent to either
θ1= (s, s, s, sr, s, sr−1) or θ2= (s, s, s, sr, sr, s).(4.1)
Note that θ1and θ2are equivalent under Φ5and that, in turn, θ1is equivalent to (3.2) under
the action of φ−1,0◦Φ3.
Type 2. Assume that n36= 0.As before, by considering the automorphism φm3,0,we can assume
n3= 1.It follows, by Claim 1, that n4= 0 or n4= 1.The first case can be disregarded, since φ−1,0◦Φ3
provides an equivalence with (4.1). Now, if n4= 1 then θis equivalent to
θu= (s, s, sr, sr, sru, sru) for some u∈ {0, . . . , q −1}.
8 MILAGROS IZQUIERDO, SEBASTI ´
AN REYES-CAROCCA, AND ANITA M. ROJAS
(1) if u6=±1 then define αuand βuby αu(1 −u)≡1 mod qand βu(1 + u)≡1 mod q. The
transformation φβu,0◦Φβu
4◦Φ5◦Φ3◦Φαu
4shows that θuis equivalent to (3.2).
(2) if u= 1 or u=−1 then we consider the transformations
Φ4◦Φ3◦Φ2◦Φ1◦Φ5◦Φ4◦Φ3◦Φ2◦φ−1,1and Φ4◦Φ2
5◦Φ3◦Φα
4
respectively (where 2α= 1) to see that θuis equivalent to (3.2).
The proof of the proposition is done.
We shall denote the equisymmetric stratum corresponding to the action (3.2) by Fg,1. With this
terminology the previous proposition can be rephrased as
g−1 odd prime =⇒ Fg,1=Fg.
In order to state the following proposition we need some notation. For each integer n>2 we write
Ω(n) = {d∈Z:ddivides nand 1 6d < n}
and for each n>2 even we write
ˆ
Ω(n) = {d∈Z:ddivides nand 1 6d < n
2}.
Let ϕdenote the Euler function.
Proposition 3. Let g>4be an integer.
(a) If g−1is odd and S∈ Fgthen JS decomposes, up to isogeny, as
JS ∼A×Πd∈Ω(g−1)B2
d
where Ais an abelian surface and Bdis an abelian variety of dimension 1
2ϕ(g−1
d).Moreover
JShri∼Aand JShsi∼Πd∈Ω(g−1)Bd
and therefore JS ∼JShri×JS2
hsi.
(b) If g−1is even and S∈ Fg,1then JS decomposes, up to isogeny, as
JS ∼E×A×Πd∈ˆ
Ω(g−1)B2
d,
where Ais an abelian surface, Bdis an abelian variety of dimension 1
2ϕ(g−1
d)and Eis an
elliptic curve. Moreover,
JShri∼A, JShsi∼Πd∈ˆ
Ω(g−1)Bdand J Shsri∼E×Πd∈ˆ
Ω(g−1)Bd
and therefore JS ∼JShri×JShsi×JShsri.
Proof. We write n:= g−1.We assume that nis odd. It is well-known that the complex irreducible
representations of Dn=hr, s :rn=s2= (sr)2= 1iare, up to equivalence:
(1) two of degree 1: the trivial representation denoted by χ1and χ2:r7→ 1, s 7→ −1.
(2) n−1
2of degree 2, given by
ψj:r7→ diag(ωj,¯ωj) and s7→ (0 1
1 0 ),
where ωis a primitive n-root of unity and j= 1,...,n−1
2.
For d∈Ω(n),we denote by Kdthe character field of ψd(an extension of Qof degree 1
2ϕ(n
d)) and define
Wd:= ⊕σ∈Gdψσ
d,(4.2)
where Gdstands for the Galois group associated to Q6Kd.Following for example [21, Section 2], up
to equivalence, the rational irreducible representations of Dnare χ1, χ2and Wdwith d∈Ω(n).
ON FAMILIES OF RIEMANN SURFACES WITH AUTOMORPHISMS 9
We recall that the Schur index of each representation of a dihedral group equals 1. Thus, if S∈ Fg
then the group algebra decomposition of J S with respect to Gis
JS ∼B2×Πd∈Ω(n)B2
d,(4.3)
where the factor B1is disregarded since the genus of SGis zero.
Note that as nis assumed to be odd, all the involutions of Dnare pairwise conjugate and therefore
the dimension of the corresponding fixed subspaces agree. This simple fact implies that the dimension
of each factor in (4.3) does not depend on the equisymmetric stratum to which Sbelongs. Then, in
order to apply the formula (2.8) we only need to compute the dimension of the fixed subspaces of χ2
and ψdunder the action of hsi.In the former case we have that
χhsi
2= 0 and therefore dim B2=−1 + 1
2(6(1 −0)) = 2,
meanwhile in the latter case, for each d∈Ω(n),we have
ψhsi
d= 1 and therefore dim Bd=1
2ϕ(n
d)(−2 + 1
2(6(2 −1)) = 1
2ϕ(n
d).
Finally, we apply the induced isogeny (2.6) with H=hriand H=hsito obtain that JShri∼B2
and JShsi∼Πd∈Ω(n)Bdrespectively. The proof of the statement (a) follows after setting A=B2.
We now assume that nis even and proceed analogously. The complex irreducible representations of
Dnare, up to equivalence, the trivial one χ1,
χ2:r7→ 1, s 7→ −1, χ3:r7→ −1, s 7→ 1 and χ4:r7→ −1, s 7→ −1.
and n
2−1 of degree 2, given by ψjwith j= 1,...,n
2−1.
Up to equivalence, the rational irreducible representations of Dnare χ1, χ2, χ3, χ4and Wdwith
d∈ˆ
Ω(n).If S∈ Fg,1then the group algebra decomposition of J S with respect to Gis
JS ∼B2×B3×B4×Πd∈ˆ
Ω(n)B2
d,
where, as before, B1is not considered. Note that
χhsi
2=χhsri
2= 0, χhsi
3= 1, χhsri
3= 0 and χhsi
4= 0, χhsri
4= 1
and for each d∈ˆ
Ω(n) we have that ψhsi
d=ψhsri
d= 1.Then, we apply (2.8) to conclude that
dim B2= 2,dim B3= 0,dim B4= 1 and dim Bd=1
2ϕ(n
d).
Finally, we consider the induced isogeny (2.6) with H=hri,H=hsiand H=hsrito obtain that
JShri∼B2, JShsi∼Πd∈ˆ
Ω(n)Bdand JShsri∼B4×Πd∈ˆ
Ω(n)Bdrespectively. The proof of the statement
(b) follows after setting E=B4and A=B2.
Remark 1. We end this section by pointing out some remarks concerning the family Fg.
(1) Note that if g−1 is an odd prime (or, more generally, odd) then Dg−1does not contain central
subgroups of order two. Thus, generically, each S∈ Fgis non-hyperelliptic.
(2) If g−1 is prime then Fgcorresponds to the family of Riemann surfaces of genus gthat are
cyclic unbranched covers of Riemann surfaces of genus two.
(3) The family F3consists of two equisymmetric strata: one of them represented by (3.2) and the
other represented by (s, s, r, r, sr, sr). See [7, Table 5, 3.h].
(4) If g−1 is not prime then Proposition 2is not longer true. For instance, if g−1 is even, then
θc:= (r(g−1)/2, r(g−1)/2, s, s, sr, sr) defines an action which is non-equivalent to (3.2).
10 MILAGROS IZQUIERDO, SEBASTI´
AN REYES-CAROCCA, AND ANITA M. ROJAS
(5) For each g>2,the (closed) family Fgcontains the complex two-dimensional family with the
maximal possible number of automorphisms (see [33]), which does not lie in the interior of Fg
(see Subsection §2.2 for the definition of interior; see also [8]).
(6) The group algebra decomposition of Jacobians of Riemann surfaces which belong to the same
family but lying in different strata may differ radically. For instance, if g≡3 mod 4 and S
belongs to the stratum defined by θc, then the group algebra decomposition of J S has three
factors of dimension one, instead of only one as in the stratum (3.2).
(7) Note that in Proposition 3(a) the Jacobian varieties JShsiand JShsriare isomorphic. Thus,
independently of the parity of g, if S∈ Fg,1then J S is isogenous to J Shri×JShsi×JShsri.
(8) Kani and Rosen in [24] provided conditions under which the Jacobian of a Riemann surface S
decomposes, up to isogeny, as a product of Jacobians of quotients of S. In spite of the fact that
JS ∼JShri×JShsi×JShsrifor each Sas in Proposition 3, the previous isogeny cannot be
derived from Kani-Rosen’s result (the reason is that hsiand hsrido not permute). It is worth
mentioning that the isogenies of Proposition 3(and the ones of Proposition 6stated later) can
be also obtained by applying the main result of [34].
(9) In [31] it was proved that the maximal order of a nilpotent group of automorphisms of a three-
dimensional family of Riemann surfaces of genus gis 2(g−1).If g−1 is a power of 2 then Dg−1
is nilpotent, showing that Fgattains the aforesaid upper bound for infinitely many values of g.
5. The four-dimensional case
Lemma 5.1. Let g>4be an even integer. If g−1is a prime number then there are no complex four-
dimensional families of compact Riemann surfaces of genus gwith strictly more than gautomorphisms.
Proof. Assume the existence of a complex four-dimensional family of Riemann surfaces of genus g
with a group of automorphisms Gof order strictly greater than g. If the signature of the action is
(h;m1, . . . , ml),then the Riemann-Hurwitz formula ensures that
2(g−1) > g(2h−2 + l−Σl
j=1 1
mj)
and, after straightforward computations, one can see that necessarily h= 0 and l= 7.Thus,
Σ7
j=1 1
mj>3 + 2
g(5.1)
If vis the number of periods mjthat are different from 2 then (5.1) implies that v∈ {0,1,2}.
If v= 0 then the signature of the action is (0; 2,7
. . ., 2) and the order of Gis 4
3(g−1).However, as
g−1 is assumed to be prime, we obtain that g= 4,and this contradicts the assumption that the order
of Gis strictly greater than the genus.
If v= 1 then the signature of the action is (0; 2,6
. . ., 2, a) for some a>3 which satisfies, by (5.1), the
inequality 2a < g. Note that the order of Gis 2a
2a−1(g−1),but, as g−1 is assumed to be prime, we see
that necessarily 2a=g; a contradiction.
Finally, if v= 2 then the signature of the action is (0; 2,5
. . ., 2, a, b) for some a, b >3 that, by (5.1),
satisfy 1
a+1
b>1
2.It follows that the signature of the action is either
(0; 2,5
. . ., 2,3,3),(0; 2,5
. . ., 2,3,4) or (0; 2,5
. . ., 2,3,5)
and, consequently, the order of Gis either 12
11 (g−1),24
23 (g−1) or 60
59 (g−1).
Note that, as before, the assumption that g−1 is prime, implies that gequals 12, 24 or 60 respectively.
The contradiction is obtained after noticing that, in every case, the order of Gagrees with the genus.
ON FAMILIES OF RIEMANN SURFACES WITH AUTOMORPHISMS 11
Lemma 5.2. For each even integer g>4,there is a complex four-dimensional family of compact
Riemann surfaces Sof genus gwith a group of automorphisms Gisomorphic to the dihedral group of
order gsuch that the signature of the action of Gon Sis (0; 2,6
. . ., 2,g
2).
Proof. Let ∆ be a Fuchsian group of signature (0;2,6
. . ., 2,g
2) with canonical presentation
∆ = hx1, . . . , x7:x2
1=· · · =x2
6=x
g
2
7=x1· · · x7= 1i,
and consider the dihedral group Dg
2=hr, s :rg
2=s2= (sr)2= 1iof order g. Note that
∆→Dg
2given by x1, . . . , x57→ s, x67→ sr−1and x77→ r
is a surface-kernel epimorphism of signature (0; 2,6
. . ., 2,g
2). In addition, the equality
2(g−1) = g[0 −2 + 6(1 −1
2) + (1 −2
g)]
shows that the Riemann-Hurwitz formula is satisfied for a g-fold regular covering map from a Riemann
surface of genus gonto the projective line with six branch values marked with 2 and with one branch
value marked with g
2. The existence of the family follows from Riemann’s existence theorem.
Notation. From now on, we shall denote the family of all those surfaces Sof genus g>4 with a
group of automorphisms Gisomorphic to the dihedral group of order gsuch that the signature of the
action of Gon Sis (0; 2,6
. . ., 2,g
2) by Vg.
Lemma 5.3. Let g>3be an odd integer. If g−1is a power of two then there are no complex
four-dimensional families of Riemann surfaces of genus gwith strictly more than g−1automorphisms.
Proof. Assume the existence of a complex four-dimensional family of Riemann surfaces of genus gwith
a group of automorphisms Gof order strictly greater than g−1,and denote the signature of the action
by (h;m1, . . . , ml).Then the Riemann-Hurwitz formula ensures that
4>2h+l−Σl
j=1 1
mj;
showing that h= 0 and l= 7,and consequently 3 <Σ7
j=1 1
mj67
2.By proceeding analogously as done
in the proof of Lemma 5.1 one sees that the signature is either
(0; 2,7
. . ., 2),(0; 2,6
. . ., 2, a),(0; 2,5
. . ., 2,3,3),(0; 2,5
. . ., 2,3,4) or (0; 2,5
. . ., 2,3,5)
for some a>3.It follows that the order of Gis either 4
3(g−1),2a
2a−1(g−1),12
11 (g−1),24
23 (g−1) or
60
59 (g−1).The contradiction is obtained after noticing that if g−1 is a power of 2,then the aforementioned
fractions are not integers.
Lemma 5.4. Let g>3be an odd integer. There are:
(a) a complex four-dimensional family of compact Riemann surfaces Sof genus gwith a group of
automorphisms Gisomorphic to the cyclic group of order g−1such that the signature of the
action of Gon Sis (1; 2,4
. . ., 2), and
(b) a complex four-dimensional family of compact Riemann surfaces Sof genus gwith a group of
automorphisms Gisomorphic to the dihedral group of order g−1such that the signature of the
action of Gon Sis (1; 2,4
. . ., 2).
Proof. Let ∆ be a Fuchsian group of signature (1;2,4
. . ., 2) with canonical presentation
∆ = hα1, β1, x1, x2, x3, x4:x2
1=x2
2=x2
3=x2
4=α1β1α−1
1β−1
1x1x2x3x4= 1i,
and consider the cyclic group Cg−1=ht:tg−1= 1iand the dihedral group Dg−1
2=hr, s :rg−1
2=s2=
(sr)2= 1i.The homomorphisms
∆→Cg−1given by α17→ t, β17→ 1, x1, x2, x3, x47→ tg−1
2(5.2)
12 MILAGROS IZQUIERDO, SEBASTI´
AN REYES-CAROCCA, AND ANITA M. ROJAS
∆→Dg−1
2given by α1, β17→ 1, x1, x27→ s, x3, x47→ sr
are surface-kernel epimorphisms of signature (1; 2,4
. . ., 2). In addition, the equality
2(g−1) = (g−1)[2 −2 + 4(1 −1
2)]
shows that the Riemann-Hurwitz formula is satisfied for a (g−1)-fold regular covering map from a
Riemann surface of genus gonto a Riemann surface of genus 1 with four branch values marked with 2.
Thus, the existence of the desired families follows from Riemann’s existence theorem.
Notation. From now on, we shall denote the family of all those surfaces Sof genus g>3 with a
group of automorphisms Gisomorphic to the cyclic group of order g−1 (to the dihedral group of order
g−1) such that the signature of the action of Gon Sis (1; 2,4
. . ., 2) by U1
g(by U2
g).
Theorem 2. If B={g∈N:g>3}then N4(g, B )does not exist.
Proof. We shall proceed by contradiction. Let us assume that N4(g, B) exists and that
N4(g, B) = ag +bfor suitable (and fixed) a, b ∈Z.
We claim that a= 1.Indeed:
(1) Clearly acannot be zero (consider Lemma 5.2 with g=b+ 1).
(2) If awere negative (and therefore bmust be positive) then for each g>−b
a+ 1 the number
N4(g, B) would be negative; a contradiction.
(3) If awere strictly greater than 1 then for
g > 2 if b>0
−b
a−1if b < 0
the number N4(g, B) would exceed g; this fact contradicts Lemmata 5.1 and 5.3.
Furthermore, by Lemma 5.3, we see that necessarily b6−1.
It follows that for every g>2,there is a complex four-dimensional family of Riemann surfaces Sof
genus gwith a group of automorphisms Gof order g+b. If the signature of the action of Gon Sis
(h;m1, . . . , ml) then each period mjmust equal g+b, since otherwise g≡ −bmod mjfor some mj,
contradicting the fact that the family exists for all g>2.In particular, we obtain that Gis necessarily
isomorphic to the cyclic group and the Riemann-Hurwitz formula implies that
2(g−1) = (g+b)[2h−2 + l(1 −1
g+b)].
Hence b= 1 −3
5gif h= 0, b =1
2(1 −g) if h= 1 and b=−1
3(g+ 1) if h= 2,showing that the existence
of the family fails to be true for all genus.
Once the non-existence of N4(g, B) has been proved, it makes sense to state the following theorem.
Theorem 3. Let A1={g∈N:g>3is odd}and A2={g∈N:g>4is even}.Then
N4(g, A1) = g−1and N4(g, A2) = g.
Proof. The proof follows directly from Lemmata 5.1,5.2,5.3 and 5.4.
Remark 2. It is worth observing that the phrase for at least one g∈Ajin the second statement of
the definition of N4(g, Aj) is not vacuous. Indeed, it is not a difficult task to verify the following facts.
(1) For each g>7 such that g≡3 mod 4 there exists a complex four-dimensional family of Riemann
surfaces of genus gwith a group of automorphisms isomorphic to the dihedral group of order
g+ 1 such that the signature of the action is (0; 2,6
. . ., 2,g+1
4).
ON FAMILIES OF RIEMANN SURFACES WITH AUTOMORPHISMS 13
(2) For each g>4 such that g≡4 mod 6 there exists a complex four-dimensional family of Riemann
surfaces of genus gwith a group of automorphisms isomorphic to the dihedral group of order
4
3(g−1) such that the signature of the action is (0; 2,7
. . ., 2).
6. The family Vg
Proposition 4. Let g>4be an even integer. If g
2is a prime number then Vgis the unique complex
four-dimensional family of compact Riemann surfaces of genus gwith gautomorphisms.
Proof. Let Vbe a complex four-dimensional family of Riemann surfaces Sof genus gwith a group of
automorphisms Gof order gacting with signature (h;m1, . . . , ml).As argued in the proof of Lemma
5.1, we observe that h= 0 and l= 7 and therefore
Σ7
j=1 1
mj= 3 + 2
g.(6.1)
We denote the number of periods mjthat are different from 2 by v. Clearly, v= 0 if and only if
g= 4.We now assume q=g
2to be prime and notice that this fact implies that if some mjis different
from 2 then mj>q. We claim that v= 1 provided that g>6.Indeed, if v>2 then (6.1) implies that
3 + 1
q6v
q+7−v
2⇐⇒ v−1
26v−1
q
and then g= 4.Thus, the only possible signature of the action of Gon Sis (0; 2,6
. . ., 2, q).
If Sdoes not belong to Vgthen G∼
=2q. However, this situation is impossible since there are no
surjective homomorphisms from a Fuchsian group of signature (0; 2,6
. . ., 2, q) onto C2q; thus V=Vg.
Proposition 5. Let g>6be an even integer such that g
2is prime. Then then family Vgconsists of at
most g+2
4equisymmetric strata.
Proof. Set g>6 such that q=g
2is prime. Let θ: ∆ →Dq=hr, s :rq=s2= (sr)2= 1ibe a
surface-kernel epimorphism representing an action of Gon S∈Vg,with ∆ canonically presented as
in the proof of Lemma 5.2. Similarly as done in the proof of Proposition 2we shall introduce some
notation. We write m∈ {1, . . . , q −1}and nj∈ {0, . . . , q −1}for j= 1,...,6 such that
srnj=θ(xj) for j= 1,...,6 and rm=θ(x7).
If nj6= 0 then we shall denote by mjits inverse in the field of qelements. The automorphism of Dq
given by (r, s)7→ (rα, srβ) is denoted by φα,β ,for 1 6α6q−1 and 0 6β6q−1.We also restate
two claims which were proved in the proof of Proposition 2.
Claim 1. If nj= 1 and nk= 0 for k < j then we can assume nj+1 = 0 or nj+1 = 1.
Claim 2. Up to equivalence, we can assume n1=n2= 0.
We shall proceed by studying separately the cases n3= 0 and n36= 0.
Type 1. Suppose n3= 0.
Assume n4= 0.If n56= 0 then we consider the automorphism φm5,0to notice that, up to equivalence,
n5= 1.Thus, θis equivalent to either (s, s, s, s, s, sru, r−u) or (s, s, s, s, sr, srv, r 1−v) where u6= 0 and
v6= 1,according to n5= 0 or n5= 1. Note that in the first case, as u6= 0,the epimorphism is equivalent
to the one in which u= 1; namely, equivalent to
(s, s, s, s, s, sr, r−1) (6.2)
Meanwhile, in the latter case, by Claim 2, the epimorphism is equivalent to
(s, s, s, s, sr, s, r) and, in turn, equivalent to (s, s, s, s, s, sr−1, r ).(6.3)
Now, the transformation φ−1,0◦Φ5provides an equivalence between (6.2) and (6.3).
14 MILAGROS IZQUIERDO, SEBASTI´
AN REYES-CAROCCA, AND ANITA M. ROJAS
Assume n46= 0.We then consider the automorphism φm4,0to notice that, up to equivalence, n4= 1
and, consequently, by Claim 2, we have that n5= 0 or n5= 1.Thereby, θis equivalent to either
(s, s, s, sr, s, sru, r−1−u) or θv= (s, s, s, sr, sr, srv, r−v)
where u6=−1 and v6= 0.The first case is equivalent to (6.3); indeed, we can consider the transformation
φ−1,0◦Φ4to see that θis equivalent to (s, s, s, s, sr, sr−u, r1+u) and therefore, by Claim 2, we can assume
u= 0.For the second case, consider Φ6◦Φ6to notice that θvand θ−vare equivalent. It follows that
there are at most q−1
2pairwise non-equivalent actions given by
(s, s, s, sr, sr, srv, r−v) for some v∈ {1,..., q−1
2}.(6.4)
Type 2. Suppose n36= 0.As before, consider the automorphism φm3,0to assume n3= 1.Moreover,
again by Claim 2, we see that, up to equivalence, n4= 0 or n4= 1.However, we only need to
consider the case n4= 1 due to the fact that, if n4= 0 then the transformation φ−1,0◦Φ3provides an
equivalence between θand either (6.3) or some (6.4). Thus, we assume that n4= 1.If n5= 0 then the
transformation Φ4◦Φ5◦φ−1,0shows that the epimorphism is equivalent to either (6.3) or some (6.4).
Then, we can assume n56= 0 and therefore the epimorphism is equivalent to one of the form
θu,v = (s, s, sr, sr, sru, sru−v, rv)
where u, v ∈ {1, . . . , q −1}.Note that the powers of Φ5provide the equivalences θu,v ∼
=θu−λv,v where
λ∈ {1, . . . , q −1}.We choose λ=u−v
vto conclude that θis equivalent to
θv,v = (s, s, sr, sr, srv, s, rv) for some v∈ {1, . . . , q −1}.
Now, consider φ−1,0◦Φ3◦Φ4◦Φ5to conclude that θv,v is equivalent to (6.4).
All the above says that θis equivalent to either (6.2) or some (6.4). Hence Vgconsists of at most
q−1
2+ 1 = g+2
4equisymmetric strata, as claimed.
Proposition 6. Let g>6be an even integer such that g
2is odd. If S∈Vgthen the Jacobian variety
JS decomposes, up to isogeny, as
JS ∼A×Πd∈Ω( g
2)B2
d,
where Ais an abelian surface and Bdis an abelian variety of dimension ϕ(g
2d).Moreover
A∼JShriand Πd∈Ω( g
2)Bd∼JShsi
and therefore JS ∼JShri×JS2
hsi.
Proof. Let S∈Vgwith g>6 and n=g
2odd. As noticed in the proof of Proposition 3and keeping
the same notations as in there, the non-trivial rational irreducible representations of Dnare χ2and Wd
with d∈Ω(n) and therefore the group algebra decomposition of each JS with respect to Gis given by
JS ∼B2×Πd∈Ω(n)B2
d.(6.5)
The fact that the involutions of Dnare conjugate implies that the dimension of B2and Bdin
(6.5) does not depend on the stratum to which Sbelongs. So, we assume the action of Gon Sto be
represented by (s, s, s, s, s, rs, r).Consider the equation (2.8) to see that dim B2= 2 and dim Bd=ϕ(n
d).
In addition, we consider the induced isogeny (2.6) with H=hriand H=hsito see that
B2∼JShriand Πd∈Ω(n)Bd∼JShsi
respectively, and therefore the proof follows after setting A=B2.
Remark 3. We end this section by remarking two facts concerning the family Vg.
ON FAMILIES OF RIEMANN SURFACES WITH AUTOMORPHISMS 15
(1) The behavior for g= 4 is completely different. Indeed, as noticed in [14] (see also [4]) the family
V4consists of two strata, represented by θ1= (r, r, r, r, r, s, sr) and θ2= (r, r, r, s, s, s, sr).By
proceeding analogously as done in the proof of Proposition 6, one sees that if S∈V4then:
(a) if Sbelongs to the stratum defined by θ1then JS ∼A1×A2,where A1∼J Shsiand
A2∼JShsriare abelian surfaces, and
(b) if Sbelongs to the stratum defined by θ2then JS ∼E1×E2×A, where E1∼JShriand
E2∼JShsiare elliptic curves and A∼JShsriis an abelian surface.
(2) As the reader could expect, if n=g
2is even then Propositions 5and 6are not longer true. For
instance, the stratum defined by η= (rn
2, r n
2, s, s, s, rs, r) is not equivalent to any of the actions
determined in Proposition 5. Furthermore, if Sbelongs to the stratum defined by ηthen, by
proceeding as in the proof of Proposition 6, one sees that if n≡0 mod 4 then JS contains two
elliptic curves, and if n≡2 mod 4 then JS contains two elliptic curves and an abelian surface.
7. The families U1
gand U2
g
Proposition 7. Let g>11 be an odd integer such that g−1is twice a prime number. Then U1
gand
U2
gare the unique complex four-dimensional families with g−1automorphisms.
Proof. Let g>11 be an odd integer and write g−1=2qwhere q>5 is a prime number. As the cyclic
and dihedral group are the unique groups of order 2q, we only need to verify that (1; 2,4
. . ., 2) is the
only possible signature for the action of a group Gof order 2qon a complex-four dimensional family of
Riemann surfaces of genus 1 + 2q.
A short computation shows that if signature of the action is not (1; 2,4
. . ., 2) then it is (h;m1, . . . , ml)
where either (h, l) = (2,1) or (h, l) = (0,7).It is straightforward to see that the former case is impossible.
So, we assume (h, l) = (0,7) and then Σ7
j=1 1
mj= 3.As argued in the proof of Lemma 5.1 and 5.3, one
sees that the number vof periods mjthat are different from 2 are either two of three.
(1) If v= 2 then the signature of the action is (0; 2,5
. . ., 2, a, b) where a, b >3 satisfy
1
a+1
b=1
2and therefore a=b= 4 or a= 3, b = 6.
(2) If v= 3 then the signature of the action is (0; 2,4
. . ., 2, a, b, c) where a, b, c >3 satisfies
1
a+1
b+1
c= 1 and therefore a=b=c= 3.
It follows that the order of the group is divisible by 3,4 or 6. Thus, q= 2 or 3 and therefore the
genus equals g= 5 or g= 7; a contradiction.
Remark 4. The exceptional signatures appearing in the proof of the proposition above are realized for
the unconsidered cases g= 5 and 7 (see, for example, [4, Lemma 8] for g= 5).
Proposition 8. Let g>3be an odd integer. The family U1
gis equisymmetric.
Proof. For g= 3 we refer to [7, Table 5, 3.b]. Assume g>5.Let ∆ be a Fuchsian group of signature
(1; 2,4
. . ., 2) canonically presented as in the proof of Lemma 5.4 and let θ: ∆ →G=ht:tg−1= 1i
be a surface-kernel epimorphism representing an action of Gon S∈U1
g.We have to prove that θis
equivalent to the surface-kernel epimorphism (5.2) in the proof of Lemma 5.4. Clearly
θ(xj) = t(g−1)/2for j= 1,2,3,4.
If we write θ(α1) = tuand θ(β1) = tvthen as, θis surjective, without loss of generality, we can assume
u= 1.Consider the transformation A1,−v(see §2.2) to see that θis equivalent to (5.2), as desired.
For n>2 even, let Λ(n) = {16d < n
2:ddivides nand dn
26≡ 0 mod n}.
16 MILAGROS IZQUIERDO, SEBASTI´
AN REYES-CAROCCA, AND ANITA M. ROJAS
Proposition 9. Let g>3be an odd integer. If S∈U1
gthen the Jacobian variety JS decomposes, up
to isogeny, as follows.
(1) If g−1
2is even then
JS ∼E×Πd∈Λ(g−1)Bd
where Eis an elliptic curve isogenous to JSGand Bdis an abelian variety of dimension 2ϕ(g−1
d).
(2) If g−1
2is odd then
JS ∼E×A×Πd∈Λ(g−1)Bd
where Ais an abelian surface and Eand Bdare as before.
Proof. Set n=g−1 and let ωbe a primitive n-th root of unity. For each 0 6j6n−1, we denote
by χjthe complex irreducible representation of G=ht:tn= 1i=Cndefined as χj:t7→ ωj.After
a routine computation, one sees that the collection {χd}where 1 6d6n
2and ddivides nyields a
maximal collection of non-trivial rational irreducible representations of G, up to equivalence.
Let Bddenote the factor associated to χdin the group algebra decomposition of J S with respect to
G. Clearly, B0is an elliptic curve isogenous to JSG.In addition, we observe that
dim Bd=ϕ(n
d)1
2·4(1 −χhtn
2i
d)
and therefore Bd= 0 if and only if χj(tn
2) = ωnd
2= 1 or, equivalently nd
2≡0 mod n. Hence, the group
algebra decomposition of JS with respecto to Gis
JS ∼JSG×Bn
2×Πd∈Λ(n)Bd
where, for each d∈Λ(n),the dimension of Bdis 2ϕ(n
d).Finally, as χn
2(t) = −1 we see that
dim Bn
2=1
2·4(1 −χhtn
2i
n
2) = 2 if n
2is odd
0 if n
2is even
and the proof follows after setting E=B0and A=Bn
2when n
2is odd.
Proposition 10. Let g>5be an odd integer such that g−1
2is a prime number. Then the family U2
g
consists of at most two equisymmetric strata.
Proof. Set q=g−1
2and assume qto be prime. Let ∆ be a Fuchsian group of signature (1;2,4
. . ., 2)
canonically presented as in Lemma 5.4 and let θ: ∆ →G=Dq=hr, s :rq=s2= (sr)2= 1ibe a
surface-kernel epimorphism representing an action of Gon S∈U2
g.We write a=θ(α1), b =θ(β1) and
srni=θ(xi) where ni∈ {0, . . . , q −1}for i= 1,2,3,4 and identify θwith (a, b;srn1, srn2, srn3, srn4).
Claim. Up to equivalence, we can assume (a, b) = (1,1) or (a, b) = (1, r).
There are four cases to consider; namely (a, b) equals to either
(ru, rv),(sru, rv),(ru, srv) or (sru, srv) for some u, v ∈ {0, . . . , q −1}.
First of all, note the third and fourth cases can be disregarded, since A1,1◦A2,1and A1,1respectively
(see §2.2), transform them into the second case.
Assume that a=ruand b=rv.
(1) If u= 0 then, up to an automorphism, we can assume (a, b) = (1,1) or (1, r).
(2) If u6= 0 and ˜uis its inverse in the field of qelements, then the transformation φ−˜u,1◦A2,1◦
A1,−1◦A1,−v˜uallows us to assume that, up to equivalence, (a, b) = (1, r).
Assume that a=sruand b=rv
(1) If v= 0 then, up to an automorphism, we can assume (a, b) = (s, 1).
(2) If v6= 0 and ˆvis its inverse in the field of qelements, then φˆv,0◦A2,−uˆvshows that, up to
equivalence, we can assume (a, b)=(s, r).
ON FAMILIES OF RIEMANN SURFACES WITH AUTOMORPHISMS 17
The proof of the claim follows after noticing that the cases (s, r) and (1, s) are equivalent to the first
case under the action of the transformations C1,4and C2,4respectively.
If (a, b) = (1,1) then we can assume n1= 0 and n2= 1.Thus, θis equivalent to (1,1; s, sr, srn3, srn3−1).
Now, we apply transformation Φ3to see that θis equivalent to
(1,1; s, sr, sr, s) and therefore equivalent to (1,1; s, s, sr, sr).(7.1)
If (a, b) = (1, r) then we can assume n1= 0 and therefore θis equivalent to (1, r;s, srn2, srn3, srn4).
(1) If n2= 0 then n3=n4.If follows that θis equivalent to (1, r;s, s, s, s) or to
θj:= (1, rj;s, s, sr, sr) for some j∈ {1, . . . , q −1}.
The latter case is equivalent to (7.1), since C2,3◦C2,2identifies θjwith θj−1.
(2) If n26= 0 then θis equivalent to (1, rj;s, sr, srn3, srn3−1) for some j∈ {1, . . . , q −1}.Note that
Φ3shows that θis equivalent to (1, rj;s, sr, sr, s); then, we see that θis equivalent to θj.
All the above says that there are at most 2 strata given by Θ1= (1,1; s, s, sr, sr) and Θ2= (1, r;s, s, s, s).
Proposition 11. Let g>7be an odd integer such that g−1
2is odd. If S∈U2
gthen the Jacobian
variety JS decomposes, up to isogeny, as
JS ∼E×A×Πd∈Ω( g−1
2)B2
d,
where Ais an abelian surface, Bdis an abelian variety of dimension ϕ(g−1
2d)and Eis an elliptic curve
isogenous to JSG. Furthermore
Prym(Shri→SG)∼Aand Prym(Shsi→SG)∼Πd∈Ω( g−1
2)Bd
and therefore JS ∼JSG×Prym(Shri→SG)×Prym(Shsi→SG)2.
Proof. Set n=g−1
2and assume that nis odd. Similarly as noticed in the proof of Proposition 3(a),
the dimension of the factors arising in the group algebra decomposition of JS does not depend on the
stratum to which Sbelongs. So, we assume the action to be represented by Θ2.Now, keeping the same
notation as before, the rational irreducible representations of Gare χ1, χ2and ψdwith d∈Ω(n) and
JS ∼B1×B2×Πd∈Ω(n)B2
d,
where B1∼JSGis an elliptic curve. As χhsi
2= 0 and ψhsi
d= 1 for d∈Ω(n) one sees that dim B2= 2
and dim Bd=ϕ(n
d).Now, we consider the induced isogeny (2.7) with H1=hriand H2=Gto see that
B2∼Prym(Shri→SG).
Similarly, consider the induced isogeny (2.7) with H1=hsiand H2=Gto see that
Πd∈Ω(n)Bd∼Prym(Shsi→SG),
and the result follows by setting E=B1and A=B2.
Remark 5. The isogeny decomposition of the Jacobian varieties JS for S∈U2
5differs from the stated
in Proposition 11 for the case g>7.Furthermore, the decomposition depends on the stratum to which
Sbelongs (due to the fact that the involutions of D2are non-conjugate). Indeed
(1) If Sbelongs to the stratum represented by Θ1then JS ∼E1×E2×E3×A, where E1∼
JSG, E2∼JShsiand E3∼JShsriare elliptic curves and A∼JShriis an abelian surface.
(2) If Sbelongs to the stratum represented by Θ2then JS ∼E×A1×A2,where E∼JSGis an
elliptic curve and A1∼JShri, A2∼J Shsriare abelian surfaces.
18 MILAGROS IZQUIERDO, SEBASTI´
AN REYES-CAROCCA, AND ANITA M. ROJAS
Acknowledgments. The authors are very grateful to SageMath (www.sagemath.org) and its devel-
opers for generously providing a useful software which allows the authors to perform helpful experiments
throughout the preparation of this manuscript, towards obtaining general results. The authors are also
very grateful to the referee for useful suggestions and comments.
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Matematiska institutionen, Link¨
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E-mail address:milagros.izquierdo@liu.se
Departamento de Matem´
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´
ıstica, Universidad de La Frontera, Temuco, Chile.
E-mail address:sebastian.reyes@ufrontera.cl
Departamento de Matem´
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Nu˜
noa, Santiago, Chile.
E-mail address:anirojas@uchile.cl