On Riemann surfaces of genus $g$ with $4g-4$ automorphisms

TL;DR

This paper classifies compact Riemann surfaces of genus g with automorphism groups of order 4g-4, especially when g-1 is prime, and analyzes their Jacobian varieties.

Contribution

It provides a complete classification of such surfaces and determines the isogeny decompositions of their Jacobians under specific conditions.

Findings

Classified Riemann surfaces with automorphism group order 4g-4 when g-1 is prime.

Determined isogeny decompositions of Jacobian varieties for these surfaces.

Enhanced understanding of automorphism groups in relation to surface genus.

Abstract

In this article we study compact Riemann surfaces with a non-large group of automorphisms of maximal order; namely, compact Riemann surfaces of genus $g$ with a group of automorphisms of order $4g-4.$ Under the assumption that $g-1$ is prime, we provide a complete classification of them and determine isogeny decompositions of the corresponding Jacobian varieties.

Equations

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

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

$$(0;2,2,2,n)\text{mbox}forn⩾3\text{mbox}or(0;2,2,3,n)\text{mbox}for3⩽n⩽5.$$

$$⟨r,s:r^{2(g-1)}=s^2=(sr{)}^2=1⟩,$$

$$⟨a,b:a^{g-1}=b^4=1,ba{b}^{-1}=a^r⟩,$$

$$\partial ({\bar{\mathcal{F}}}_g^t)={\bar{\mathcal{F}}}_g^t\setminus {\mathcal{F}}_g^t$$

$$\partial(\bar{\mathcal{F}}_{g}^{1})=\left\{\begin{array}[]{ll}\{X_{1},X_{2}\}&\textrm{if $g\equiv 2\mbox{ mod }8$}\\ \,\,\,\,\,\,\,\,\,\,\emptyset&\textrm{if $g\not\equiv 2\mbox{ mod }8,$}\par\end{array}\right.$$

$$\partial(\bar{\mathcal{F}}_{g}^{2})=\left\{\begin{array}[]{ll}\{Y_{1},Y_{2}\}&\textrm{if $g\equiv 2\mbox{ mod }3$}\\ \,\,\,\,\,\,\,\,\,\,\emptyset&\textrm{if $g\not\equiv 2\mbox{ mod }3,$}\par\end{array}\right.$$

$${\pi }^{\ast }:J{S}_H\to JS$$

$$JS\sim J{S}_{⟨a⟩}\times (J{S}_{⟨b⟩}{)}^4.$$

$$JS\sim J{S}_{⟨r⟩}\times J{S}_{⟨s⟩}\times J{S}_{⟨sr⟩}.$$

$$\mathbb{H}=\{z\in \mathbb{C}:\text{mbox}Im(z)>0\}.$$

$$\sigma (\Delta )=(h;m_1,\dots ,m_l),$$

$$x_1^{m_1}=\cdots =x_l^{m_l}={\Pi }_{i=1}^h[a_i,b_i]{\Pi }_{i=1}^l{x}_i=1,$$

$$\mu (\Delta )=2\pi [2h-2+{\Sigma }_{i=1}^l(1-\frac{1}{m_i})].$$

$$\mu (\Delta )=[\Gamma :\Delta ]\cdot \mu (\Gamma ).$$

$$\theta :\Delta \to G\text{mbox}suchthat\text{mbox}ker(\theta )=\Gamma .$$

$$\Theta :{\Delta }^{\prime }\to G^{\prime }\text{mbox}insuchawaythat\Theta {|}_{\Delta }=\theta .$$

$${\psi }_2(g)=f{\psi }_1(\omega (g))f^{-1}\text{mbox}forallg\in G.$$

$$((\omega ,f^{\ast }),\theta )\mapsto \omega \circ \theta \circ (f^{\ast }{)}^{-1}.$$

$$\Delta =⟨x_1,\dots ,x_l:x_1^{m_1}=\cdots =x_l^{m_l}=x_1\cdots x_l=1⟩,$$

$$x_i\mapsto x_{i+1},x_{i+1}\mapsto x_{i+1}{x}_i{x}_{i+1}^{-1}\text{mbox}and{x}_j\mapsto x_j\text{mbox}whenj\ne i,i+1$$

$$\text{mbox}Sing({\mathcal{M}}_g)={\cup }_{G\ne 1,\theta }{\bar{\mathcal{M}}}_g^{G,\theta }.$$

$$JS\sim A_1\times \dots \times A_r$$

$$d_V(h-1)+\frac{1}{2}{\Sigma }_{i=1}^l(d_V-d_V^{⟨\theta (x_i)⟩})\ne 0,$$

$$d_V^{H_1}+\cdots +d_V^{H_t}⩽d_V\text{mbox}foreachV\in {\mathfrak{J}}_{\theta },$$

$$JS\sim {\Pi }_{i=1}^{t}J{S}_{H_i}\times P$$

$$⟨a,b:a^{g-1}=b^4=1,ba{b}^{-1}=a^r⟩=C_{g-1}{⋊}_4{C}_4,$$

$$\Delta =⟨x_1,x_2,x_3,x_4:x_1^2=x_2^2=x_3^4=x_4^4=x_1{x}_2{x}_3{x}_4=1⟩.$$

$$\Theta (x_1)=b^2,\Theta (x_2)=a{b}^2,\Theta (x_3)=ab,\Theta (x_4)=b^3$$

Full text

Partially supported by Fondecyt Grant 11180024, Redes Etapa Inicial Grant 2017-170071 and Anillo ACT1415 PIA-CONICYT Grant

1. Introduction and statement of the results

The classification of finite groups of automorphisms of compact Riemann surfaces is a classical and interesting problem which has attracted a considerable interest, going back to contributions of Wiman, Klein, Schwartz and Hurwitz, among others.

It is classically known that the full automorphism group of a compact Riemann surface of genus $g\geqslant 2$ is finite, and that its order is bounded by $84g-84.$ This bound is sharp for infinitely many values of $g,$ and a Riemann surface with this maximal number of automorphisms is characterized as a branched regular cover of the projective line with three branch values, marked with 2, 3 and 7.

A result due to Wiman asserts that the largest cyclic group of automorphisms of a Riemann surface of genus $g\geqslant 2$ has order at most $4g+2.$ Moreover, the curve

[TABLE]

shows that this upper bound is sharp for each genus; see [46] and [18].

Besides, Accola and Maclachlan independently proved that for fixed $g\geqslant 2$ the maximum among the orders of the full automorphism group of Riemann surfaces of genus $g$ is at least $8g+8.$ Moreover, the curve

[TABLE]

shows that this lower bound is sharp for each genus; see [1] and [31].

An interesting problem is to understand the extent to which the order of the full automorphism group determines the Riemann surface. In this regard, Kulkarni proved that for $g$ sufficiently large the curve (1.1) is the unique Riemann surface with an automorphism of order $4g+2,$ and for $g\not\equiv-1\mbox{ mod }4$ sufficiently large the curve (1.2) is the unique Riemann surface with $8g+8$ automorphisms; see [25].

Let $a,b\in\mathbb{Z}.$ Following [26], the sequence $ag+b$ for $g=2,3,\ldots$ is called admissible if for infinitely many values of $g$ there is a compact Riemann surface of genus $g$ with a group of automorphisms of order $ag+b.$

In addition to the mentioned admissible sequences $84g-84,4g+2$ and $8g+8,$ very recently the cases $4g+4$ and $4g$ have been studied; see [8], [15] and also [37].

Let $a\geqslant 7.$ The admissible sequence $ag-a$ has been considered by Belolipetsky and Jones in [3]. Concretely, they succeeded in proving that under the assumption that $g-1$ is a sufficiently large prime number, a compact Riemann surface of genus $g$ with a group of automorphisms of order $ag-a$ lies in one of six infinite well-described sequences of examples. The cases $a=5$ and $a=6$ have been recently classified by Izquierdo and the author in [22].

All the aforementioned cases are examples of compact Riemann surfaces possessing a so-called large group of automorphisms: namely, a group of automorphisms of order greater than $4g-4,$ where $g$ is the genus. In this case, it is known that the Riemann surface is either quasiplatonic (it does not admit non-trivial deformations in the moduli space with its automorphisms) or belongs to a complex one-dimensional family in such a way that the signature of the action is

[TABLE]

Riemann surfaces with large groups of automorphisms have been considered from different points of view; see, for example, [13], [27], [30], [33], [44], [45] and [46].

In this article we consider compact Riemann surfaces admitting a non-large group of automorphisms of maximal order; concretely, we study and classify those compact Riemann surfaces of genus $g$ with a group of automorphisms of order $4g-4,$ under the assumption that $g-1$ is a prime number.

Theorem 1**.**

Let $g\geqslant 8$ such that $g-1$ is prime, and let $S$ be a compact Riemann surface of genus $g$ with a group of automorphisms of order $4g-4$.

If $g\equiv 0\mbox{ mod }4$ then $S$ belongs to $\bar{\mathcal{F}}_{g}^{2}$, where $\bar{\mathcal{F}}_{g}^{2}$ is the complex two-dimensional equisymmetric family of compact Riemann surfaces of genus $g$ with a group of automorphisms $G_{2}$ isomorphic to the dihedral group

[TABLE]

such that the signature of the action of $G_{2}$ is $(0;2,2,2,2,2).$

If $g\equiv 2\mbox{ mod }4$ then $S$ belongs to either $\bar{\mathcal{F}}_{g}^{2}$ or $\bar{\mathcal{F}}_{g}^{1},$ where $\bar{\mathcal{F}}_{g}^{1}$ is the complex one-dimensional equisymmetric family of compact Riemann surfaces of genus $g$ with a group of automorphisms $G_{1}$ isomorphic to

[TABLE]

where $r$ is a $4$-th primitive root of the unity in the field of $g-1$ elements, such that the signature of the action of $G_{1}$ is $(0;2,2,4,4).$

By virtue of Dirichlet’s prime number theorem, the congruences of Theorem 1 are satisfied for infinitely many values. The genera $g=3,4$ and $6$ are exceptional in the sense that, in addition to the families before introduced, appear finitely many quasiplatonic Riemann surfaces (see [2], [5], [6], [11], [14], [28] and [32]).

As proved by Costa and Izquierdo in [15], the largest order of the full automorphism group of a complex one-dimensional family of Riemann surfaces of genus $g$, appearing in all genera, is $4g+4.$ On the other hand, $4g-4$ is the maximal possible order of the full automorphism group of a complex two-dimensional family of compact Riemann surfaces of genus $g.$ It is worth mentioning that the existence of the family $\bar{\mathcal{F}}_{g}^{2}$ shows that this upper bound is attained in each genus, with $g-1$ not necessarily prime.

Let $t=1,2.$ The family $\bar{\mathcal{F}}_{g}^{t}$ is a closed algebraic subvariety of the moduli space of compact Riemann surfaces of genus $g$; we shall denote by ${\mathcal{F}}_{g}^{t}$ its interior, and by

[TABLE]

its boundary. Based on classical results due to Singerman [43] and on Belolipetsky and Jones’ classification [3], we are able to describe the interior and (up to possibly finitely many exceptional cases in small genera) the boundary of the families $\bar{\mathcal{F}}_{g}^{t}.$

More precisely:

Theorem 2**.**

Let $g\geqslant 8$ such that $g-1$ is prime.

For $g\equiv 2\mbox{ mod }4,$ the interior ${\mathcal{F}}_{g}^{1}$ of the family $\bar{\mathcal{F}}_{g}^{1}$ consists of those Riemann surfaces for which $G_{1}$ is the full automorphism group. Moreover, there is a positive integer $\epsilon_{1}$ such that if $g\geqslant\epsilon_{1}$ then

[TABLE]

where $X_{1}$ and $X_{2}$ are the two non-isomorphic compact Riemann surfaces of genus $g$ with full automorphism group of order $8g-8$.

The interior ${\mathcal{F}}_{g}^{2}$ of the family $\bar{\mathcal{F}}_{g}^{2}$ consists of those Riemann surfaces for which $G_{2}$ is the full automorphism group. Moreover, there is a positive integer $\epsilon_{2}$ such that if $g\geqslant\epsilon_{2}$ then

[TABLE]

where $Y_{1}$ and $Y_{2}$ are the two non-isomorphic compact Riemann surfaces of genus $g$ with full automorphism group of order $12g-12$.

We recall that the Jacobian variety $JS$ of a compact Riemann surface $S$ of genus $g$ is an irreducible principally polarized abelian variety of dimension $g.$ The relevance of the Jacobian varieties lies in the well-known Torelli’s theorem, which establishes that two compact Riemann surfaces are isomorphic if and only if the corresponding Jacobian varieties are isomorphic as principally polarized abelian varieties. See, for example, [4] and [16].

If $H$ is a group of automorphisms of $S$ then the associated regular covering map $\pi:S\to S_{H}$ given by the action of $H$ on $S$ induces a homomorphism

[TABLE]

between the corresponding Jacobians; the set $\pi^{*}(JS_{H})$ is an abelian subvariety of $JS$ which is isogenous to $JS_{H}.$ We keep the same notations as in Theorem 1.

Theorem 3**.**

Let $g\geqslant 8$ such that $g-1$ is prime.

If $S\in\bar{\mathcal{F}}_{g}^{1}$ then the Jacobian variety $JS$ decomposes, up to isogeny, as the product

[TABLE]

If $S\in\bar{\mathcal{F}}_{g}^{2}$ then the Jacobian variety $JS$ decomposes, up to isogeny, as the product

[TABLE]

The article is organized as follows.

1. (1)

In Section 2 we shall review the basic background: namely, Fuchsian groups, actions on compact Riemann surfaces, the equisymmetric stratification of the moduli space and the decomposition of Jacobian varieties. 2. (2)

In Section 3 we shall prove the existence of the families $\bar{\mathcal{F}}_{g}^{1}$ and $\bar{\mathcal{F}}_{g}^{2}.$ 3. (3)

Theorems 1, 2 and 3 will be proved in Section 4, 5 and 6 respectively. 4. (4)

Finally, and for the sake of completeness, in Section 7 two examples will be constructed to show that Theorem 1 is false if $g-1$ is not prime.

2. Preliminaries

2.1. Fuchsian groups and group actions on Riemann surfaces

By a Fuchsian group we mean a discrete group of automorphisms of the upper-half plane

[TABLE]

If $\Delta$ is a Fuchsian group and 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:

[TABLE]

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

Let $\Delta$ be a Fuchsian group of signature (2.1). Then

1. (1)

$\Delta$ has a canonical presentation given by generators $a_{1},\ldots,a_{h}$, $b_{1},\ldots,b_{h},$ $x_{1},\ldots,x_{l}$ and relations

[TABLE]

where $[u,v]$ stands for the commutator $uvu^{-1}v^{-1},$ 2. (2)

the elements of $\Delta$ of finite order are conjugate to powers of $x_{1},\ldots,x_{l}.$ 3. (3)

the Teichmüller space of $\Delta$ is a complex analytic manifold homeomorphic to the complex ball of dimension $3h-3+l$, and 4. (4)

the hyperbolic area of each fundamental region of $\Delta$ is

[TABLE]

Let $\Gamma$ be a group of automorphisms of $\mathbb{H}.$ If $\Delta$ is a subgroup of $\Gamma$ of finite index then $\Gamma$ is also Fuchsian and they are related by the Riemann-Hurwitz formula

[TABLE]

Let $S$ be a compact Riemann surface and let $\mbox{Aut}(S)$ denote its full automorphism group. It is said that a finite group $G$ acts on $S$ if there is a group monomorphism $\psi:G\to\operatorname{Aut}(S).$ The space of orbits $S_{G}$ of the action of $G$ on $S$ is endowed with a Riemann surface structure such that the projection $S\to S_{G}$ is holomorphic.

Compact Riemann surfaces and group actions can be understood in terms of Fuchsian groups as follows. By uniformization theorem (see, for example, [17, p. 203]), there is a (uniquely determined, up to conjugation) surface Fuchsian group $\Gamma$ such that $S$ and $\mathbb{H}_{\Gamma}$ are isomorphic. Moreover, $G$ acts on $S\cong\mathbb{H}_{\Gamma}$ if and only if there is a Fuchsian group $\Delta$ containing $\Gamma$ together with a group epimorphism

[TABLE]

In this case, it is said that $G$ acts on $S$ with signature $\sigma(\Delta)$ and that this action is represented by the epimorphism $\theta.$ If $G$ is a subgroup of $G^{\prime}$ then the action of $G$ on $S$ is said to extend to an action of $G^{\prime}$ on $S$ if:

1. (1)

there is a Fuchsian group $\Delta^{\prime}$ containing $\Delta.$ 2. (2)

the Teichmüller spaces of $\Delta$ and $\Delta^{\prime}$ have the same dimension, and 3. (3)

there exists an epimorphism

[TABLE]

An action is termed maximal if it cannot be extended. A complete list of signatures of Fuchsian groups $\Delta$ and $\Delta^{\prime}$ for which it might be possible to have an extension as before was determined by Singerman in [43].

2.2. Actions and equisymmetric stratification

Let $\text{Hom}^{+}(S)$ denote the group of orientation preserving homeomorphisms of $S.$ Two actions $\psi_{i}:G\to\mbox{Aut}(S)$ are topologically equivalent if there exist $\omega\in\operatorname{Aut}(G)$ and $f\in\text{Hom}^{+}(S)$ such that

[TABLE]

Each homeomorphism $f$ satisfying (2.3) yields an automorphism $f^{*}$ of $\Delta$ where $\mathbb{H}_{\Delta}\cong S_{G}$. If $\mathscr{B}$ is the subgroup of $\mbox{Aut}(\Delta)$ consisting of them, then $\mbox{Aut}(G)\times\mathscr{B}$ acts on the set of epimorphisms defining actions of $G$ on $S$ with signature $\sigma(\Delta)$ by

[TABLE]

Two epimorphisms $\theta_{1},\theta_{2}:\Delta\to G$ define topologically equivalent actions if and only if they belong to the same $(\mbox{Aut}(G)\times\mathscr{B})$-orbit (see [6]; also [19] and [34]). We remark that if the genus $h=g_{S_{G}}$ of $S_{G}$ is zero and

[TABLE]

then $\mathscr{B}$ is generated by the braid transformations $\Phi_{i}\in\mbox{Aut}(\Delta)$ defined by

[TABLE]

for each $i\in\{1,\ldots,l-1\}.$ See, for example, [24, p. 31].

Let $\mathscr{M}_{g}$ denote the moduli space of compact Riemann surfaces of genus $g\geqslant 2.$ It is well-known that $\mathscr{M}_{g}$ is endowed with a structure of complex analytic space of dimension $3g-3,$ and that for $g\geqslant 4$ its singular locus $\mbox{Sing}(\mathscr{M}_{g})$ agrees with the set of points representing compact Riemann surfaces with non-trivial automorphisms.

Following [7], the singular locus $\mbox{Sing}(\mathscr{M}_{g})$ admits an equisymmetric stratification $\{\mathscr{M}_{g}^{G,\theta}\}$, where each equisymmetric stratum $\mathscr{M}_{g}^{G,\theta}$, if nonempty, corresponds to one topological class of maximal actions. More precisely:

1. (1)

the closure $\bar{\mathscr{M}}_{g}^{G,\theta}$ of $\mathscr{M}_{g}^{G,\theta}$ consists of those Riemann surfaces of genus $g$ admitting an action of the group $G$ with fixed topological class given by $\theta,$ 2. (2)

$\bar{\mathscr{M}}_{g}^{G,\theta}$ is a closed irreducible algebraic subvariety of $\mathscr{M}_{g},$ 3. (3)

if the stratum ${\mathscr{M}}_{g}^{G,\theta}$ is nonempty, then it is a smooth, connected, locally closed algebraic subvariety of $\mathscr{M}_{g}$ which is Zariski dense in $\bar{\mathscr{M}}_{g}^{G,\theta},$ 4. (4)

there are finitely many distinct strata, and

[TABLE]

Let $\bar{\mathcal{F}}$ be a (closed) family of compact Riemann surfaces of genus $g$ such that each of its members has a group of automorphisms isomorphic to $G.$ The family $\bar{\mathcal{F}}$ is termed equisymmetric if its interior $\mathcal{F}$ consists of only one stratum.

2.3. Decomposition of Jacobians

If a finite group $G$ acts on a compact Riemann surface $S$ then it is known that this action induces an isogeny decomposition

[TABLE]

which is $G$-equivariant; see [10] and [29]. The factors $A_{j}$ in (2.4) are in bijective correspondence with the rational irreducible representations of $G$. If the factor $A_{1}$ is associated with the trivial representation of $G,$ then $A_{1}\sim JS_{G}.$

The decomposition of Jacobians with group actions has been extensively studied, going back to Wirtinger, Schottky and Jung (see, for example, [42] and [47]). For decompositions of Jacobians with respect to special groups, we refer to the articles [9], [20], [21], [35], [36] and [39].

Assume that $G$ acts on a compact Riemann surface $S$ with signature (2.1), and that this action is determined by the epimorphism $\theta:\Delta\to G,$ where $\Delta$ is written with its canonical presentation (2.2). We define $\mathfrak{J}_{\theta}$ as the set of complex irreducible representations $V$ of $G$ characterized as follows:

1. (1)

the trivial representation belongs to $\mathfrak{J}_{\theta}$ if and only if $h\neq 0,$ and 2. (2)

a non-trivial representation $V$ belongs to $\mathfrak{J}_{\theta}$ if and only if

[TABLE]

where $d_{V}$ is the degree of $V$ and $d_{V}^{\langle\theta(x_{i})\rangle}$ is the dimension of the subspace of $V$ fixed under the action of the subgroup of $G$ generated by $\theta(x_{i}).$

Let $H_{1},\ldots,H_{t}$ be groups of automorphisms of $S$ such that $G$ contains $H_{i}$ for each $i$. Following [38] (and using [40, Theorem 5.12]), the collection $\{H_{1},\ldots,H_{t}\}$ is termed $G$-admissible if

[TABLE]

and is called admissible if it is $G$-admissible for some group $G$. If $\{H_{1},\ldots,H_{t}\}$ is admissible then, by [38], $JS$ decomposes, up to isogeny, as

[TABLE]

for some abelian subvariety $P$ of $JS.$ See also [23].

Notation

Let $n\geqslant 2$ be an integer. Throughout this article we shall denote by $C_{n}$ the cyclic group of order $n$ and by $\mathbf{D}_{n}$ the dihedral group of order $2n.$

3. Existence of the families $\bar{\mathcal{F}}_{g}^{1}$ and $\bar{\mathcal{F}}_{g}^{2}$

Proposition 1**.**

Let $g\geqslant 6$ such that $g-1$ is a prime number and $g\equiv 2\mbox{ mod }4.$ There exists a complex one-dimensional equisymmetric family $\bar{\mathcal{F}}_{g}^{1}$ of compact Riemann surfaces of genus $g$ with a group of automorphisms isomorphic to

[TABLE]

where $r$ is a $4$-th primitive root of the unity in the field of $g-1$ elements, such that the signature of the action is $(0;2,2,4,4).$

Proof.

Set $q=g-1,$ and let $\Delta$ be a Fuchsian group of signature $\sigma=(0;2,2,4,4)$ with canonical presentation

[TABLE]

The epimorphism $\Theta:\Delta\to C_{q}\rtimes_{4}C_{4}$ defined by

[TABLE]

guarantees the existence of a complex one-dimensional family $\bar{\mathcal{F}}_{g}^{1}$ of compact Riemann surfaces $S$ of genus $g$ with a group of automorphisms $G$ isomorphic to $C_{q}\rtimes_{4}C_{4}$ acting on $S$ with signature $\sigma.$

To prove that $\bar{\mathcal{F}}_{g}^{1}$ consists of only one stratum, we firstly notice that the involutions of $C_{q}\rtimes_{4}C_{4}$ are $a^{l}b^{2}$ and the elements of order 4 are $a^{l}b$ and $a^{l}b^{3}$ for $1\leqslant l\leqslant q.$ Then, up to a permutation, an epimorphism $\theta:\Delta\to C_{q}\rtimes_{4}C_{4}$ representing an action of $G$ on $S$ is defined by

[TABLE]

for some $1\leqslant l_{1},\ldots,l_{4}\leqslant q.$ After applying an inner automorphism of $G,$ we can assume that $l_{4}\equiv 0\mbox{ mod }q$ and therefore $l_{2}\equiv l_{1}+l_{3}\mbox{ mod }q.$ Note that if $l_{1}\equiv l_{3}\equiv 0\mbox{ mod }q$ then $\theta$ is not surjective; thus, without loss of generality, we can assume $l_{3}\not\equiv 0\mbox{ mod }q.$ Now, consider the automorphism of $G$ given by $a\mapsto a^{t_{3}},b\to b,$ where $l_{3}t_{3}\equiv 1\mbox{ mod }q,$ to obtain that $\theta$ is equivalent to the epimorphism $\theta_{n}$ defined by

[TABLE]

for some $1\leqslant n\leqslant q.$ Finally, as $\Phi_{1}\cdot\theta_{n}=\theta_{n+1},$ each $\theta_{n}$ is equivalent to $\theta_{0}=\Theta.$ ∎

Proposition 2**.**

There exists a complex two-dimensional family $\bar{\mathcal{F}}_{g}^{2}$ of compact Riemann surfaces of genus $g\geqslant 2$ with a group of automorphisms isomorphic to the dihedral group of order $4g-4$ such that the signature of the action is $(0;2,2,2,2,2).$ If, in addition, $g-1$ is prime then the family is equisymmetric.

Proof.

Set $q=g-1,$ and let $\Delta$ be a Fuchsian group of signature $\sigma=(0;2,2,2,2,2)$ with canonical presentation

[TABLE]

The epimorphism $\Theta:\Delta\to\mathbf{D}_{2q}=\langle r,s:r^{2q}=s^{2}=(sr)^{2}=1\rangle$ defined by

[TABLE]

guarantees the existence of a complex two-dimensional family $\bar{\mathcal{F}}_{g}^{2}$ of compact Riemann surfaces $S$ of genus $g$ with a group of automorphisms $G$ isomorphic to $\mathbf{D}_{2q}$ acting on $S$ with signature $\sigma.$

We now assume $q$ to be prime and proceed to prove that $\bar{\mathcal{F}}_{g}^{2}$ is equisymmetric. Let $\theta:\Delta\to\mathbf{D}_{2q}$ be an epimorphism representing an action of $G$ on $S.$ Note that the involutions of $G$ are $r^{q}$ and $sr^{t}$ for $1\leqslant t\leqslant 2q,$ and if some $\theta(x_{i})$ equals $r^{q}$ then, after considering suitable braid automorphisms, it can be supposed $i=5.$

We claim that one and only one among the elements $\theta(x_{i})$ equals $r^{q}.$ Indeed, if $l$ denotes the number of elements $\theta(x_{i})$ which are equal to $r^{q},$ then clearly $l$ is different from 4 and 5 because otherwise $\theta$ is not surjective. If $l=3$ then it can be supposed $\theta(x_{1}x_{2})=r^{q}.$ If we write $\theta(x_{i})=sr^{t_{i}}$ then $t_{2}-t_{1}\equiv q\mbox{ mod }2q,$ showing that $t_{2}-t_{1}$ is odd and therefore, after considering the automorphism of $G$ given by $r\mapsto r,s\mapsto sr,$ we can assume that $t_{1}$ is even and that $t_{2}$ is odd. Now, we apply an appropriate inner automorphism of $G$ to suppose that $t_{1}\equiv 0\mbox{ mod }2q$ and $t_{2}\equiv q\mbox{ mod }2q.$ The contradiction is obtained by noticing that $\langle s,r^{q}\rangle\cong C_{2}^{2}.$ Similarly, if $l=2$ then we can suppose $\theta(x_{1}x_{2}x_{3})=1.$ If we write $\theta(x_{i})=sr^{t_{i}}$ then $sr^{t_{1}-t_{2}+t_{3}}=1,$ which is not possible.

It follows that, up to equivalence, the epimorphism $\theta:\Delta\to\mathbf{D}_{2q}$ is given by

[TABLE]

for some $1\leqslant t_{1},\ldots,t_{4}\leqslant 2q$ which satisfy $t_{2}-t_{1}+t_{4}-t_{3}\equiv q\mbox{ mod }2q.$ Now, after considering, if necessary, braid automorphisms and the automorphism of $G$ given by $r\mapsto r,s\mapsto sr,$ we can suppose $t_{1},t_{2},t_{3}$ to be even and $t_{4}$ to be odd. Furthermore, after applying a suitable inner automorphism of $G$, we can assume that $t_{1}=0.$

If $t_{4}\neq q$ then we apply the automorphism of $G$ given by $r\mapsto r^{l_{4}},s\mapsto s,$ where $t_{4}l_{4}\equiv 1\mbox{ mod }2q,$ to see that $\theta$ is equivalent to the epimorphism $\theta_{n}$ given by

[TABLE]

for some $1\leqslant n\leqslant 2q$ even. The equality $\Phi_{2}^{2}\cdot\theta_{n}=\theta_{n-2}$ shows that $\theta_{n}$ is equivalent to $\theta_{0}=\Theta.$ Similarly, if now $t_{4}=q$ then $t_{2}\equiv t_{3}\mbox{ mod }2q;$ we write $t=t_{2}=t_{3}.$ We apply the inner automorphism of $G$ induced by $sr^{t/2}$ and then the braid automorphism $\Phi_{2}\circ\Phi_{1}$ to see that $\theta$ is equivalent to the epimorphism $\theta_{t}$ defined by

[TABLE]

where $t\not\equiv 0\mbox{ mod }2q.$ Finally, consider the automorphism of $G$ given by $r\mapsto r^{l},s\mapsto s,$ where $(q+t)l\equiv 1\mbox{ mod }2q,$ to see that $\theta_{t}$ is equivalent to $\Theta.$ ∎

4. Proof of Theorem 1

Set $q=g-1$ and let $S$ be a compact Riemann surface of genus $g\geqslant 8$ with a group of automorphisms $G$ of order $4q$ where $q$ is prime. By the Riemann-Hurwitz formula the possible signatures of the action of $G$ on $S$ are

[TABLE]

By the classical Sylow’s theorems if $q\equiv 3\mbox{ mod }4$ then $G$ is isomorphic to either

[TABLE]

and if $q\equiv 1\mbox{ mod }4$ then, in addition to these groups, $G$ can be isomorphic to

[TABLE]

where $r$ is a $4$-th primitive root of the unity in the field of $q$ elements.

The proof of Theorem 1 is a consequence of the following three claims.

Claim 1. The following statements are equivalent.

1. (1)

$S$ is a compact Riemann surface of genus $g$ with a group of automorphisms of order $4q$ acting on $S$ with signature $\sigma=(0;2,2,4,4).$ 2. (2)

$g\equiv 2\mbox{ mod }4$ and $S\in\bar{\mathcal{F}}_{g}^{1}.$

Let us assume that $S$ is a compact Riemann surface of genus $g$ with a group of automorphisms $G$ of order $4q$ acting on $S$ with signature $\sigma.$ Let $\Delta$ be a Fuchsian group of signature $\sigma$ with canonical presentation

[TABLE]

and assume the action of $G$ on $S$ to be represented by the epimorphism $\theta:\Delta\to G.$

First of all, note that $G$ cannot be isomorphic to $C_{q}\times C_{2}^{2}$ or $\mathbf{D}_{2q}$ because they do not have elements of order 4, and cannot be isomorphic to $C_{4q}$ because otherwise $C_{4q}$ would be generated by elements of order 2 and 4. We claim that $G$ cannot be isomorphic to $C_{q}\rtimes_{2}C_{4}$ either. Indeed, as $b^{2}$ is the unique involution of $C_{q}\rtimes_{2}C_{4}$ and as its elements of order 4 are $a^{t}b$ and $a^{t}b^{3}$ for $1\leqslant t\leqslant q,$ after considering the automorphism of $C_{q}\rtimes_{2}C_{4}$ given by $a\mapsto a,b\mapsto b^{3},$ the epimorphism $\theta$ could only be defined either by

1. (1)

$\theta(x_{1})=b^{2},\,\theta(x_{2})=b^{2},\,\theta(x_{3})=a^{t_{1}}b,\,\theta(x_{4})=a^{t_{2}}b,$ or 2. (2)

$\theta(x_{1})=b^{2},\,\theta(x_{2})=b^{2},\,\theta(x_{3})=a^{t_{1}}b,\,\theta(x_{4})=a^{t_{2}}b^{3},$

for some $1\leqslant t_{1},t_{2}\leqslant q$. The former case cannot yield an action because the image of $x_{1}x_{2}x_{3}x_{4}$ is different from $1$ for each possible choice of $t_{1}$ and $t_{2}$. Similarly, the latter case could give rise to an action only if $t_{1}\equiv t_{2}\mbox{ mod }q;$ however, in this case $\theta$ would not be surjective.

All of the above shows that $g\equiv 2\mbox{ mod }4$ and that $G$ is isomorphic to $C_{q}\rtimes_{4}C_{4};$ consequently $S\in\bar{\mathcal{F}}_{g}^{1}$. The converse is direct, and the proof of the claim is done.

Claim 2. There is no a compact Riemann surface $S$ of genus $g$ with a group of automorphisms of order $4q$ acting on it with signature $(1;2).$

Let $G$ be a group of order $4q$ and let $\Delta$ be a Fuchsian group of signature $(1;2).$ For every torsion-free kernel epimorphism

[TABLE]

the image of $x_{1}$ must belong to the commutator subgroup of $G.$ Thus, to conclude suffice to notice that the commutator subgroup of each group of order $4q$ does not contain involutions.

Claim 3. The following statements are equivalent.

1. (1)

$S$ is a compact Riemann surface of genus $g$ with a group of automorphisms of order $4q$ acting on $S$ with signature $\sigma=(0;2,2,2,2,2).$ 2. (2)

$S\in\bar{\mathcal{F}}_{g}^{2}.$

Let us assume that $S$ is a compact Riemann surface of genus $g$ with a group of automorphisms $G$ of order $4q$ acting on $S$ with signature $\sigma.$ Let $\Delta$ be a Fuchsian group of signature $\sigma$ with canonical presentation

[TABLE]

and assume the action of $G$ on $S$ to be represented by the epimorphism $\theta:\Delta\to G.$

The group $G$ cannot be isomorphic to $C_{q}\rtimes_{2}C_{4}$ because it has a unique involution and therefore every homomorphism $\Delta\to C_{q}\rtimes_{2}C_{4}$ is not surjective. Similarly, the group $G$ cannot be isomorphic to $C_{q}\rtimes_{4}C_{4}$ because it has exactly $q$ involutions and all of them are contained in the proper subgroup $\langle a,b^{2}\rangle\cong\mathbf{D}_{q}.$ Finally, if $G$ were abelian then $G$ would be isomorphic to a subgroup of $C_{2}^{4};$ but this is not possible.

All the above ensures that $G$ is isomorphic to the dihedral group of order $4q$ and therefore $S\in\bar{\mathcal{F}}_{g}^{2}$. The converse is direct, and the proof of the claim is done.

5. Proof of Theorem 2

Let $g\geqslant 8$ such that $q=g-1$ is prime. Assume $g\equiv 2\mbox{ mod }4$ and let $S$ be a compact Riemann surface lying in the family $\bar{\mathcal{F}}_{g}^{1}.$ We recall that $S$ has a group of automorphisms $G_{1}$ isomorphic to

[TABLE]

where $r$ is a $4$-th primitive root of the unity in the field of $q$ elements, and that the action is represented by the epimorphism $\Theta:\Delta\to C_{q}\rtimes_{4}C_{4}$ defined by

[TABLE]

where $\Delta=\langle x_{1},x_{2},x_{3},x_{4}:x_{1}^{2}=x_{2}^{2}=x_{3}^{4}=x_{4}^{4}=x_{1}x_{2}x_{3}x_{4}=1\rangle.$

By classical results due to Singerman [43], the action of $G_{1}$ on a (generic) member $S$ of $\bar{\mathcal{F}}_{g}^{1}$ can be possibly extended to only an action of a group of order $8q$ with signature $\hat{\sigma}=(0;2,2,2,4).$ However, as proved in [3] for $g\geqslant 18$ and in [12] for $g=14,$ there is no compact Riemann surfaces of genus $g$ with a group of automorphisms of order $8q$ acting with signature $\hat{\sigma}$. It follows that:

1. (1)

the interior $\mathcal{F}_{g}^{1}$ of the family $\bar{\mathcal{F}}_{g}^{1}$ consists of those Riemann surfaces for which $G_{1}$ agrees with the full automorphism group, and 2. (2)

the boundary

[TABLE]

consists of finitely many points.

Note that for each $X\in\partial(\bar{\mathcal{F}}_{g}^{1})$ its full automorphism group has order $\mu q$ where $4$ divides $\mu.$ Then, following [3], there exists a positive integer $\epsilon_{1}$ such that if $g\geqslant\epsilon_{1}$ then either:

1. (1)

$\mbox{Aut}(X)\cong C_{q}\rtimes_{8}C_{8}$ acting with signature $(0;2,8,8)$, for $g\equiv 2\mbox{ mod }8;$ or 2. (2)

$\mbox{Aut}(X)\cong(C_{q}\rtimes_{6}C_{6})\times C_{2}$ acting with signature $(0;2,6,6),$ for $g\equiv 2\mbox{ mod }3.$

The latter case is not possible because $(C_{q}\rtimes_{6}C_{6})\times C_{2}$ does not have elements of order 4, showing that if $g\not\equiv 2\mbox{ mod }8$ then $\partial(\bar{\mathcal{F}}_{g}^{1})$ is empty. Let us now assume that $g\equiv 2\mbox{ mod }8,$ and let

[TABLE]

be a Fuchsian group of signature $(0;2,8,8).$ Again, following [3], there are exactly two non-isomorphic Riemann surfaces $X_{1}$ and $X_{2}$ of genus $g\geqslant 18$ such that

[TABLE]

where $u$ is a $8$-th primitive root of the unity in the field of $q$ elements, and the action of $\mbox{Aut}(X_{i})$ on $X_{i}$ is determined by the epimorphisms $\Theta_{i}:\Delta^{\prime}\to C_{q}\rtimes_{8}C_{8}$

[TABLE]

[TABLE]

The subgroup of $\Delta^{\prime}$ generated by the elements

[TABLE]

is isomorphic to $\Delta,$ and

[TABLE]

Thus, for $i=1,2,$ the restriction of $\Theta_{i}$ to $\Delta\cong\langle\hat{x}_{1},\ldots,\hat{x}_{4}\rangle$

[TABLE]

defines an action $C_{q}\rtimes_{4}C_{4}$ on $X_{i}$ with signature $(0;2,2,4,4),$ showing that $\partial(\bar{\mathcal{F}}_{g}^{1})$ agrees with $\{X_{1},X_{2}\}$.

Now, let $S$ be a compact Riemann surface lying in the family $\bar{\mathcal{F}}_{g}^{2}.$ We recall that $S$ has a group of automorphisms $G_{2}$ isomorphic to

[TABLE]

and that the action of $G_{2}$ on $S$ is represented by the epimorphism $\Theta:\Delta\to\mathbf{D}_{2q}$

[TABLE]

where $\Delta=\langle x_{1},x_{2},x_{3},x_{4}:x_{1}^{2}=x_{2}^{2}=x_{3}^{2}=x_{4}^{2}=x_{5}^{2}=x_{1}x_{2}x_{3}x_{4}x_{5}=1\rangle.$

By [43] the action of $G_{2}$ on a generic member $S$ of $\bar{\mathcal{F}}_{g}^{2}$ cannot be extended. Thus:

1. (1)

the interior $\mathcal{F}_{g}^{2}$ of the family $\bar{\mathcal{F}}_{g}^{2}$ consists of those Riemann surfaces for which $G_{2}$ agrees with the full automorphism group, and 2. (2)

the boundary

[TABLE]

consists of finitely many points and finitely many one-dimensional families.

By [3], there exists $\epsilon_{2}$ such that if $g\geqslant\epsilon_{2}$ and $Y\in\partial(\bar{\mathcal{F}}_{g}^{2})$ then either

1. (1)

$\mbox{Aut}(Y)\cong C_{q}\rtimes_{8}C_{8}$ acting with signature $(0;2,8,8)$, for $g\equiv 2\mbox{ mod }8,$ or 2. (2)

$\mbox{Aut}(Y)\cong(C_{q}\rtimes_{6}C_{6})\times C_{2}$ acting with signature $(0;2,6,6),$ for $g\equiv 2\mbox{ mod }3.$

The former case is not possible because $C_{q}\rtimes_{8}C_{8}$ does not have elements of order $2q$; thus, if $g\not\equiv 2\mbox{ mod }3$ then $\partial(\bar{\mathcal{F}}_{g}^{2})$ is empty. Let us now assume that $g\equiv 2\mbox{ mod }3,$ and let $\Delta^{\prime}$ be a Fuchsian group of signature $(0;2,6,6)$ with canonical presentation

[TABLE]

Again, following [3], for $g\geqslant\epsilon_{2}$ there are exactly two non-isomorphic Riemann surfaces $Y_{1}$ and $Y_{2}$ of genus $g$ with full automorphism group isomorphic to

[TABLE]

where $u$ is a $6$-th primitive root of the unity in the field of $q$ elements, and the action of $\mbox{Aut}(Y_{i})$ on $Y_{i}$ is determined by the epimorphisms $\Theta_{i}:\Delta^{\prime}\to(C_{q}\rtimes_{6}C_{6})\times C_{2}$

[TABLE]

[TABLE]

The subgroup of $\Delta^{\prime}$ generated by

[TABLE]

is isomorphic to $\Delta,$ and

[TABLE]

[TABLE]

Thus, for $i=1,2,$ the restriction of $\Theta_{i}$ to $\Delta\cong\langle\tilde{x}_{1},\ldots,\tilde{x}_{4}\rangle$

[TABLE]

defines an action $\mathbf{D}_{2q}$ on $Y_{i}$ with signature $(0;2,2,2,2,2),$ showing that $\partial(\bar{\mathcal{F}}_{g}^{2})$ agrees with $\{Y_{1},Y_{2}\}$.

The proof is done.

6. Proof of Theorem 3

Let $g\geqslant 8$ such that $q=g-1$ is prime.

We recall the well-known fact that the dihedral group

[TABLE]

has, up to equivalence, 4 complex irreducible representations of degree one; namely,

[TABLE]

and $q-1$ complex irreducible representations of degree two; namely,

[TABLE]

for $1\leqslant j\leqslant q-1$ and $\omega_{t}=\mbox{exp}(\tfrac{2\pi i}{t}).$ See, for example, [41, p. 36].

Following the notations introduced in Subsection 2.3, the trivial representation $U_{1}^{+}$ does not belong to $\mathfrak{J}_{\theta},$ where $\theta$ represents the action of $\mathbf{D}_{2q}$ on each member $S$ of the family $\bar{\mathcal{F}}_{g}^{2}.$ The following table summarizes the dimension of the vector subspaces of the non-trivial complex irreducible representations of $\mathbf{D}_{2q}$ fixed under the action of the subgroups $\langle r\rangle,\langle s\rangle$ and $\langle sr\rangle$.

[TABLE]

It follows that the collection $\{\langle r\rangle,\langle s\rangle,\langle sr\rangle\}$ is admissible and therefore, by [38], if $S\in\bar{\mathcal{F}}_{g}^{2}$ then there exists an abelian subvariety $P$ of $JS$ such that

[TABLE]

The covering maps given by the action of $\langle s\rangle,\langle r\rangle$ and $\langle sr\rangle$ ramify over six, two and two values respectively; then, the Riemann-Hurwitz formula implies that

[TABLE]

It follows that $P=0$ and the desired decomposition is obtained.

Remark 1*.*

Note that if $S\in\bar{\mathcal{F}}_{g}^{2}$ then $JS$ contains an elliptic curve.

We now assume $g\equiv 2\mbox{ mod }4.$ Let $r$ be a $4$-th primitive root of the unity in the field of $q$ elements, write $m=\tfrac{q-1}{4}$ and choose $1\leqslant k_{1},\ldots,k_{m}\leqslant q-1$ such that

[TABLE]

where the symbol $\sqcup$ stands for disjoint union. Then

[TABLE]

has, up to equivalence, $m$ complex irreducible representations of degree 4, given by

[TABLE]

and four complex irreducible representations of degree 1, given by

[TABLE]

(see, for example, [41, p. 62]). Choose four pairwise different integers $t_{1},t_{2},t_{3},t_{4}\in\{1,\ldots,q-1\},$ and consider the following subgroups of $C_{q}\rtimes_{4}C_{4}$

[TABLE]

The trivial representation $U_{0}$ does not belong to $\mathfrak{J}_{\theta},$ where $\theta$ represents the action of $C_{q}\rtimes_{4}C_{4}$ on each member $S$ of the family $\bar{\mathcal{F}}_{g}^{1}.$ The dimension of the vector subspaces of the non-trivial complex irreducible representations of $C_{q}\rtimes_{4}C_{4}$ fixed under the action of the subgroups $\langle a\rangle$ and $\langle a^{t_{i}}b\rangle$ is:

[TABLE]

Thus $\{\langle a\rangle,\langle a^{t_{1}}b\rangle,\ldots,\langle a^{t_{4}}b\rangle\}$ is admissible and therefore, by [38], if $S\in\bar{\mathcal{F}}_{q}^{1}$ then

[TABLE]

for some abelian subvariety $Q$ of $JS$, where the isomorphism follows after noticing that each $\langle a^{t_{i}}b\rangle$ and $\langle b\rangle$ are conjugate. The covering map $S\to S_{\langle a\rangle}$ is unbranched, and the covering map $S\to S_{\langle b\rangle}$ ramifies over four values, two marked with 2 and two marked with 4. Then, the Riemann-Hurwitz formula implies that

[TABLE]

Thereby, $Q=0$ and the decomposition of $JS$ stated in Theorem 3 is done.

7. The case $g-1$ not prime

Example 1.

Let $n\geqslant 3$ be an integer, and consider the group

[TABLE]

where $\mathbf{Q}_{8}$ denotes the quaternion group, and let $\Delta$ be a Fuchsian group of signature $(1;2)$ with canonical presentation $\Delta=\langle a_{1},b_{1},x_{1}:[a_{1},b_{1}]x_{1}=x_{1}^{2}=1\rangle.$ For each $n$ odd, the epimorphism $\Delta\to\mathbf{Q}_{8}\times C_{n}$ given by $a_{1}\mapsto x,\,b_{1}\mapsto yz,\,x_{1}\mapsto y^{2}$ guarantees the existence of a complex one-dimensional family of compact Riemann surfaces of genus $g=1+2n$ with a group of automorphisms of order $8n=4g-4$ isomorphic to $\mathbf{Q}_{8}\times C_{n}$ acting with signature $(1;2).$

Example 2.

Let $n\geqslant 3.$ Choose $m\in\{\pm 1,2^{n-1}\pm 1\},$ consider the group

[TABLE]

and let $\Delta$ be a Fuchsian group of the signature $\sigma=(0;2,2,2,2,2)$ with presentation

[TABLE]

The epimorphism $\Delta\to\mathbf{D}_{2^{n}}\rtimes C_{2}$ given by

[TABLE]

guarantees the existence of a complex two-dimensional family of compact Riemann surfaces of genus $g=2^{n}+1,$ with a group of automorphisms of order $2^{n+2}=4g-4$ isomorphic to $\mathbf{D}_{2^{n}}\rtimes C_{2}$ acting with signature $\sigma.$ Two different choices of $m$ yield non-isomorphic groups, showing that if $g-1=2^{n}$ then there exist at least four pairwise non-isomorphic groups of order $4g-4$ acting on compact Riemann surfaces of genus $g$ with the same signature $\sigma.$

Acknowledgments

The author is grateful to his colleague Angel Carocca who generously told him how to construct Example 2 in Section 7.