# On families of Riemann surfaces with automorphisms

Milagros Izquierdo  Matematiska institutionen, Linköpings Universitet,
Linköping, Sweden<EMAIL_ADDRESS>,  Sebastián Reyes-Carocca
Departamento de Matemática y Estadística, Universidad de La Frontera, Temuco,
Chile<EMAIL_ADDRESS>and  Anita M. Rojas  Departamento de
Matemáticas, Facultad de Ciencias, Universidad de Chile, Ñuñoa, Santiago,
Chile<EMAIL_ADDRESS>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 $a$ and $b$ are 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.

###### Key words and phrases:

Riemann surfaces, Fuchsian groups, Group actions, Jacobian varieties

###### 2010 Mathematics Subject Classification:

30F10, 14H37, 14H30, 14H40

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.

##  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 $g\geqslant 2$ is finite and its order is at most $84g-84.$

Riemann surfaces of genus $g$ with a group of automorphisms of order of the
form

$ag+b\,\,\mbox{ where }a,b\mbox{ are integers}$

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 $g\geqslant 2$ has order at most
$4g+2.$ Furthermore, the Riemann surface given by

$y^{2}=x^{2g+1}-1$

shows that, for each value of $g,$ this upper bound is attained; see also
[18]. Kulkarni in [25] proved that, for $g$ sufficiently large, the
aforementioned curve is the unique Riemann surface of genus $g$ with an
automorphism of order $4g+2.$

Riemann surfaces with $4g$ automorphisms 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 $ag-a$ has 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 automorphism group of the form $ag+b,$ where $a$ and $b$ are
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.

######  Definition.

For each $d\geqslant 0$ and $A\subseteq\mathbb{N}-\\{1\\}$ we define
$N_{d}(g,A)$ to be the unique integer of the form $ag+b$ where
$a,b\in\mathbb{Z}$, if exists, which satisfies:

  1. (1)

for each $g\in A$ there is a complex $d$-dimensional family of Riemann
surfaces of genus $g$ with a group of automorphisms of order $N_{d}(g,A),$ and

  2. (2)

for at least one $g\in A,$ there is no a complex $d$-dimensional family of
Riemann surfaces of genus $g$ with strictly more than $N_{d}(g,A)$
automorphisms.

If $A=\mathbb{N}-\\{1\\}$ then we simply write $N_{d}(g)$ instead of
$N_{d}(g,A).$

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

$N_{0}(g)=8g+8$

by considering the Riemann surface given by the curve $y^{2}=x^{2g+2}-1.$
Later, the uniqueness problem was addressed by Kulkarni in [25]. Concretely,
he succeeded in proving that for $g\not\equiv 3\mbox{ mod }4$ sufficiently
large, the aforementioned curve is the unique Riemann surface of genus $g$
with $8g+8$ automorphisms.

The one-dimensional case was studied in [15]. For each $g\geqslant 2,$ there
is a complex one-dimensional family of Riemann surfaces of genus $g$ with a
group of automorphisms isomorphic to $\mathbf{D}_{g+1}\times C_{2}$ and

$N_{1}(g)=4g+4.$

The uniqueness problem was also studied by noticing that for $g\equiv 3\mbox{
mod }4,$ 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 $g$ endowed with a
maximal non-large group of automorphisms was studied. By means of this
classification, it was noticed that

$N_{2}(g)=4g-4$

due to the existence, for all $g\geqslant 2,$ 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

$N_{3}(g)=2g-2$

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 $B=\\{g:g\geqslant 3\\}$ then

$N_{4}(g,B)\,\,\mbox{does not exist.}$

In proving the non-existence of $N_{4}(g,B)$, we obtain the following facts,
which are interesting in their own right. If $A_{1}$ and $A_{2}$ consist of
those values of $g\geqslant 3$ that are odd and even respectively, then

$N_{4}(g,A_{1})=g-1\,\,\mbox{ and }\,\,N_{4}(g,A_{2})=g.$

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. (1)

we address the uniqueness problem by providing conditions under which they
turn into unique,

  2. (2)

we describe the families themselves as subsets of the moduli space, and

  3. (3)

we provide an isogeny decomposition of the corresponding families of Jacobian
varieties.

Some computations will be done with the help of SageMath [38].

Section §2 is devoted to review the basic preliminaries. The three-dimensional
case will be considered in Sections §3 and §4. The four-dimensional case will
be considered in Sections §5, §6 and §7.

##  2\. Preliminaries

###  2.1. Fuchsian groups and group actions

Let $\Delta$ be a Fuchsian group, namely, a discrete group of automorphisms of
the upper half-plane $\mathbb{H}.$ If the orbit space $\mathbb{H}_{\Delta}$
given by the action of $\Delta$ on $\mathbb{H}$ is compact, then the algebraic
structure of $\Delta$ is determined by its signature; namely, the tuple

$\sigma(\Delta)=(h;m_{1},\ldots,m_{l}),$ (2.1)

where $h$ denotes the genus of the quotient surface $\mathbb{H}_{\Delta}$ and
$m_{1},\ldots,m_{l}$ the branch indices in the associated universal projection
$\mathbb{H}\to\mathbb{H}_{\Delta}.$ If $l=0$ then it is said that $\Delta$ is
a surface Fuchsian group.

If $\Delta$ is a Fuchsian group of signature (2.1) then $\Delta$ has a
canonical presentation with generators $\alpha_{1},\ldots,\alpha_{h}$,
$\beta_{1},\ldots,\beta_{h},$ $x_{1},\ldots,x_{l}$ and relations

$x_{1}^{m_{1}}=\cdots=x_{l}^{m_{l}}=\Pi_{i=1}^{h}[\alpha_{i},\beta_{i}]\Pi_{j=1}^{l}x_{j}=1,$
(2.2)

where $[u,v]$ stands for the commutator $uvu^{-1}v^{-1}.$ The Teichmüller
space of $\Delta$ is a complex analytic manifold homeomorphic to the complex
ball of dimension $3h-3+l$.

Let $\Delta_{2}$ be a group of automorphisms of $\mathbb{H}.$ If $\Delta$ is a
subgroup of $\Delta_{2}$ of finite index then $\Delta_{2}$ is also Fuchsian.
Moreover, if the signature of $\Delta_{2}$ is $(h_{2};n_{1},\ldots,n_{s})$
then

$2h-2+\Sigma_{i=1}^{l}(1-\tfrac{1}{m_{i}})=[\Delta_{2}:\Delta](2h_{2}-2+\Sigma_{i=1}^{s}(1-\tfrac{1}{n_{i}})).$

This equality is called the Riemann-Hurwitz formula. We refer to [19], [37]
and [40] for more details.

Let $S$ be a compact Riemann surface and let $\mbox{Aut}(S)$ denote its
automorphism group. A finite group $G$ acts on $S$ if there is a group
monomorphism $G\to\operatorname{Aut}(S).$ The orbit space $S_{G}$ is endowed
with a Riemann surface structure such that the canonical projection $S\to
S_{G}$ is holomorphic.

By uniformization theorem, there is a surface Fuchsian group $\Gamma$ such
that $S$ and $\mathbb{H}_{\Gamma}$ are isomorphic. Moreover, Riemann’s
existence theorem ensures that $G$ acts on $S\cong\mathbb{H}_{\Gamma}$ if and
only if there is a Fuchsian group $\Delta$ containing $\Gamma$ together with a
group epimorphism

$\theta:\Delta\to G\,\,\mbox{ such that }\,\,\mbox{ker}(\theta)=\Gamma.$

Note that $S_{G}\cong\mathbb{H}_{\Delta}.$ It is said that $G$ acts on $S$
with signature $\sigma(\Delta)$ and that this action is represented by the
surface-kernel epimorphism $\theta$. For the sake of simplicity, we usually
identify $\theta$ with the tuple of its images or generating vector: (see, for
example, [7] and [37])

$\theta=(\theta(\alpha_{1}),\ldots,\theta(\alpha_{h}),\theta(\beta_{1}),\ldots,\theta(\beta_{h}),\theta(x_{1}),\ldots,\theta(x_{l})).$

###  2.2. Equivalence of actions

Let $S_{1}$ and $S_{2}$ be two compact Riemann surfaces of the same genus. Two
actions $\psi_{i}:G\to\mbox{Aut}(S_{i})$, for $i=1,2,$ are topologically
equivalent if there exist $\omega\in\mbox{Aut}(G)$ and an orientation-
preserving homeomorphism $f:S_{1}\to S_{2}$ such that

$\psi_{2}(g)=f\psi_{1}(\omega(g))f^{-1}\mbox{ for all }g\in G.$ (2.3)

Observe that if we write $(S_{i})_{G}\cong\mathbb{H}_{\Delta_{i}}$ then each
homeomorphism $f$ satisfying (2.3) induces an isomorphism
$f^{*}:\Delta_{1}\to\Delta_{2}$. Thus, in order to describe the topological
equivalence classes of a given group, one may assume in the above that
$\Delta_{1}=\Delta_{2}.$ We shall write $\Delta$ instead of $\Delta_{i}$ and
$S$ instead of $S_{i}.$

We denote the subgroup of $\mbox{Aut}(\Delta)$ consisting of those $f^{*}$ by
$\mathfrak{B}$. It is known that $\theta_{1},\theta_{2}:\Delta\to G$ define
topologically equivalent actions if and only if there are
$\omega\in\mbox{Aut}(G)$ and $f^{*}\in\mathfrak{B}$ such that
$\theta_{2}=\omega\circ\theta_{1}\circ f^{*}$ (see [7] and [19]). We recall
for later use that, with the notations (2.2), if the genus $h$ of $S_{G}$ is
zero, then $\mathfrak{B}$ is generated by the braid transformations $\Phi_{i}$
defined by:

$\Phi_{i}:x_{i}\mapsto x_{i+1},\hskip 8.5359ptx_{i+1}\mapsto
x_{i+1}^{-1}x_{i}x_{i+1}\hskip 8.5359pt\mbox{ and }\hskip 8.5359ptx_{j}\mapsto
x_{j}\mbox{ when }j\neq i,i+1$ (2.4)

for each $i\in\\{1,\ldots,l-1\\}.$ Meanwhile, if $h=1$, then, in addition to
(2.4), $\mathfrak{B}$ contains

$A_{1,n}:\alpha_{1}\mapsto\alpha_{1},\,\,\beta_{1}\mapsto\beta_{1}\alpha_{1}^{n},\,\,x_{j}\to
x_{j},\hskip
14.22636ptA_{2,n}:\alpha_{1}\mapsto\alpha_{1}\beta_{1}^{n},\,\,\beta_{1}\mapsto\beta_{1},\,\,x_{j}\to
x_{j}$

where $n\in\mathbb{Z},$ and the transformations

$C_{1,i}:\alpha_{1}\mapsto
x_{1}\alpha_{1},\,\,\beta_{1}\mapsto\beta_{1},\,\,x_{i}\mapsto
y_{1}x_{i}y_{1}^{-1},\,\,x_{j}\mapsto x_{j}\,\,\mbox{ for each }\,j\neq i$
$C_{2,i}:\alpha_{1}\mapsto\alpha_{1},\,\,\beta_{1}\mapsto
x_{2}\beta_{1},\,\,x_{i}\mapsto y_{2}x_{i}y_{2}^{-1},\,\,x_{j}\mapsto
x_{j}\,\,\mbox{ for each }\,j\neq i$

for $i\in\\{1,\ldots,l\\}$, where
$x_{1}=\beta_{1}^{-1}wz,y_{1}=z\beta_{1}^{-1}w,x_{2}=wz\alpha_{1},y_{2}=z\alpha_{1}w$,
$w=\Pi_{k<i}x_{k}$ and $z=\Pi_{k>i}x_{k}.$ See, for example, [3], [7] and
[19].

Let $\mathscr{B}_{g}$ denote the locus of orbifold-singular points of the
moduli space $\mathscr{M}_{g}$ of Riemann surfaces of genus $g.$ It was proved
in [8] (see also [19]) that $\mathscr{B}_{g}$ admits an equisymmetric
stratification,

$\mathscr{B}_{g}=\cup_{G,\theta}\bar{\mathscr{M}}_{g}^{G,\theta}$

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. (1)

the equisymmetric stratum ${\mathscr{M}}_{g}^{G,\theta}$ consists of those
Riemann surfaces $S$ of genus $g$ with (full) automorphism group isomorphic to
$G$ such that the action is topologically equivalent to $\theta$,

  2. (2)

the closure $\bar{\mathscr{M}}_{g}^{G,\theta}$ of
${\mathscr{M}}_{g}^{G,\theta}$ is a closed irreducible algebraic subvariety of
$\mathscr{M}_{g}$ and consists of those Riemann surfaces $S$ of genus $g$ with
a group of automorphisms isomorphic to $G$ such that the action is
topologically equivalent to $\theta$, and

  3. (3)

if the equisymmetric stratum ${\mathscr{M}}_{g}^{G,\theta}$ is non-empty then
it is a smooth, connected, locally closed algebraic subvariety of
$\mathscr{M}_{g}$ which is Zariski dense in
$\bar{\mathscr{M}}_{g}^{G,\theta}.$

######  Definition.

Let $G$ be a group and let $\sigma$ be a signature. The subset of
$\mathscr{M}_{g}$ consisting of all those Riemann surfaces of genus $g$
endowed with a group of automorphisms isomorphic to $G$ acting with signature
$\sigma$ will be called a closed family or simply a family.

We recall that:

  1. (1)

the complex dimension of the family agrees with the complex dimension of the
Teichmüller space associated to a Fuchsian group of signature $\sigma,$

  2. (2)

the interior of a family, if non-empty, consists of those Riemann surfaces
whose (full) automorphism group is isomorphic to $G$ and is formed by finitely
many equisymmetric strata which are in correspondence with the pairwise non-
equivalent topological actions of $G,$ and

  3. (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 $S$ be a compact Riemann surface of genus $g\geqslant 2.$ We denote by
$\mathscr{H}^{1}(S,\mathbb{C})^{*}$ the dual of the $g$-dimensional complex
vector space of 1-forms on $S,$ and by $H_{1}(S,\mathbb{Z})$ the first
integral homology group of $S.$ We recall that the Jacobian variety of $S,$
defined by

$JS=\mathscr{H}^{1}(S,\mathbb{C})^{*}/H_{1}(S,\mathbb{Z}),$

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.

If $H\leqslant\mbox{Aut}(S)$ then the associated regular covering map
$\pi:S\to S_{H}$ induces a homomorphism

$\pi^{*}:JS_{H}\to JS$

between the associated Jacobians. The image of $\pi^{*}$ is an abelian
subvariety of $JS$ isogenous to $JS_{H}$. Thereby, the classical Poincaré’s
Reducibility theorem implies that there exists an abelian subvariety of $JS,$
henceforth denoted by $\mbox{Prym}(S\to S_{H})$ and called the Prym variety
associated to $\pi,$ such that

$JS\sim JS_{H}\times\mbox{Prym}(S\to S_{H}),$

where $\sim$ stands for isogeny. See [6] for more details.

If $G$ acts on a compact Riemann surface $S$ then this action induces a
$\mathbb{Q}$-algebra homomorphism

$\rho:\mathbb{Q}[G]\to\mbox{End}_{\mathbb{Q}}(JS)=\mbox{End}(JS)\otimes_{\mathbb{Z}}\mathbb{Q},$

from the rational group algebra of $G$ to the rational endomorphism algebra of
$JS.$ For $\alpha\in{\mathbb{Q}}[G]$, set

$A_{\alpha}:={\textup{I}m}(\alpha)=\rho(n\alpha)(JS)\subseteq JS$

where $n$ is a suitable positive integer chosen in such a way that
$n\alpha\in{\mathbb{Z}}[G]$.

Let $W_{1},\ldots,W_{r}$ be the rational irreducible representations of $G$
and for each $W_{l}$ denote by $V_{l}$ a complex irreducible representation of
$G$ associated to it. Following [26], the decomposition of $1$ as the sum
$e_{1}+\cdots+e_{r}$ in $\mathbb{Q}[G],$ where $e_{l}$ is central idempotent
associated to $W_{l},$ yields a $G$-equivariant isogeny

$JS\sim A_{e_{1}}\times\cdots\times A_{e_{r}}.$

Moreover, for each $l$ there are idempotents $f_{l1},\dots,f_{ln_{l}}$ such
that $e_{l}=f_{l1}+\dots+f_{ln_{l}}$ where $n_{l}=d_{V_{l}}/s_{V_{l}}$ is the
quotient of the degree $d_{V_{l}}$ of $V_{l}$ and its Schur index $s_{V_{l}}$.
These idempotents provide $n_{l}$ pairwise isogenous subvarieties of $JS;$ let
$B_{l}$ be one of them, for each $l.$ Thus, the following isogeny is obtained

$JS\sim B_{1}^{n_{1}}\times\cdots\times B_{r}^{n_{r}}$ (2.5)

and is called the group algebra decomposition of $JS$ with respect to $G$. See
[13] and also [35].

If $W_{1}$ denotes the trivial representation then $n_{1}=1$ and $B_{1}\sim
JS_{G}$.

If $H\leqslant G$ then we denote by $d_{V_{l}}^{H}$ the dimension of the
vector subspace $V_{l}^{H}$ of $V_{l}$ of the elements fixed under $H.$
Following [13], the group algebra decomposition (2.5) induces the following
isogenies.

  1. (1)

The Jacobian variety $JS_{H}$ of the quotient $S_{H}$ decomposes as

$JS_{H}\sim B_{1}^{{n}_{1}^{H}}\times\cdots\times
B_{r}^{n_{r}^{H}}\,\,\,\mbox{ where
}\,\,\,{n}_{l}^{H}=d_{V_{l}}^{H}/s_{V_{l}}.$ (2.6)

  2. (2)

Let $H_{1}\leqslant H_{2}$ be subgroups of $G.$ The Prym variety associated to
$S_{H_{1}}\to S_{H_{2}}$ decomposes as

$\mbox{Prym}(S_{H_{1}}\to S_{H_{2}})\sim
B_{1}^{{n}_{1}^{H_{1},H_{2}}}\times\cdots\times
B_{r}^{{n}_{r}^{H_{1},H_{2}}}\,\,\,\mbox{ where
}\,\,\,{n}_{l}^{H_{1},H_{2}}=n_{l}^{H_{1}}-n_{l}^{H_{2}}.$ (2.7)

The previous induced isogenies have been useful to provide decomposition of
Jacobian varieties $JS$ whose factors are isogenous to Jacobians of quotients
of $S$ and Pryms of intermediate coverings; see, for example, [10], [11] and
[34].

Assume that $(\gamma;m_{1},\ldots,m_{l})$ is the signature of the action of
$G$ on $S$ and that this action is represented by the surface-kernel
epimorphism $\theta:\Delta\to G,$ with $\Delta$ canonically presented as in
(2.2). As proved in [37, Theorem 5.12], the dimension of $B_{i}$ in (2.5) for
$i\geqslant 2$ is given by

$\dim
B_{i}=k_{V_{i}}[d_{V_{i}}(\gamma-1)+\frac{1}{2}\Sigma_{k=1}^{l}(d_{V_{i}}-d_{V_{i}}^{\langle\theta(x_{k})\rangle})]$
(2.8)

where $k_{V_{i}}$ is the degree of the extension $\mathbb{Q}\leq L_{V_{i}}$
with $L_{V_{i}}$ denoting a minimal field of definition for $V_{i}.$ Note that
the dimension of $B_{1}$ equals $\gamma.$

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].

### Notation

We denote the cyclic group of order $n$ by $C_{n}$ and the dihedral group of
order $2n$ by $\mathbf{D}_{n}.$

##  3\. The three-dimensional case

######  Theorem 1.

$N_{3}(g)=2g-2.$

The proof of the theorem will follow directly from Lemmata 3.1 and 3.2 stated
and proved below.

######  Lemma 3.1.

Let $g\geqslant 2$ be an integer. There are no complex three-dimensional
families of compact Riemann surfaces of genus $g$ with strictly more than
$2(g-1)$ automorphisms.

###### Proof.

Assume the existence of a complex three-dimensional family of Riemann surfaces
$S$ of genus $g$ with a group of automorphisms $G$ of order strictly greater
that $2(g-1).$ If the signature of the action of $G$ on $S$ is
$(h;m_{1},\ldots,m_{l})$ then, by the Riemann-Hurwitz formula, we have that

$2(g-1)>2(g-1)[2h-2+\Sigma_{j=1}^{l}(1-\tfrac{1}{m_{j}})],$

or, equivalently, $\Sigma_{j=1}^{l}\tfrac{1}{m_{j}}>2h+l-3.$ As the dimension
$3h-3+l$ of the family is assumed to be 3,

$\Sigma_{j=1}^{l}\tfrac{1}{m_{j}}>1+\tfrac{l}{3}\,\,\mbox{ where
}\,\,l\in\\{0,3,6\\}.$ (3.1)

If $l=0$ then (3.1) turns into $0>1.$ Besides, if $l=3$ or $l=6$ then (3.1)
turns into $\Sigma_{j=1}^{3}\tfrac{1}{m_{j}}>2$ and
$\Sigma_{j=1}^{6}\tfrac{1}{m_{j}}>3$ respectively. In both cases this
contradicts the fact that each $m_{j}$ is at least 2. ∎

######  Lemma 3.2.

Let $g\geqslant 2$ be an integer. There is a complex three-dimensional family
of compact Riemann surfaces $S$ of genus $g$ with a group of automorphisms $G$
isomorphic to the dihedral group of order $2(g-1)$ such that the signature of
the action of $G$ on $S$ is $(0;2,\stackrel{{\scriptstyle 6}}{{\ldots}},2).$

###### Proof.

Let $\Delta$ be a Fuchsian group of signature $(0;2,\stackrel{{\scriptstyle
6}}{{\ldots}},2)$ with canonical presentation

$\Delta=\langle x_{1},\ldots,x_{6}:x_{1}^{2}=\cdots=x_{6}^{2}=x_{1}\cdots
x_{6}=1\rangle,$

and consider the dihedral group $\mathbf{D}_{g-1}=\langle
r,s:r^{g-1}=s^{2}=(sr)^{2}=1\rangle.$ Note that if $g\geqslant 3$ then

$\Delta\to\mathbf{D}_{g-1}\,\mbox{ given by }\,x_{1},\ldots,x_{4}\mapsto
s\,\mbox{ and }\,x_{5},x_{6}\mapsto sr$ (3.2)

is a surface-kernel epimorphism of signature $(0;2,\stackrel{{\scriptstyle
6}}{{\ldots}},2)$. If $g=2$ then the group is $C_{2}=\langle s\rangle$ and the
surface-kernel epimorphism can be chosen to be $x_{j}\mapsto s$ for each
$1\leqslant j\leqslant 6.$ In addition, for each $g\geqslant 2,$ the equality

$2(g-1)=2(g-1)[0-2+6(1-\tfrac{1}{2})]$

shows that the Riemann-Hurwitz formula is satisfied for a $2(g-1)$-fold
regular covering map from a Riemann surface of genus $g$ onto 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 $S$ of
genus $g\geqslant 2$ with a group of automorphisms $G$ isomorphic to the
dihedral group of order $2(g-1)$ such that the signature of the action of $G$
on $S$ is $(0;2,\stackrel{{\scriptstyle 6}}{{\ldots}},2)$ by
$\mathcal{F}_{g}$.

##  4\. The family $\mathcal{F}_{g}$

######  Proposition 1.

Let $g\geqslant 3$ be an integer. If $g-1$ is a prime number then
$\mathcal{F}_{g}$ is the unique complex three-dimensional family of compact
Riemann surfaces of genus $g$ with $2(g-1)$ automorphisms.

###### Proof.

Set $q=g-1.$ Let $\mathcal{F}$ be a complex three-dimensional family of
Riemann surfaces of genus $g$ with a group of automorphisms $G$ of 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

$1=2h-2+\Sigma_{j=1}^{l}(1-\tfrac{1}{m_{j}})$

is $h=0,l=6$ and $m_{j}=2$ for each $1\leqslant j\leqslant 6$. Thus, the
signature of the action of $G$ on each $S\in\mathcal{F}$ is necessarily equal
to $(0;2,\stackrel{{\scriptstyle 6}}{{\ldots}},2).$ If we now assume $q$ to be
prime then $G$ is 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 $\Delta\to C_{2q}$ where $\Delta$ is a
Fuchsian group of signature $(0;2,\stackrel{{\scriptstyle 6}}{{\ldots}},2).$
This, in turn, would imply that $C_{2q}$ can be generated by involutions; a
contradiction. It follows that $G$ is isomorphic to the dihedral group and
therefore $\mathcal{F}$ agrees with the family $\mathcal{F}_{g}$ as desired. ∎

######  Proposition 2.

Let $g\geqslant 4$ be an integer. If $g-1$ is a prime number then
$\mathcal{F}_{g}$ is equisymmetric.

###### Proof.

Set $q=g-1$ and assume $q$ to be prime. Let
$\theta:\Delta\to\mathbf{D}_{q}=\langle r,s:r^{q}=s^{2}=(sr)^{2}=1\rangle$ be
a surface-kernel epimorphism representing an action of $G$ on
$S\in\mathcal{F}_{g}.$ What we need to prove is that $\theta$ is equivalent to
the surface-kernel epimorphism (3.2) of Lemma 3.2. To accomplish this task we
shall introduce some notation. We write

$sr^{n_{j}}=\theta(x_{j})\,\mbox{ where }\,j=1,\ldots,6\,\mbox{ and
}\,n_{j}\in\\{0,\ldots,q-1\\},$

and if $n_{j}\neq 0$ then we shall denote by $m_{j}$ its inverse in the field
of $q$ elements. Also, we denote by $\phi_{\alpha,\beta}$ the automorphism of
$\mathbf{D}_{q}$ given by $(r,s)\mapsto(r^{\alpha},sr^{\beta})$ for
$1\leqslant\alpha\leqslant q-1$ and $0\leqslant\beta\leqslant q-1.$

Claim 1. Let $j\in\\{1,\ldots,5\\}$ fixed. If $n_{j}=1$ and $n_{k}=0$ for all
$k<j$ then, up to equivalence, we can assume that $n_{j+1}=0$ or $n_{j+1}=1.$

Assume $n_{j+1}\neq 0.$ Then the transformation
$\phi_{m_{j+1},0}\circ\Phi_{j}$ induces the correspondence

$(s,\stackrel{{\scriptstyle
j-1}}{{\ldots}},s,sr,sr^{n_{j+1}})\to(s,\stackrel{{\scriptstyle
j-1}}{{\ldots}},s,sr,sr^{f(n_{j+1})})\,\,\mbox{ where }f(u)=2-\tfrac{1}{u}.$

The claim follows by noting that the rule $u\mapsto f(u)$ fixes 1 and has an
orbit of length is $q-1.$

Claim 2. Up to equivalence, we can assume $n_{1}=n_{2}=0.$

Note that if $n_{1}=n_{2}$ then it is enough to consider $\phi_{1,-n_{1}}$ to
obtain the claim. Thus, we shall assume that $n_{1}\neq n_{2}.$ If
$\alpha:=(n_{2}-n_{1})^{-1}$ and $\beta:=n_{1}(n_{1}-n_{2})^{-1}$ (where the
inverses are taken in the field of $q$ elements) then the automorphism
$\phi_{\alpha,\beta}$ ensures that, up to equivalence, $n_{1}=0$ and
$n_{2}=1.$ Now:

  1. (a)

if $n_{3}=0$ then $\Phi_{2}$ shows that we can assume $n_{1}=n_{2}=0,$ and

  2. (b)

if $n_{3}\neq 0$ then, by Claim 1, we can assume $n_{3}=1.$ We now apply
$\Phi_{2}\circ\Phi_{1}\circ\phi_{-1,1}$ to obtain that, up to equivalence,
$n_{1}=n_{2}=0.$

The proof of the claim is done.

We proceed by studying two cases separately, according to $n_{3}=0$ or
$n_{3}\neq 0.$

Type 1. Assume that $n_{3}=0.$

  1. (a)

If $n_{4}=0$ then necessarily $n_{5}$ and $n_{6}$ are equal and different from
zero. We consider $\phi_{m_{5},0}$ to obtain that $\theta$ is equivalent to
(3.2).

  2. (b)

If $n_{4}\neq 0$ then, we consider $\phi_{m_{4},0}$ to assume $n_{4}=1.$ Now,
by Claim 1, we can ensure that $n_{5}=0$ or $n_{6}=1;$ thus, $\theta$ is
equivalent to either

$\theta_{1}=(s,s,s,sr,s,sr^{-1})\,\,\mbox{ or
}\,\,\theta_{2}=(s,s,s,sr,sr,s).$ (4.1)

Note that $\theta_{1}$ and $\theta_{2}$ are equivalent under $\Phi_{5}$ and
that, in turn, $\theta_{1}$ is equivalent to (3.2) under the action of
$\phi_{-1,0}\circ\Phi_{3}.$

Type 2. Assume that $n_{3}\neq 0.$ As before, by considering the automorphism
$\phi_{m_{3},0},$ we can assume $n_{3}=1.$ It follows, by Claim 1, that
$n_{4}=0$ or $n_{4}=1.$ The first case can be disregarded, since
$\phi_{-1,0}\circ\Phi_{3}$ provides an equivalence with (4.1). Now, if
$n_{4}=1$ then $\theta$ is equivalent to

$\theta_{u}=(s,s,sr,sr,sr^{u},sr^{u})\,\,\mbox{ for some
}\,\,u\in\\{0,\ldots,q-1\\}.$

  1. (1)

if $u\neq\pm 1$ then define $\alpha_{u}$ and $\beta_{u}$ by
$\alpha_{u}(1-u)\equiv 1\mbox{ mod }q$ and $\beta_{u}(1+u)\equiv 1\mbox{ mod
}q.$ The transformation
$\phi_{\beta_{u},0}\circ\Phi_{4}^{\beta_{u}}\circ\Phi_{5}\circ\Phi_{3}\circ\Phi_{4}^{\alpha_{u}}$
shows that $\theta_{u}$ is equivalent to (3.2).

  2. (2)

if $u=1$ or $u=-1$ then we consider the transformations

$\Phi_{4}\circ\Phi_{3}\circ\Phi_{2}\circ\Phi_{1}\circ\Phi_{5}\circ\Phi_{4}\circ\Phi_{3}\circ\Phi_{2}\circ\phi_{-1,1}\,\,\mbox{
and }\,\,\Phi_{4}\circ\Phi_{5}^{2}\circ\Phi_{3}\circ\Phi_{4}^{\alpha}$

respectively (where $2\alpha=1$) to see that $\theta_{u}$ is equivalent to
(3.2).

The proof of the proposition is done. ∎

We shall denote the equisymmetric stratum corresponding to the action (3.2) by
$\mathcal{F}_{g,1}$. With this terminology the previous proposition can be
rephrased as

$g-1\,\mbox{ odd prime }\implies\mathcal{F}_{g,1}=\mathcal{F}_{g}.$

In order to state the following proposition we need some notation. For each
integer $n\geqslant 2$ we write

$\Omega(n)=\\{d\in\mathbb{Z}:d\text{ divides }n\text{ and }1\leqslant d<n\\}$

and for each $n\geqslant 2$ even we write

$\hat{\Omega}(n)=\\{d\in\mathbb{Z}:d\text{ divides }n\text{ and }1\leqslant
d<\tfrac{n}{2}\\}.$

Let $\varphi$ denote the Euler function.

######  Proposition 3.

Let $g\geqslant 4$ be an integer.

  1. (a)

If $g-1$ is odd and $S\in\mathcal{F}_{g}$ then $JS$ decomposes, up to isogeny,
as

$JS\sim A\times\Pi_{d\in\Omega(g-1)}B_{d}^{2}$

where $A$ is an abelian surface and $B_{d}$ is an abelian variety of dimension
$\tfrac{1}{2}\varphi(\tfrac{g-1}{d}).$ Moreover

$JS_{\langle r\rangle}\sim A\,\,\mbox{ and }\,\,JS_{\langle
s\rangle}\sim\Pi_{d\in\Omega(g-1)}B_{d}$

and therefore $JS\sim JS_{\langle r\rangle}\times JS_{\langle s\rangle}^{2}.$

  2. (b)

If $g-1$ is even and $S\in\mathcal{F}_{g,1}$ then $JS$ decomposes, up to
isogeny, as

$JS\sim E\times A\times\Pi_{d\in\hat{\Omega}(g-1)}B_{d}^{2},$

where $A$ is an abelian surface, $B_{d}$ is an abelian variety of dimension
$\tfrac{1}{2}\varphi(\tfrac{g-1}{d})$ and $E$ is an elliptic curve. Moreover,

$JS_{\langle r\rangle}\sim A,\,\,JS_{\langle
s\rangle}\sim\Pi_{d\in\hat{\Omega}(g-1)}B_{d}\,\,\mbox{ and }\,\,JS_{\langle
sr\rangle}\sim E\times\Pi_{d\in\hat{\Omega}(g-1)}B_{d}$

and therefore $JS\sim JS_{\langle r\rangle}\times JS_{\langle s\rangle}\times
JS_{\langle sr\rangle}.$

###### Proof.

We write $n:=g-1.$ We assume that $n$ is odd. It is well-known that the
complex irreducible representations of $\mathbf{D}_{n}=\langle
r,s:r^{n}=s^{2}=(sr)^{2}=1\rangle$ are, up to equivalence:

  1. (1)

two of degree $1$: the trivial representation denoted by $\chi_{1}$ and
$\chi_{2}:r\mapsto 1,\,\,s\mapsto-1.$

  2. (2)

$\frac{n-1}{2}$ of degree $2$, given by

$\psi_{j}:r\mapsto\mbox{diag}(\omega^{j},\bar{\omega}^{j})\,\,\mbox{ and
}\,\,s\mapsto\left(\begin{smallmatrix}0&1\\\ 1&0\\\ \end{smallmatrix}\right),$

where $\omega$ is a primitive $n$-root of unity and
$j=1,\ldots,\tfrac{n-1}{2}.$

For $d\in\Omega(n),$ we denote by $K_{d}$ the character field of $\psi_{d}$
(an extension of $\mathbb{Q}$ of degree $\frac{1}{2}\varphi(\frac{n}{d})$) and
define

$W_{d}:=\oplus_{\sigma\in G_{d}}\psi_{d}^{\sigma},$ (4.2)

where $G_{d}$ stands for the Galois group associated to $\mathbb{Q}\leqslant
K_{d}.$ Following for example [21, Section 2], up to equivalence, the rational
irreducible representations of $\mathbf{D}_{n}$ are $\chi_{1},\chi_{2}$ and
$W_{d}$ with $d\in\Omega(n).$

We recall that the Schur index of each representation of a dihedral group
equals 1. Thus, if $S\in\mathcal{F}_{g}$ then the group algebra decomposition
of $JS$ with respect to $G$ is

$JS\sim B_{2}\times\Pi_{d\in\Omega(n)}B_{d}^{2},$ (4.3)

where the factor $B_{1}$ is disregarded since the genus of $S_{G}$ is zero.

Note that as $n$ is assumed to be odd, all the involutions of $\mathbf{D}_{n}$
are 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 $S$ belongs. Then,
in order to apply the formula (2.8) we only need to compute the dimension of
the fixed subspaces of $\chi_{2}$ and $\psi_{d}$ under the action of $\langle
s\rangle.$ In the former case we have that

$\chi_{2}^{\langle s\rangle}=0\,\,\mbox{ and therefore}\,\,\dim
B_{2}=-1+\tfrac{1}{2}(6(1-0))=2,$

meanwhile in the latter case, for each $d\in\Omega(n),$ we have

$\psi_{d}^{\langle s\rangle}=1\,\,\mbox{ and therefore}\,\,\dim
B_{d}=\tfrac{1}{2}\varphi(\tfrac{n}{d})(-2+\tfrac{1}{2}(6(2-1))=\tfrac{1}{2}\varphi(\tfrac{n}{d}).$

Finally, we apply the induced isogeny (2.6) with $H=\langle r\rangle$ and
$H=\langle s\rangle$ to obtain that $JS_{\langle r\rangle}\sim B_{2}$ and
$JS_{\langle s\rangle}\sim\Pi_{d\in\Omega(n)}B_{d}$ respectively. The proof of
the statement (a) follows after setting $A=B_{2}.$

We now assume that $n$ is even and proceed analogously. The complex
irreducible representations of $\mathbf{D}_{n}$ are, up to equivalence, the
trivial one $\chi_{1}$,

$\chi_{2}:r\mapsto 1,\,\,s\mapsto-1,\,\,\chi_{3}:r\mapsto-1,s\mapsto 1\,\mbox{
and }\,\chi_{4}:r\mapsto-1,s\mapsto-1.$

and $\frac{n}{2}-1$ of degree $2$, given by $\psi_{j}$ with
$j=1,\ldots,\tfrac{n}{2}-1.$

Up to equivalence, the rational irreducible representations of
$\mathbf{D}_{n}$ are $\chi_{1},\chi_{2},\chi_{3},\chi_{4}$ and $W_{d}$ with
$d\in\hat{\Omega}(n).$ If $S\in\mathcal{F}_{g,1}$ then the group algebra
decomposition of $JS$ with respect to $G$ is

$JS\sim B_{2}\times B_{3}\times
B_{4}\times\Pi_{d\in\hat{\Omega}(n)}B_{d}^{2},$

where, as before, $B_{1}$ is not considered. Note that

$\chi_{2}^{\langle s\rangle}=\chi_{2}^{\langle
sr\rangle}=0,\,\,\chi_{3}^{\langle s\rangle}=1,\,\chi_{3}^{\langle
sr\rangle}=0\,\,\mbox{ and }\,\,\chi_{4}^{\langle
s\rangle}=0,\,\chi_{4}^{\langle sr\rangle}=1$

and for each $d\in\hat{\Omega}(n)$ we have that $\psi_{d}^{\langle
s\rangle}=\psi_{d}^{\langle sr\rangle}=1.$ Then, we apply (2.8) to conclude
that

$\dim B_{2}=2,\,\,\dim B_{3}=0,\,\,\dim B_{4}=1\,\mbox{ and }\,\dim
B_{d}=\tfrac{1}{2}\varphi(\tfrac{n}{d}).$

Finally, we consider the induced isogeny (2.6) with $H=\langle r\rangle$,
$H=\langle s\rangle$ and $H=\langle sr\rangle$ to obtain that $JS_{\langle
r\rangle}\sim B_{2},JS_{\langle s\rangle}\sim\Pi_{d\in\hat{\Omega}(n)}B_{d}$
and $JS_{\langle sr\rangle}\sim B_{4}\times\Pi_{d\in\hat{\Omega}(n)}B_{d}$
respectively. The proof of the statement (b) follows after setting $E=B_{4}$
and $A=B_{2}.$ ∎

######  Remark 1.

We end this section by pointing out some remarks concerning the family
$\mathcal{F}_{g}.$

  1. (1)

Note that if $g-1$ is an odd prime (or, more generally, odd) then
$\mathbf{D}_{g-1}$ does not contain central subgroups of order two. Thus,
generically, each $S\in\mathcal{F}_{g}$ is non-hyperelliptic.

  2. (2)

If $g-1$ is prime then $\mathcal{F}_{g}$ corresponds to the family of Riemann
surfaces of genus $g$ that are cyclic unbranched covers of Riemann surfaces of
genus two.

  3. (3)

The family $\mathcal{F}_{3}$ consists 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. (4)

If $g-1$ is not prime then Proposition 2 is not longer true. For instance, if
$g-1$ is even, then $\theta_{c}:=(r^{(g-1)/2},r^{(g-1)/2},s,s,sr,sr)$ defines
an action which is non-equivalent to (3.2).

  5. (5)

For each $g\geqslant 2,$ the (closed) family $\mathcal{F}_{g}$ contains the
complex two-dimensional family with the maximal possible number of
automorphisms (see [33]), which does not lie in the interior of
$\mathcal{F}_{g}$ (see Subsection §2.2 for the definition of interior; see
also [8]).

  6. (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 $g\equiv 3\mbox{ mod }4$ and $S$ belongs to the stratum defined
by $\theta_{c}$, then the group algebra decomposition of $JS$ has three
factors of dimension one, instead of only one as in the stratum (3.2).

  7. (7)

Note that in Proposition 3(a) the Jacobian varieties $JS_{\langle s\rangle}$
and $JS_{\langle sr\rangle}$ are isomorphic. Thus, independently of the parity
of $g,$ if $S\in\mathcal{F}_{g,1}$ then $JS$ is isogenous to $JS_{\langle
r\rangle}\times JS_{\langle s\rangle}\times JS_{\langle sr\rangle}.$

  8. (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 $JS\sim JS_{\langle
r\rangle}\times JS_{\langle s\rangle}\times JS_{\langle sr\rangle}$ for each
$S$ as in Proposition 3, the previous isogeny cannot be derived from Kani-
Rosen’s result (the reason is that $\langle s\rangle$ and $\langle sr\rangle$
do not permute). It is worth mentioning that the isogenies of Proposition 3
(and the ones of Proposition 6 stated later) can be also obtained by applying
the main result of [34].

  9. (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 $g$
is $2(g-1).$ If $g-1$ is a power of 2 then $\mathbf{D}_{g-1}$ is nilpotent,
showing that $\mathcal{F}_{g}$ attains the aforesaid upper bound for
infinitely many values of $g.$

##  5\. The four-dimensional case

######  Lemma 5.1.

Let $g\geqslant 4$ be an even integer. If $g-1$ is a prime number then there
are no complex four-dimensional families of compact Riemann surfaces of genus
$g$ with strictly more than $g$ automorphisms.

###### Proof.

Assume the existence of a complex four-dimensional family of Riemann surfaces
of genus $g$ with a group of automorphisms $G$ of order strictly greater than
$g.$ If the signature of the action is $(h;m_{1},\ldots,m_{l}),$ then the
Riemann-Hurwitz formula ensures that

$2(g-1)>g(2h-2+l-\Sigma_{j=1}^{l}\tfrac{1}{m_{j}})$

and, after straightforward computations, one can see that necessarily $h=0$
and $l=7.$ Thus,

$\Sigma_{j=1}^{7}\tfrac{1}{m_{j}}>3+\tfrac{2}{g}$ (5.1)

If $v$ is the number of periods $m_{j}$ that are different from 2 then (5.1)
implies that $v\in\\{0,1,2\\}.$

If $v=0$ then the signature of the action is $(0;2,\stackrel{{\scriptstyle
7}}{{\ldots}},2)$ and the order of $G$ is $\tfrac{4}{3}(g-1).$ However, as
$g-1$ is assumed to be prime, we obtain that $g=4,$ and this contradicts the
assumption that the order of $G$ is strictly greater than the genus.

If $v=1$ then the signature of the action is $(0;2,\stackrel{{\scriptstyle
6}}{{\ldots}},2,a)$ for some $a\geqslant 3$ which satisfies, by (5.1), the
inequality $2a<g.$ Note that the order of $G$ is $\tfrac{2a}{2a-1}(g-1),$ 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
$(0;2,\stackrel{{\scriptstyle 5}}{{\ldots}},2,a,b)$ for some $a,b\geqslant 3$
that, by (5.1), satisfy $\tfrac{1}{a}+\tfrac{1}{b}>\tfrac{1}{2}.$ It follows
that the signature of the action is either

$(0;2,\stackrel{{\scriptstyle
5}}{{\ldots}},2,3,3),\,\,(0;2,\stackrel{{\scriptstyle
5}}{{\ldots}},2,3,4)\,\,\mbox{ or }\,\,(0;2,\stackrel{{\scriptstyle
5}}{{\ldots}},2,3,5)$

and, consequently, the order of $G$ is either
$\tfrac{12}{11}(g-1),\,\,\tfrac{24}{23}(g-1)\,\,\mbox{ or
}\,\,\tfrac{60}{59}(g-1).$

Note that, as before, the assumption that $g-1$ is prime, implies that $g$
equals 12, 24 or 60 respectively. The contradiction is obtained after noticing
that, in every case, the order of $G$ agrees with the genus. ∎

######  Lemma 5.2.

For each even integer $g\geqslant 4,$ there is a complex four-dimensional
family of compact Riemann surfaces $S$ of genus $g$ with a group of
automorphisms $G$ isomorphic to the dihedral group of order $g$ such that the
signature of the action of $G$ on $S$ is $(0;2,\stackrel{{\scriptstyle
6}}{{\ldots}},2,\frac{g}{2})$.

###### Proof.

Let $\Delta$ be a Fuchsian group of signature $(0;2,\stackrel{{\scriptstyle
6}}{{\ldots}},2,\frac{g}{2})$ with canonical presentation

$\Delta=\langle
x_{1},\ldots,x_{7}:x_{1}^{2}=\cdots=x_{6}^{2}=x_{7}^{\frac{g}{2}}=x_{1}\cdots
x_{7}=1\rangle,$

and consider the dihedral group $\mathbf{D}_{\frac{g}{2}}=\langle
r,s:r^{\frac{g}{2}}=s^{2}=(sr)^{2}=1\rangle$ of order $g.$ Note that

$\Delta\to\mathbf{D}_{\frac{g}{2}}\,\mbox{ given by
}\,x_{1},\ldots,x_{5}\mapsto s,x_{6}\mapsto sr^{-1}\,\mbox{ and
}\,x_{7}\mapsto r$

is a surface-kernel epimorphism of signature $(0;2,\stackrel{{\scriptstyle
6}}{{\ldots}},2,\frac{g}{2})$. In addition, the equality

$2(g-1)=g[0-2+6(1-\tfrac{1}{2})+(1-\tfrac{2}{g})]$

shows that the Riemann-Hurwitz formula is satisfied for a $g$-fold regular
covering map from a Riemann surface of genus $g$ onto the projective line with
six branch values marked with 2 and with one branch value marked with
$\tfrac{g}{2}$. The existence of the family follows from Riemann’s existence
theorem. ∎

Notation. From now on, we shall denote the family of all those surfaces $S$ of
genus $g\geqslant 4$ with a group of automorphisms $G$ isomorphic to the
dihedral group of order $g$ such that the signature of the action of $G$ on
$S$ is $(0;2,\stackrel{{\scriptstyle 6}}{{\ldots}},2,\frac{g}{2})$ by
$\mathscr{V}_{g}.$

######  Lemma 5.3.

Let $g\geqslant 3$ be an odd integer. If $g-1$ is a power of two then there
are no complex four-dimensional families of Riemann surfaces of genus $g$ with
strictly more than $g-1$ automorphisms.

###### Proof.

Assume the existence of a complex four-dimensional family of Riemann surfaces
of genus $g$ with a group of automorphisms $G$ of order strictly greater than
$g-1,$ and denote the signature of the action by $(h;m_{1},\ldots,m_{l}).$
Then the Riemann-Hurwitz formula ensures that

$4>2h+l-\Sigma_{j=1}^{l}\tfrac{1}{m_{j}};$

showing that $h=0$ and $l=7,$ and consequently
$3<\Sigma_{j=1}^{7}\tfrac{1}{m_{j}}\leqslant\tfrac{7}{2}.$ By proceeding
analogously as done in the proof of Lemma 5.1 one sees that the signature is
either

$(0;2,\stackrel{{\scriptstyle 7}}{{\ldots}},2),\,(0;2,\stackrel{{\scriptstyle
6}}{{\ldots}},2,a),\,(0;2,\stackrel{{\scriptstyle
5}}{{\ldots}},2,3,3),\,(0;2,\stackrel{{\scriptstyle 5}}{{\ldots}},2,3,4)\mbox{
or }(0;2,\stackrel{{\scriptstyle 5}}{{\ldots}},2,3,5)$

for some $a\geqslant 3.$ It follows that the order of $G$ is either
$\tfrac{4}{3}(g-1),\,\tfrac{2a}{2a-1}(g-1),\,\tfrac{12}{11}(g-1),\,\tfrac{24}{23}(g-1)$
or $\tfrac{60}{59}(g-1).$ 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 $g\geqslant 3$ be an odd integer. There are:

  1. (a)

a complex four-dimensional family of compact Riemann surfaces $S$ of genus $g$
with a group of automorphisms $G$ isomorphic to the cyclic group of order
$g-1$ such that the signature of the action of $G$ on $S$ is
$(1;2,\stackrel{{\scriptstyle 4}}{{\ldots}},2)$, and

  2. (b)

a complex four-dimensional family of compact Riemann surfaces $S$ of genus $g$
with a group of automorphisms $G$ isomorphic to the dihedral group of order
$g-1$ such that the signature of the action of $G$ on $S$ is
$(1;2,\stackrel{{\scriptstyle 4}}{{\ldots}},2)$.

###### Proof.

Let $\Delta$ be a Fuchsian group of signature $(1;2,\stackrel{{\scriptstyle
4}}{{\ldots}},2)$ with canonical presentation

$\Delta=\langle\alpha_{1},\beta_{1},x_{1},x_{2},x_{3},x_{4}:x_{1}^{2}=x_{2}^{2}=x_{3}^{2}=x_{4}^{2}=\alpha_{1}\beta_{1}\alpha_{1}^{-1}\beta_{1}^{-1}x_{1}x_{2}x_{3}x_{4}=1\rangle,$

and consider the cyclic group $C_{g-1}=\langle t:t^{g-1}=1\rangle$ and the
dihedral group $\mathbf{D}_{\frac{g-1}{2}}=\langle
r,s:r^{\frac{g-1}{2}}=s^{2}=(sr)^{2}=1\rangle.$ The homomorphisms

$\Delta\to C_{g-1}\,\mbox{ given by }\,\alpha_{1}\mapsto t,\,\beta_{1}\mapsto
1,\,x_{1},x_{2},x_{3},x_{4}\mapsto t^{\frac{g-1}{2}}$ (5.2)
$\Delta\to\mathbf{D}_{\frac{g-1}{2}}\,\mbox{ given by
}\,\alpha_{1},\beta_{1}\mapsto 1,\,x_{1},x_{2}\mapsto s,\,x_{3},x_{4}\mapsto
sr$

are surface-kernel epimorphisms of signature $(1;2,\stackrel{{\scriptstyle
4}}{{\ldots}},2)$. In addition, the equality

$2(g-1)=(g-1)[2-2+4(1-\tfrac{1}{2})]$

shows that the Riemann-Hurwitz formula is satisfied for a $(g-1$)-fold regular
covering map from a Riemann surface of genus $g$ onto 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 $S$ of
genus $g\geqslant 3$ with a group of automorphisms $G$ isomorphic to the
cyclic group of order $g-1$ (to the dihedral group of order $g-1$) such that
the signature of the action of $G$ on $S$ is $(1;2,\stackrel{{\scriptstyle
4}}{{\ldots}},2)$ by $\mathscr{U}_{g}^{1}$ (by $\mathscr{U}_{g}^{2}$).

######  Theorem 2.

If $B=\\{g\in\mathbb{N}:g\geqslant 3\\}$ then $N_{4}(g,B)$ does not exist.

###### Proof.

We shall proceed by contradiction. Let us assume that $N_{4}(g,B)$ exists and
that

$N_{4}(g,B)=ag+b\,\,\mbox{ for suitable (and fixed) }\,\,a,b\in\mathbb{Z}.$

We claim that $a=1.$ Indeed:

  1. (1)

Clearly $a$ cannot be zero (consider Lemma 5.2 with $g=b+1$).

  2. (2)

If $a$ were negative (and therefore $b$ must be positive) then for each
$g\geqslant-\tfrac{b}{a}+1$ the number $N_{4}(g,B)$ would be negative; a
contradiction.

  3. (3)

If $a$ were strictly greater than 1 then for

$g>\left\\{\begin{array}[]{ll}\,\,\,2&\textrm{if $b\geqslant 0$}\\\
\tfrac{-b}{a-1}&\textrm{if $b<0$}\end{array}\right.$

the number $N_{4}(g,B)$ would exceed $g;$ this fact contradicts Lemmata 5.1
and 5.3.

Furthermore, by Lemma 5.3, we see that necessarily $b\leqslant-1.$

It follows that for every $g\geqslant 2,$ there is a complex four-dimensional
family of Riemann surfaces $S$ of genus $g$ with a group of automorphisms $G$
of order $g+b.$ If the signature of the action of $G$ on $S$ is
$(h;m_{1},\ldots,m_{l})$ then each period $m_{j}$ must equal $g+b,$ since
otherwise $g\equiv-b\mbox{ mod }m_{j}$ for some $m_{j},$ contradicting the
fact that the family exists for all $g\geqslant 2.$ In particular, we obtain
that $G$ is necessarily isomorphic to the cyclic group and the Riemann-Hurwitz
formula implies that

$2(g-1)=(g+b)[2h-2+l(1-\tfrac{1}{g+b})].$

Hence $b=1-\tfrac{3}{5}g$ if $h=0,$ $b=\tfrac{1}{2}(1-g)$ if $h=1$ and
$b=\tfrac{-1}{3}(g+1)$ if $h=2,$ showing that the existence of the family
fails to be true for all genus. ∎

Once the non-existence of $N_{4}(g,B)$ has been proved, it makes sense to
state the following theorem.

######  Theorem 3.

Let $A_{1}=\\{g\in\mathbb{N}:g\geqslant 3\mbox{ is odd}\\}$ and
$A_{2}=\\{g\in\mathbb{N}:g\geqslant 4\mbox{ is even}\\}.$ Then

$N_{4}(g,A_{1})=g-1\,\,\,\mbox{ and }\,\,\,\,N_{4}(g,A_{2})=g.$

###### 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 $g\in A_{j}$ in the
second statement of the definition of $N_{4}(g,A_{j})$ is not vacuous. Indeed,
it is not a difficult task to verify the following facts.

  1. (1)

For each $g\geqslant 7$ such that $g\equiv 3\mbox{ mod }4$ there exists a
complex four-dimensional family of Riemann surfaces of genus $g$ with a group
of automorphisms isomorphic to the dihedral group of order $g+1$ such that the
signature of the action is $(0;2,\stackrel{{\scriptstyle
6}}{{\ldots}},2,\tfrac{g+1}{4})$.

  2. (2)

For each $g\geqslant 4$ such that $g\equiv 4\mbox{ mod }6$ there exists a
complex four-dimensional family of Riemann surfaces of genus $g$ with a group
of automorphisms isomorphic to the dihedral group of order $\tfrac{4}{3}(g-1)$
such that the signature of the action is $(0;2,\stackrel{{\scriptstyle
7}}{{\ldots}},2)$.

##  6\. The family $\mathscr{V}_{g}$

######  Proposition 4.

Let $g\geqslant 4$ be an even integer. If $\tfrac{g}{2}$ is a prime number
then $\mathscr{V}_{g}$ is the unique complex four-dimensional family of
compact Riemann surfaces of genus $g$ with $g$ automorphisms.

###### Proof.

Let $\mathscr{V}$ be a complex four-dimensional family of Riemann surfaces $S$
of genus $g$ with a group of automorphisms $G$ of order $g$ acting with
signature $(h;m_{1},\ldots,m_{l}).$ As argued in the proof of Lemma 5.1, we
observe that $h=0$ and $l=7$ and therefore

$\Sigma_{j=1}^{7}\tfrac{1}{m_{j}}=3+\tfrac{2}{g}.$ (6.1)

We denote the number of periods $m_{j}$ that are different from 2 by $v$.
Clearly, $v=0$ if and only if $g=4.$ We now assume $q=\tfrac{g}{2}$ to be
prime and notice that this fact implies that if some $m_{j}$ is different from
2 then $m_{j}\geqslant q.$ We claim that $v=1$ provided that $g\geqslant 6.$
Indeed, if $v\geqslant 2$ then (6.1) implies that

$3+\tfrac{1}{q}\leqslant\tfrac{v}{q}+\tfrac{7-v}{2}\iff\tfrac{v-1}{2}\leqslant\tfrac{v-1}{q}$

and then $g=4.$ Thus, the only possible signature of the action of $G$ on $S$
is $(0;2,\stackrel{{\scriptstyle 6}}{{\ldots}},2,q).$

If $S$ does not belong to $\mathscr{V}_{g}$ then $G\cong 2q.$ However, this
situation is impossible since there are no surjective homomorphisms from a
Fuchsian group of signature $(0;2,\stackrel{{\scriptstyle 6}}{{\ldots}},2,q)$
onto $C_{2q}$; thus $\mathscr{V}=\mathscr{V}_{g}.$ ∎

######  Proposition 5.

Let $g\geqslant 6$ be an even integer such that $\tfrac{g}{2}$ is prime. Then
then family $\mathscr{V}_{g}$ consists of at most $\frac{g+2}{4}$
equisymmetric strata.

###### Proof.

Set $g\geqslant 6$ such that $q=\tfrac{g}{2}$ is prime. Let
$\theta:\Delta\to\mathbf{D}_{q}=\langle r,s:r^{q}=s^{2}=(sr)^{2}=1\rangle$ be
a surface-kernel epimorphism representing an action of $G$ on
$S\in\mathscr{V}_{g},$ with $\Delta$ canonically presented as in the proof of
Lemma 5.2. Similarly as done in the proof of Proposition 2 we shall introduce
some notation. We write $m\in\\{1,\ldots,q-1\\}$ and
$n_{j}\in\\{0,\ldots,q-1\\}$ for $j=1,\ldots,6$ such that

$sr^{n_{j}}=\theta(x_{j})\,\,\mbox{ for }\,\,j=1,\ldots,6\,\,\mbox{ and
}\,\,r^{m}=\theta(x_{7}).$

If $n_{j}\neq 0$ then we shall denote by $m_{j}$ its inverse in the field of
$q$ elements. The automorphism of $\mathbf{D}_{q}$ given by
$(r,s)\mapsto(r^{\alpha},sr^{\beta})$ is denoted by $\phi_{\alpha,\beta},$ for
$1\leqslant\alpha\leqslant q-1$ and $0\leqslant\beta\leqslant q-1.$ We also
restate two claims which were proved in the proof of Proposition 2.

Claim 1. If $n_{j}=1$ and $n_{k}=0$ for $k<j$ then we can assume $n_{j+1}=0$
or $n_{j+1}=1.$

Claim 2. Up to equivalence, we can assume $n_{1}=n_{2}=0.$

We shall proceed by studying separately the cases $n_{3}=0$ and $n_{3}\neq 0.$

Type 1. Suppose $n_{3}=0.$

Assume $n_{4}=0.$ If $n_{5}\neq 0$ then we consider the automorphism
$\phi_{m_{5},0}$ to notice that, up to equivalence, $n_{5}=1.$ Thus, $\theta$
is equivalent to either $(s,s,s,s,s,sr^{u},r^{-u})$ or
$(s,s,s,s,sr,sr^{v},r^{1-v})$ where $u\neq 0$ and $v\neq 1,$ according to
$n_{5}=0$ or $n_{5}=1$. Note that in the first case, as $u\neq 0,$ 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

$(s,s,s,s,sr,s,r)\,\,\mbox{ and, in turn, equivalent to
}\,\,(s,s,s,s,s,sr^{-1},r).$ (6.3)

Now, the transformation $\phi_{-1,0}\circ\Phi_{5}$ provides an equivalence
between (6.2) and (6.3).

Assume $n_{4}\neq 0.$ We then consider the automorphism $\phi_{m_{4},0}$ to
notice that, up to equivalence, $n_{4}=1$ and, consequently, by Claim 2, we
have that $n_{5}=0$ or $n_{5}=1.$ Thereby, $\theta$ is equivalent to either

$(s,s,s,sr,s,sr^{u},r^{-1-u})\,\,\mbox{ or
}\,\,\theta_{v}=(s,s,s,sr,sr,sr^{v},r^{-v})$

where $u\neq-1$ and $v\neq 0.$ The first case is equivalent to (6.3); indeed,
we can consider the transformation $\phi_{-1,0}\circ\Phi_{4}$ to see that
$\theta$ is equivalent to $(s,s,s,s,sr,sr^{-u},r^{1+u})$ and therefore, by
Claim 2, we can assume $u=0.$ For the second case, consider
$\Phi_{6}\circ\Phi_{6}$ to notice that $\theta_{v}$ and $\theta_{-v}$ are
equivalent. It follows that there are at most $\tfrac{q-1}{2}$ pairwise non-
equivalent actions given by

$(s,s,s,sr,sr,sr^{v},r^{-v})\,\,\mbox{ for some
}\,\,v\in\\{1,\ldots,\tfrac{q-1}{2}\\}.$ (6.4)

Type 2. Suppose $n_{3}\neq 0.$ As before, consider the automorphism
$\phi_{m_{3},0}$ to assume $n_{3}=1.$ Moreover, again by Claim 2, we see that,
up to equivalence, $n_{4}=0$ or $n_{4}=1.$ However, we only need to consider
the case $n_{4}=1$ due to the fact that, if $n_{4}=0$ then the transformation
$\phi_{-1,0}\circ\Phi_{3}$ provides an equivalence between $\theta$ and either
(6.3) or some (6.4). Thus, we assume that $n_{4}=1.$ If $n_{5}=0$ then the
transformation $\Phi_{4}\circ\Phi_{5}\circ\phi_{-1,0}$ shows that the
epimorphism is equivalent to either (6.3) or some (6.4). Then, we can assume
$n_{5}\neq 0$ and therefore the epimorphism is equivalent to one of the form

$\theta_{u,v}=(s,s,sr,sr,sr^{u},sr^{u-v},r^{v})$

where $u,v\in\\{1,\ldots,q-1\\}.$ Note that the powers of $\Phi_{5}$ provide
the equivalences $\theta_{u,v}\cong\theta_{u-\lambda v,v}$ where
$\lambda\in\\{1,\ldots,q-1\\}.$ We choose $\lambda=\tfrac{u-v}{v}$ to conclude
that $\theta$ is equivalent to

$\theta_{v,v}=(s,s,sr,sr,sr^{v},s,r^{v})\,\,\mbox{ for some
}\,\,v\in\\{1,\ldots,q-1\\}.$

Now, consider $\phi_{-1,0}\circ\Phi_{3}\circ\Phi_{4}\circ\Phi_{5}$ to conclude
that $\theta_{v,v}$ is equivalent to (6.4).

All the above says that $\theta$ is equivalent to either (6.2) or some (6.4).
Hence $\mathscr{V}_{g}$ consists of at most $\tfrac{q-1}{2}+1=\tfrac{g+2}{4}$
equisymmetric strata, as claimed. ∎

######  Proposition 6.

Let $g\geqslant 6$ be an even integer such that $\tfrac{g}{2}$ is odd. If
$S\in\mathscr{V}_{g}$ then the Jacobian variety $JS$ decomposes, up to
isogeny, as

$JS\sim A\times\Pi_{d\in\Omega(\frac{g}{2})}B_{d}^{2},$

where $A$ is an abelian surface and $B_{d}$ is an abelian variety of dimension
$\varphi(\frac{g}{2d}).$ Moreover

$A\sim JS_{\langle r\rangle}\,\,\mbox{ and
}\,\,\Pi_{d\in\Omega(\frac{g}{2})}B_{d}\sim JS_{\langle s\rangle}$

and therefore $JS\sim JS_{\langle r\rangle}\times JS_{\langle s\rangle}^{2}.$

###### Proof.

Let $S\in\mathscr{V}_{g}$ with $g\geqslant 6$ and $n=\tfrac{g}{2}$ odd. As
noticed in the proof of Proposition 3 and keeping the same notations as in
there, the non-trivial rational irreducible representations of
$\mathbf{D}_{n}$ are $\chi_{2}$ and $W_{d}$ with $d\in\Omega(n)$ and therefore
the group algebra decomposition of each $JS$ with respect to $G$ is given by

$JS\sim B_{2}\times\Pi_{d\in\Omega(n)}B_{d}^{2}.$ (6.5)

The fact that the involutions of $\mathbf{D}_{n}$ are conjugate implies that
the dimension of $B_{2}$ and $B_{d}$ in (6.5) does not depend on the stratum
to which $S$ belongs. So, we assume the action of $G$ on $S$ to be represented
by $(s,s,s,s,s,rs,r).$ Consider the equation (2.8) to see that $\dim B_{2}=2$
and $\dim B_{d}=\varphi(\tfrac{n}{d}).$ In addition, we consider the induced
isogeny (2.6) with $H=\langle r\rangle$ and $H=\langle s\rangle$ to see that

$B_{2}\sim JS_{\langle r\rangle}\,\,\mbox{ and
}\,\,\Pi_{d\in\Omega(n)}B_{d}\sim JS_{\langle s\rangle}$

respectively, and therefore the proof follows after setting $A=B_{2}.$ ∎

######  Remark 3.

We end this section by remarking two facts concerning the family
$\mathscr{V}_{g}.$

  1. (1)

The behavior for $g=4$ is completely different. Indeed, as noticed in [14]
(see also [4]) the family $\mathscr{V}_{4}$ consists of two strata,
represented by $\theta_{1}=(r,r,r,r,r,s,sr)$ and
$\theta_{2}=(r,r,r,s,s,s,sr).$ By proceeding analogously as done in the proof
of Proposition 6, one sees that if $S\in\mathscr{V}_{4}$ then:

    1. (a)

if $S$ belongs to the stratum defined by $\theta_{1}$ then $JS\sim A_{1}\times
A_{2},$ where $A_{1}\sim JS_{\langle s\rangle}$ and $A_{2}\sim JS_{\langle
sr\rangle}$ are abelian surfaces, and

    2. (b)

if $S$ belongs to the stratum defined by $\theta_{2}$ then $JS\sim E_{1}\times
E_{2}\times A,$ where $E_{1}\sim JS_{\langle r\rangle}$ and $E_{2}\sim
JS_{\langle s\rangle}$ are elliptic curves and $A\sim JS_{\langle sr\rangle}$
is an abelian surface.

  2. (2)

As the reader could expect, if $n=\tfrac{g}{2}$ is even then Propositions 5
and 6 are not longer true. For instance, the stratum defined by
$\eta=(r^{\frac{n}{2}},r^{\frac{n}{2}},s,s,s,rs,r)$ is not equivalent to any
of the actions determined in Proposition 5. Furthermore, if $S$ belongs to the
stratum defined by $\eta$ then, by proceeding as in the proof of Proposition
6, one sees that if $n\equiv 0\mbox{ mod }4$ then $JS$ contains two elliptic
curves, and if $n\equiv 2\mbox{ mod }4$ then $JS$ contains two elliptic curves
and an abelian surface.

##  7\. The families $\mathscr{U}^{1}_{g}$ and $\mathscr{U}^{2}_{g}$

######  Proposition 7.

Let $g\geqslant 11$ be an odd integer such that $g-1$ is twice a prime number.
Then $\mathscr{U}_{g}^{1}$ and $\mathscr{U}_{g}^{2}$ are the unique complex
four-dimensional families with $g-1$ automorphisms.

###### Proof.

Let $g\geqslant 11$ be an odd integer and write $g-1=2q$ where $q\geqslant 5$
is a prime number. As the cyclic and dihedral group are the unique groups of
order $2q,$ we only need to verify that $(1;2,\stackrel{{\scriptstyle
4}}{{\ldots}},2)$ is the only possible signature for the action of a group $G$
of order $2q$ on a complex-four dimensional family of Riemann surfaces of
genus $1+2q$.

A short computation shows that if signature of the action is not
$(1;2,\stackrel{{\scriptstyle 4}}{{\ldots}},2)$ then it is
$(h;m_{1},\ldots,m_{l})$ 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 $\Sigma_{j=1}^{7}\tfrac{1}{m_{j}}=3.$ As argued in the
proof of Lemma 5.1 and 5.3, one sees that the number $v$ of periods $m_{j}$
that are different from 2 are either two of three.

  1. (1)

If $v=2$ then the signature of the action is $(0;2,\stackrel{{\scriptstyle
5}}{{\ldots}},2,a,b)$ where $a,b\geqslant 3$ satisfy

$\tfrac{1}{a}+\tfrac{1}{b}=\tfrac{1}{2}\,\,\mbox{ and therefore
}\,\,a=b=4\,\mbox{ or }\,a=3,b=6.$

  2. (2)

If $v=3$ then the signature of the action is $(0;2,\stackrel{{\scriptstyle
4}}{{\ldots}},2,a,b,c)$ where $a,b,c\geqslant 3$ satisfies

$\tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}=1\,\,\mbox{ and therefore
}\,\,a=b=c=3.$

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 $g\geqslant 3$ be an odd integer. The family $\mathscr{U}_{g}^{1}$ is
equisymmetric.

###### Proof.

For $g=3$ we refer to [7, Table 5, 3.b]. Assume $g\geqslant 5.$ Let $\Delta$
be a Fuchsian group of signature $(1;2,\stackrel{{\scriptstyle
4}}{{\ldots}},2)$ canonically presented as in the proof of Lemma 5.4 and let
$\theta:\Delta\to G=\langle t:t^{g-1}=1\rangle$ be a surface-kernel
epimorphism representing an action of $G$ on $S\in\mathscr{U}_{g}^{1}.$ We
have to prove that $\theta$ is equivalent to the surface-kernel epimorphism
(5.2) in the proof of Lemma 5.4. Clearly

$\theta(x_{j})=t^{(g-1)/2}\,\,\mbox{ for }\,\,j=1,2,3,4.$

If we write $\theta(\alpha_{1})=t^{u}$ and $\theta(\beta_{1})=t^{v}$ then as,
$\theta$ is surjective, without loss of generality, we can assume $u=1.$
Consider the transformation $A_{1,-v}$ (see §2.2) to see that $\theta$ is
equivalent to (5.2), as desired. ∎

For $n\geqslant 2$ even, let $\Lambda(n)=\\{1\leqslant d<\tfrac{n}{2}:d\text{
divides }n\text{ and }\tfrac{dn}{2}\not\equiv 0\mbox{ mod }n\\}.$

######  Proposition 9.

Let $g\geqslant 3$ be an odd integer. If $S\in\mathscr{U}_{g}^{1}$ then the
Jacobian variety $JS$ decomposes, up to isogeny, as follows.

  1. (1)

If $\tfrac{g-1}{2}$ is even then

$JS\sim E\times\Pi_{d\in\Lambda(g-1)}B_{d}$

where $E$ is an elliptic curve isogenous to $JS_{G}$ and $B_{d}$ is an abelian
variety of dimension $2\varphi(\tfrac{g-1}{d}).$

  2. (2)

If $\tfrac{g-1}{2}$ is odd then

$JS\sim E\times A\times\Pi_{d\in\Lambda(g-1)}B_{d}$

where $A$ is an abelian surface and $E$ and $B_{d}$ are as before.

###### Proof.

Set $n=g-1$ and let $\omega$ be a primitive $n$-th root of unity. For each
$0\leqslant j\leqslant n-1$, we denote by $\chi_{j}$ the complex irreducible
representation of $G=\langle t:t^{n}=1\rangle=C_{n}$ defined as
$\chi_{j}:t\mapsto\omega^{j}.$ After a routine computation, one sees that the
collection $\\{\chi_{d}\\}$ where $1\leqslant d\leqslant\tfrac{n}{2}$ and $d$
divides $n$ yields a maximal collection of non-trivial rational irreducible
representations of $G$, up to equivalence.

Let $B_{d}$ denote the factor associated to $\chi_{d}$ in the group algebra
decomposition of $JS$ with respect to $G.$ Clearly, $B_{0}$ is an elliptic
curve isogenous to $JS_{G}.$ In addition, we observe that

$\dim B_{d}=\varphi(\tfrac{n}{d})\tfrac{1}{2}\cdot 4(1-\chi_{d}^{\langle
t^{\frac{n}{2}}\rangle})$

and therefore $B_{d}=0$ if and only if
$\chi_{j}(t^{\frac{n}{2}})=\omega^{\frac{nd}{2}}=1$ or, equivalently
$\tfrac{nd}{2}\equiv 0\mbox{ mod }n.$ Hence, the group algebra decomposition
of $JS$ with respecto to $G$ is

$JS\sim JS_{G}\times B_{\frac{n}{2}}\times\Pi_{d\in\Lambda(n)}B_{d}$

where, for each $d\in\Lambda(n),$ the dimension of $B_{d}$ is
$2\varphi(\frac{n}{d}).$ Finally, as $\chi_{\frac{n}{2}}(t)=-1$ we see that

$\dim B_{\frac{n}{2}}=\tfrac{1}{2}\cdot 4(1-\chi_{\frac{n}{2}}^{\langle
t^{\frac{n}{2}}\rangle})=\left\\{\begin{array}[]{ll}2&\textrm{if
$\tfrac{n}{2}$ is odd}\\\ 0&\textrm{if $\tfrac{n}{2}$ is
even}\end{array}\right.$

and the proof follows after setting $E=B_{0}$ and $A=B_{\frac{n}{2}}$ when
$\tfrac{n}{2}$ is odd. ∎

######  Proposition 10.

Let $g\geqslant 5$ be an odd integer such that $\tfrac{g-1}{2}$ is a prime
number. Then the family $\mathscr{U}_{g}^{2}$ consists of at most two
equisymmetric strata.

###### Proof.

Set $q=\tfrac{g-1}{2}$ and assume $q$ to be prime. Let $\Delta$ be a Fuchsian
group of signature $(1;2,\stackrel{{\scriptstyle 4}}{{\ldots}},2)$ canonically
presented as in Lemma 5.4 and let $\theta:\Delta\to G=\mathbf{D}_{q}=\langle
r,s:r^{q}=s^{2}=(sr)^{2}=1\rangle$ be a surface-kernel epimorphism
representing an action of $G$ on $S\in\mathscr{U}_{g}^{2}.$ We write
$a=\theta(\alpha_{1}),b=\theta(\beta_{1})$ and $sr^{n_{i}}=\theta(x_{i})$
where $n_{i}\in\\{0,\ldots,q-1\\}$ for $i=1,2,3,4$ and identify $\theta$ with
$(a,b;sr^{n_{1}},sr^{n_{2}},sr^{n_{3}},sr^{n_{4}}).$

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

$(r^{u},r^{v}),\,\,(sr^{u},r^{v}),\,\,(r^{u},sr^{v})\,\,\mbox{ or
}\,\,(sr^{u},sr^{v})\,\,\,\,\mbox{ for some}\,\,u,v\in\\{0,\ldots,q-1\\}.$

First of all, note the third and fourth cases can be disregarded, since
$A_{1,1}\circ A_{2,1}$ and $A_{1,1}$ respectively (see §2.2), transform them
into the second case.

Assume that $a=r^{u}$ and $b=r^{v}.$

  1. (1)

If $u=0$ then, up to an automorphism, we can assume $(a,b)=(1,1)$ or $(1,r).$

  2. (2)

If $u\neq 0$ and $\tilde{u}$ is its inverse in the field of $q$ elements, then
the transformation $\phi_{-\tilde{u},1}\circ A_{2,1}\circ A_{1,-1}\circ
A_{1,-v\tilde{u}}$ allows us to assume that, up to equivalence, $(a,b)=(1,r).$

Assume that $a=sr^{u}$ and $b=r^{v}$

  1. (1)

If $v=0$ then, up to an automorphism, we can assume $(a,b)=(s,1).$

  2. (2)

If $v\neq 0$ and $\hat{v}$ is its inverse in the field of $q$ elements, then
$\phi_{\hat{v},0}\circ A_{2,-u\hat{v}}$ shows that, up to equivalence, we can
assume $(a,b)=(s,r).$

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 $C_{1,4}$ and $C_{2,4}$ respectively.

If $(a,b)=(1,1)$ then we can assume $n_{1}=0$ and $n_{2}=1.$ Thus, $\theta$ is
equivalent to $(1,1;s,sr,sr^{n_{3}},sr^{n_{3}-1}).$ Now, we apply
transformation $\Phi_{3}$ to see that $\theta$ is equivalent to

$(1,1;s,sr,sr,s)\,\,\mbox{ and therefore equivalent to }\,\,(1,1;s,s,sr,sr).$
(7.1)

If $(a,b)=(1,r)$ then we can assume $n_{1}=0$ and therefore $\theta$ is
equivalent to $(1,r;s,sr^{n_{2}},sr^{n_{3}},sr^{n_{4}}).$

  1. (1)

If $n_{2}=0$ then $n_{3}=n_{4}.$ If follows that $\theta$ is equivalent to
$(1,r;s,s,s,s)$ or to

$\theta_{j}:=(1,r^{j};s,s,sr,sr)\,\,\mbox{ for some }j\in\\{1,\ldots,q-1\\}.$

The latter case is equivalent to (7.1), since $C_{2,3}\circ C_{2,2}$
identifies $\theta_{j}$ with $\theta_{j-1}.$

  2. (2)

If $n_{2}\neq 0$ then $\theta$ is equivalent to
$(1,r^{j};s,sr,sr^{n_{3}},sr^{n_{3}-1})$ for some $j\in\\{1,\ldots,q-1\\}.$
Note that $\Phi_{3}$ shows that $\theta$ is equivalent to
$(1,r^{j};s,sr,sr,s);$ then, we see that $\theta$ is equivalent to
$\theta_{j}.$

All the above says that there are at most 2 strata given by
$\Theta_{1}=(1,1;s,s,sr,sr)$ and $\Theta_{2}=(1,r;s,s,s,s).$ ∎

######  Proposition 11.

Let $g\geqslant 7$ be an odd integer such that $\tfrac{g-1}{2}$ is odd. If
$S\in\mathscr{U}_{g}^{2}$ then the Jacobian variety $JS$ decomposes, up to
isogeny, as

$JS\sim E\times A\times\Pi_{d\in\Omega(\frac{g-1}{2})}B_{d}^{2},$

where $A$ is an abelian surface, $B_{d}$ is an abelian variety of dimension
$\varphi(\frac{g-1}{2d})$ and $E$ is an elliptic curve isogenous to $JS_{G}$.
Furthermore

$\mbox{Prym}(S_{\langle r\rangle}\to S_{G})\sim A\,\,\mbox{ and
}\,\,\mbox{Prym}(S_{\langle s\rangle}\to
S_{G})\sim\Pi_{d\in\Omega(\frac{g-1}{2})}B_{d}$

and therefore $JS\sim JS_{G}\times\mbox{Prym}(S_{\langle r\rangle}\to
S_{G})\times\mbox{Prym}(S_{\langle s\rangle}\to S_{G})^{2}.$

###### Proof.

Set $n=\tfrac{g-1}{2}$ and assume that $n$ is 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 $S$
belongs. So, we assume the action to be represented by $\Theta_{2}.$ Now,
keeping the same notation as before, the rational irreducible representations
of $G$ are $\chi_{1},\chi_{2}$ and $\psi_{d}$ with $d\in\Omega(n)$ and

$JS\sim B_{1}\times B_{2}\times\Pi_{d\in\Omega(n)}B_{d}^{2},$

where $B_{1}\sim JS_{G}$ is an elliptic curve. As $\chi_{2}^{\langle
s\rangle}=0$ and $\psi_{d}^{\langle s\rangle}=1$ for $d\in\Omega(n)$ one sees
that $\dim B_{2}=2$ and $\dim B_{d}=\varphi(\tfrac{n}{d}).$ Now, we consider
the induced isogeny (2.7) with $H_{1}=\langle r\rangle$ and $H_{2}=G$ to see
that

$B_{2}\sim\mbox{Prym}(S_{\langle r\rangle}\to S_{G}).$

Similarly, consider the induced isogeny (2.7) with $H_{1}=\langle s\rangle$
and $H_{2}=G$ to see that

$\Pi_{d\in\Omega(n)}B_{d}\sim\mbox{Prym}(S_{\langle s\rangle}\to S_{G}),$

and the result follows by setting $E=B_{1}$ and $A=B_{2}.$ ∎

######  Remark 5.

The isogeny decomposition of the Jacobian varieties $JS$ for
$S\in\mathscr{U}_{5}^{2}$ differs from the stated in Proposition 11 for the
case $g\geqslant 7.$ Furthermore, the decomposition depends on the stratum to
which $S$ belongs (due to the fact that the involutions of $\mathbf{D}_{2}$
are non-conjugate). Indeed

  1. (1)

If $S$ belongs to the stratum represented by $\Theta_{1}$ then $JS\sim
E_{1}\times E_{2}\times E_{3}\times A,$ where $E_{1}\sim JS_{G},E_{2}\sim
JS_{\langle s\rangle}$ and $E_{3}\sim JS_{\langle sr\rangle}$ are elliptic
curves and $A\sim JS_{\langle r\rangle}$ is an abelian surface.

  2. (2)

If $S$ belongs to the stratum represented by $\Theta_{2}$ then $JS\sim E\times
A_{1}\times A_{2},$ where $E\sim JS_{G}$ is an elliptic curve and $A_{1}\sim
JS_{\langle r\rangle},A_{2}\sim JS_{\langle sr\rangle}$ are abelian surfaces.

Acknowledgments. The authors are very grateful to SageMath (www.sagemath.org)
and its developers 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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