### dr. Ruben Hidalgo

##### Títulos profesionales pregrado:

Licenciado en Matemáticas, Universidad de Santiago

##### Grados Académicos postgrado:

Mg. en Ciencias, mención Matemáticas, Universidad Técnica Federico Santa María

PhD. in Mathematics, The State University New York

Habilitation, Universitaet Bielefeld, Germany

PhD. in Mathematics, The State University New York

Habilitation, Universitaet Bielefeld, Germany

##### Información de Contacto:

Fono: +56 452 59 6698

email: ruben.hidalgo@ufrontera.cl

email: ruben.hidalgo@ufrontera.cl

##### Áreas de investigación

Grupos Kleinianos, Superficies de Riemann

Cuerpos de moduli y de definición

Uniformización numérica y Sistemas Dinámicos

Cuerpos de moduli y de definición

Uniformización numérica y Sistemas Dinámicos

### Publicaciones

##### Publicaciones indexada:

**2016**

**(with Saul Quispe)**Fields of moduli of some special curves.

Journal of Pure and Applied Algebra. 220 (2016), 55-60. DOI information: 10.1016/j.jsc.2014.09.042

In this paper we provide necessary conditions for a curve to be definable over its field of moduli. These conditions generalize the results known for the hyperelliptic case by B. Huggins and for normal cyclic $p$-gonal curves by A. Kontergeorgis.

**(with M. Carvacho and S. Quispe)**Jacobian variety of generalized Fermat curves.

Quarterly Journal of Math. 67 (2016), 261-284. DOI information: 10.1093/qmath/haw009

The isogenous decomposition of the Jacobian variety of classical Fermat curve of prime degree $p \geq 5$ has been obtained by Aoki using techniques of number theory, by Barraza and Rojas in terms of decompositions of the algebra of groups and by Hidalgo and Rodr\'{\i}guez using Kani-Rosen results. In the last, it was seen that all factors in the isogenous decomposition are Jacobian varieties of certain cyclic $p$-gonal curves. The highest Abelian branched covers of an orbifold of genus zero with exactly $n+1$ branch points, each one of order $p$, are provided by the so called generalized Fermat curves of type $(p,n)$; these being a suitable fiber product of $n-1$ Fermat curves of degree $p$. In this paper, we provide a decomposition, up to isogeny, of the Jacobian variety of a generalized Fermat curve $S$ of type $(p,n)$ as a product of Jacobian varieties of certain cyclic $p$-gonal curves whose explicit equations are provided in terms of the equations for $S$. As a consequence of this decomposition, we are able to provide explicit positive-dimensional families of closed Riemann surfaces whose Jacobian variety is isogenous to the product of elliptic curves.

**(with M. Carvacho, G. Gromadzki and CB. Baginski)**On periodic self-homeomorphisms of closed orientable surfaces determined by their orders.

Collectanea Mathematica 67 (2016), 415-429. doi:10.1007/s13348-015-0151-1

The fundamentals for the topological classification of periodic orientation preserving self-homeomorphisms of a closed orientable topological surface $X=X_g$ of genus $g \geq 2$ have been established, by Nielsen, in the thirties of the last century. Here we consider two concepts related to this classification; rigidity and weak rigidity. A cyclic action $G$ of order $N$ on $X$ is said to be {topologically rigid} if any other cyclic action of order $N$ on $X$ is topologically conjugate to it. If this assertion holds for arbitrary other action having, in addition, the same orbit genus and the same structure of singular orbits, then $G$ is said to be {weakly topologically rigid}. Here we give a precise description of rigid and weakly rigid quasi-platonic actions.

**(with A. F. Costa)**Automorphisms of non-cyclic p-gonal Riemann surfaces.

Moscow Mathematical Journal 16 (2016), 659-674.

In this paper we prove that the order of a holomorphic automorphism of a non-cyclic $p$-gonal compact Riemann surface $S$ of genus $g>(p-1)^{2}$ is bounded above by $2(g+p-1)$. We also show that this maximal order is attained for infinitely many genera. This generalises the similar result for the particular case $p=3$ recently obtained by Costa-Izquierdo. Moreover, we also observe that the full group of holomorphic automorphisms of $S$ is either the trivial group or is a finite cyclic group or a dihedral group or one of the Platonic groups ${\mathcal A}_{4}$, ${\mathcal A}_{5}$ and $\Sigma_{4}$. Examples in each case are also provided. In the case that $S$ admits a holomorphic automorphism of order $2(g+p-1)$, then its full group of automorphisms is the cyclic group generated by it and every $p$-gonal map of $S$ is necessarily simply branched. Finally, we note that each pair $(S,\pi)$, where $S$ is a non-cyclic $p$-gonal Riemann surface and $\pi$ is a $p$-gonal map, can be defined over its field of moduli. Also, if the group of automorphisms of $S$ is different from a non-trivial cyclic group and $g>(p-1)^{2}$, then $S$ can be also be defined over its field of moduli.

In this paper we provide necessary conditions for a curve to be definable over its field of moduli. These conditions generalize the results known for the hyperelliptic case by B. Huggins and for normal cyclic $p$-gonal curves by A. Kontergeorgis.

**2017**

**(with L. Jimenez, S. Quispe and S. reyes-Carocca)**Quasiplatonic curves with symmetry group $\mathbb{Z}_2^2 \rtimes \mathbb{Z}_m$ are definable over $\mathbb{Q}$.

Bull. of the London Math. Soc. 49 (2017), 165-183.

It is well known that every closed Riemann surface $S$ of genus $g \geq 2$, admitting a group $G$ of conformal automorphisms so that $S/G$ has triangular signature, can be defined over a finite extension of ${\mathbb Q}$. It is interesting to know, in terms of the algebraic structure of $G$, if $S$ can in fact be defined over ${\mathbb Q}$. This is the situation if $G$ is either abelian or isomorphic to $A \rtimes {\mathbb Z}_{2}$, where $A$ is an abelian group. On the other hand, as shown by Streit and Wolfart, if $G \cong {\mathbb Z}_{p} \rtimes {\mathbb Z}_{q}$ where $p,q>3$ are prime integers, then $S$ is not necessarily definable over ${\mathbb Q}$. In this paper, we observe that if $G\cong{\mathbb Z}_{2}^{2} \rtimes {\mathbb Z}_{m}$ with $m \geq 3$, then $S$ can be defined over ${\mathbb Q}$. Moreover, we describe explicit models for $S$, the corresponding groups of automorphisms and an isogenous decomposition of their Jacobian varieties as product of Jacobians of hyperelliptic Riemann surfaces.

**(with S. Quispe)**Regular dessins d'enfants with field of moduli $\mathbb{Q}(\sqrt[p]{2})$.

ARS Mathematica Contemporanea 13 No. 2 (2017), 323-330.

Herradon has recently provided an example of a regular dessin d'enfant whose field of moduli is the non-abelian extension ${\mathbb Q}(\sqrt[3]{2})$ answering in this way a question due to Conder, Jones, Streit and Wolfart. In this paper we observe that Herradon's example belongs naturally to an infinite series of such kind of examples; for each prime integer $p \geq 3$ we construct a regular dessin d'enfant whose field of moduli is the non-abelian extension ${\mathbb Q}(\sqrt[p]{2})$; for $p=3$ it coincides with Herradon's example.

**(with A. Kontogeorgis, M. Leyton and P. Paramantzouglou)**Automorphisms of generalized Fermat curves.

Journal of Pure and Applied Agebra 221 (2017), 2312-2337

Let $K$ be an algebraically closed field of characteristic $p \geq 0$. A generalized Fermat curve of type $(k,n)$, where $k,n \geq 2$ are integers (for $p \neq 0$ we also assume that $k$ is relatively prime to $p$), is a non-singular irreducible projective algebraic curve $F_{k,n}$ defined over $K$ admitting a group of automorphisms $H \cong {\mathbb Z}_{k}^{n}$ so that $F_{k,n}/H$ is the projective line with exactly $(n+1)$ cone points, each one of order $k$. Such a group $H$ is called a generalized Fermat group of type $(k,n)$.

If $(n-1)(k-1)>2$, then $F_{k,n}$ has genus $g_{n,k}>1$ and it is known to be non-hyperelliptic. In this paper, we prove that every generalized Fermat curve of type $(k,n)$ has a unique generalized Fermat group of type $(k,n)$ if $(k-1)(n-1)>2$ (for $p>0$ we also assume that $k-1$ is not a power of $p$). Generalized Fermat curves of type $(k,n)$ can be described as a suitable fiber product of $(n-1)$ classical Fermat curves of degree $k$. We prove that, for $(k-1)(n-1)>2$ (for $p>0$ we also assume that $k-1$ is not a power of $p$), each automorphism of such a fiber product curve can be extended to an automorphism of the ambient projective space. In the case that $p>0$ and $k-1$ is a power of $p$, we use tools from the theory of complete projective intersections in order to prove that, for $k$ and $n+1$ relatively prime, every automorphism of the fiber product curve can also be extended to an automorphism of the ambient projective space. In this article we also prove that the set of fixed points of the non-trivial elements of the generalized Fermat group coincide with the hyper-osculating points of the fiber product model under the assumption that the characteristic $p$ is either zero or $p>k^{n-1}$.Let $K$ be an algebraically closed field of characteristic $p \geq 0$. A generalized Fermat curve of type $(k,n)$, where $k,n \geq 2$ are integers (for $p \neq 0$ we also assume that $k$ is relatively prime to $p$), is a non-singular irreducible projective algebraic curve $F_{k,n}$ defined over $K$ admitting a group of automorphisms $H \cong {\mathbb Z}_{k}^{n}$ so that $F_{k,n}/H$ is the projective line with exactly $(n+1)$ cone points, each one of order $k$. Such a group $H$ is called a generalized Fermat group of type $(k,n)$.

If $(n-1)(k-1)>2$, then $F_{k,n}$ has genus $g_{n,k}>1$ and it is known to be non-hyperelliptic. In this paper, we prove that every generalized Fermat curve of type $(k,n)$ has a unique generalized Fermat group of type $(k,n)$ if $(k-1)(n-1)>2$ (for $p>0$ we also assume that $k-1$ is not a power of $p$).

**(with M. Artebani, M. Carvacho, and S. Quispe)**A Tower of Riemann Surfaces which cannot be defined over their Field of Moduli.

Glasgow Math. J. 59 No. 2 (2017), 379-393.

Explicit examples of both hyperelliptic and non-hyperelliptic curves which cannot be defined over their field of moduli are known in the literature.

In this paper, we construct a tower of explicit examples of such kind of curves.

In that tower there are both hyperelliptic curves and non-hyperelliptic curves3.

In this paper we provide necessary conditions for a curve to be definable over its field of moduli. These conditions generalize the results known for the hyperelliptic case by B. Huggins and for normal cyclic $p$-gonal curves by A. Kontergeorgis.

**(with C. Baginski and G. Gromadzki)**On purely non-free finite actions of abelian groups on compact surfaces.

Archiv der Mathematik 109 (2017), 311-321

A finite group of conformal automorphisms of a closed orientable Riemann

surface is said to act on it {\it purely non-freely} if each of its elements has a fixed point; we also called it a {\it gpnf}-action. In this paper we observe that {\it gpnf}-actions exist for an arbitrary finite group and we discuss on the minimum genus problem for such actions; we solve it for cyclic and abelian non-cyclic groups. In the first case we prove that this minimal {\it gpnf}-action genus coincides with Harvey's minimal genus.

**2018**

**(with S. Quispe)**A note on the connectedness of the branch locus of rational points.

Glasgow Math. J. 60 No. 1 (2018), 199-207. DOI: https://doi.org/10.1017/S0017089516000665

Milnor proved that the moduli space ${\rm M}_{d}$ of rational maps of degree $d \geq 2$ has a complex orbifold structure of dimension $2(d-1)$. Let us denote by ${\mathcal S}_{d}$ the singular locus of ${\rm M}_{d}$ and by ${\mathcal B}_{d}$ the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify ${\rm M}_2$ with ${\mathbb C}^2$ and, within that identification, that ${\mathcal B}_{2}$ is a cubic curve; so ${\mathcal B}_{2}$ is connected and ${\mathcal S}_{2}=\emptyset$. If $d \geq 3$, then it is well known that ${\mathcal S}_{d}={\mathcal B}_{d}$. In this paper we use simple arguments to prove the connectivity of ${\mathcal S}_{d}$.

**(with A. F. Costa)**On the connectedness of the set of Riemann surfaces with real moduli.

Archiv der Mathematik 110 (2018), 305-310.

The moduli space ${\mathcal{M}}_{g}$, of genus $g\geq2$ closed Riemann surfaces, is a complex orbifold of dimension $3(g-1)$ which carries a natural real structure i.e. it admits an anti-holomorphic involution $\sigma$. The involution $\sigma$ maps each point corresponding to a Riemann surface $S$ to its complex conjugate $\overline{S}$. The fixed point set of $\sigma$ consists of the isomorphism classes of closed Riemann surfaces admitting an anticonformal automorphism. Inside $\mathrm{Fix}(\sigma)$ is the locus ${\mathcal{M}}_{g}(\mathbb{R})$, the set of real Riemann surfaces, which is known to be connected by results due to P. Buser, M. Sepp\"{a}l\"{a} and R. Silhol. The complement $\mathrm{Fix}(\sigma)-{\mathcal{M}}_{g}(\mathbb{R})$ consists of the so called pseudo-real Riemann surfaces, which is known to be non-connected. In this short note we provide a simple argument to observe that $\mathrm{Fix}(\sigma)$ is connected.

**About the Fricke-Macbeath curve.**European Journal of Math. 4 (2018), 313-325. DOI 10.1007/s40879-017-0166-0

A closed Riemann surface of genus $g \geq 2$ is called a Hurwitz curve if its group of conformal automorphisms has order $84(g-1)$. In 1895, A. Wiman noticed that there is no Hurwitz curve of genus $g=2,4,5,6$ and, up to isomorphisms, there is a unique Hurwitz curve of genus $g=3$; this being Klein's plane quartic curve. Later, in 1965, A. Macbeath proved the existence, up to isomorphisms, of a unique Hurwitz curve of genus $g=7$; this known as the Fricke-Macbeath curve. Equations were also provided; that being the fiber product of suitable three elliptic curves. In the same year, W. Edge constructed such a genus seven Hurwitz curve by elementary projective geometry. Such a construction was provided by first constructing a $4$-dimensional family of genus seven closed Riemann surfaces $S_{\mu}$ admitting a group $G_{\mu} \cong {\mathbb Z}_{2}^{3}$ of conformal automorphisms so that $S_{\mu}/G_{\mu}$ has genus zero. In this paper we discuss the above curves in terms of fiber products of classical Fermat curves and we provide a geometrical explanation of the three elliptic curves in Macbeath's description. We also observe that the jacobian variety of each $S_{\mu}$ is isogenous to the product of seven elliptic curves (explicitly given) and, for the particular Fricke-Macbeath curve, we obtain the well known fact that its jacobian variety is isogenous to $E^{7}$ for a suitable elliptic curve $E$.

**(with E. Badr and S. Quispe)**Riemann surfaces defined over the reals.

Archiv der Mathematik 110 No. 4 (2018), 351-362 ISSN 0003-889X

https://doi.org/10.1007/s00013-017-1146-9

The known (explicit) examples of Riemann surfaces not definable over their field of moduli are not real whose field of moduli is a subfield of the reals. In this paper we provide explicit families of real Riemann surfaces which cannot be defined over the field of moduli.

**(with T. Shaska)**On the field of moduli of superelliptic curves.

Contemporary Mathematics 703 (2018), 47-62. ISBN: 978-1-4704-2856-3

Beshaj and Thompson have proved that a superelliptic curve $\X$ can always be defined over a quadratic extension of its field of moduli. If $\X$ has no extra automorphisms, then equations over the minimal field of definition can be determined. In this case, using ideas of Clebsh, it can be decided if $\X$ can be defined over its field of moduli. If $\X$ has extra automorphisms, then to determine if $\X$ can be defined over its field of moduli is more difficult. In this case, Beshaj and Thompson provided equations over the minimal field of definition using the dihedral (or Shaska) invariants. In this paper we observe that if the reduced group is different from the trivial or cyclic group, then $\X$ can be defined over its field of moduli; in the cyclic situation we provide a sufficient condition for this to happen. We also determine those of genus at most $10$ which might not be definable over their field of moduli.

**(with L. Beshaj, S. Kruk, A. Malmendier, S. Quispe and T. Shaska)**Rational points in the moduli space of genus two.

Contemporary Mathematics 703 (2018), 83-115. ISBN: 978-1-4704-2856-3

A database of over 2 million isomorphism classes of genus 2 curves defined over ${\mathbb Q}$ ordered based on their minimal absolute height, moduli height. In addition for each isomorphism class, a minimal equation over the minimal field of definition is provided, the automorphism group, the conductor, and all the twists. Some interesting statistics about the distribution of rational points in the moduli space $M_2$ are also presented. Such database organized in a dictionary in Sagemath is made public including a search utility.

**p-groups acting on Riemann surfaces.**Journal of Pure and Applied Algebra. 222 (2018), 4173-4188. ISSN: 0022-4049

Let $p$ be a prime integer and let $r \geq 3$ be an integer so that $p \geq 5r-7$ and $p \neq 5r-6$. We show that a closed Riemann surface $S$ of genus $g \geq 2$ has at most one $p$-group $H$ of conformal automorphisms so that $S/H$ has genus zero and exactly $r$ cone points. This, in particular, asserts that, for $r=3$ and $p \geq 11$, the minimal field of definition of $S$ coincides with that of $(S,H)$.

Another application of this fact, for the case that $S$ is pseudo-real, is that ${\rm Aut}(S)/H$ must be either trivial or a cyclic group and that $r$ is necessarily even. This generalizes a result due to Bujalance-Costa for the case of pseudo-real cyclic $p$-gonal Riemann surfaces.

**Totally degenerate extended Kleinian groups.**Revista CUBO 19 No. 3 (2018), 69-77. ISSN: 0716-7776

The theoretical existence of totally degenerate Kleinian groups is originally due to Bers and Maskit. In fact, Maskit proved that for any co-compact non-triangle Fuchsian group acting on the hyperbolic plane ${\mathbb H}^{2}$ there is a totally degenerate Kleinian group algebraically isomorphic to it. In this paper, by making a subtle modification to Maskit's construction, we show that for any non-Euclidean crystallographic group $F$, such that ${\mathbb H}^{2}/F$ is not homeomorphic to a pant, there exists an extended Kleinian group $G$ which is algebraically isomorphic to $F$ and whose orientation-preserving half is a totally degenerate Kleinian group. Moreover, such an isomorphism is provided by conjugation by an orientation-preserving homeomorphism $\phi:{\mathbb H}^{2} \to \Omega$, where $\Omega$ is the region of discontinuity of $G$. In particular, this also provides another proof to Miyachi's existence of totally degenerate finitely generated Kleinian groups whose limit set contains arcs of Euclidean circles.

**On the minimal field of definition of rational maps: rational maps of odd signature.**Annales Academiae Scientiarum Fennicae. 43 (2018), 685-692. ISSN 1239-629X

The field of moduli of a rational map is an invariant under conjugation by M\"obius transformations. J. H. Silverman proved that a rational map, either of even degree or equivalent to a polynomial, is definable over its field of moduli and he also provided examples of rational maps of odd degree for which such a property fails. We introduce the notion for a rational map to have odd signature and prove that this condition ensures for the field of moduli to be a field of definition. Rational maps being either of even degree or equivalent to polynomials are examples of odd signature ones.

**(wih Sebastian Sarmiento)**Real Structures on Marked Schottky Space.

Journal of the London Math. Soc. 98 (2) (2018) , 253-274. ISSN:1469-7750

Marked Schottky space $\MS$ is an explicit model of the quasiconformal deformation space of a Schottky group of rank $g$, this being isomorphic to the punctured unit disc for $g=1$ and, for $g \geq 2$, a connected complex manifold of dimension $3(g-1)$. This space is an intermediate non-Galois cover of moduli space of genus $g$ curves. In this paper we provide a description of the real structures of $\MS$, up to holomorphic automorphisms, together with their real parts.

**Uniformizations of stable $(\gamma,n)$-gonal Riemann surfaces.**Analysis and Mathematical Physics 8 (2018), 655-677. ISSN: 1664-2368

A $(\gamma,n)$-gonal pair is a pair $(S,f)$, where $S$ is a closed Riemann surface and $f:S \to R$ is a degree $n$ holomorphic map onto a closed Riemann surface $R$ of genus $\gamma$. If the signature of $(S,f)$ is of hyperbolic type, then it admits a uniformizing pair $(\Gamma,G)$, where $G$ is a Fuchsian group acting on the unit disc ${\mathbb D}$ containing $\Gamma$ as an index $n$ subgroup, such that $f$ is induced by the inclusion $\Gamma \leq G$. The uniformizing pair is uniquely determined by $(S,f)$, up to conjugation by holomorphic automorphisms of ${\mathbb D}$, and it permits to provide a natural complex orbifold structure on the Hurwitz space parametrizing (twisted) isomorphic classes of pairs topologically equivalent to $(S,f)$. In order to produce certain compactifications of these Hurwitz spaces, one needs to consider the so called

stable $(\gamma,n)$-gonal pairs, which are natural geometrical deformations of $(\gamma,n)$-gonal pairs. Due to the above, it seems interesting to search for uniformizations of stable $(\gamma,n)$-gonal pairs, in terms of certain class of Kleinian groups. In this paper we review such uniformizations by using noded Fuchsian groups, obtained from the noded Beltrami differentials of Fuchsian groups that were previously studied by Alexander Vasil'ev and the author, and which provide uniformizations of stable Riemann orbifolds. These uniformizations permit to obtain a compactification of the Hurwitz spaces together a complex orbifold structure, these being quotients of the augmented Teichm\"uller space of $G$ by a suitable finite index subgroup of its modular group.

**(with G. Gromadzki)**On Macbeath's formula for hyperbolic manifolds.

Albanian Journal of Mathematics 12 (1) (2018), 15-23. ISSN: 1930-1235

Around 1973, Macbeath provided a formula for the number of fixed points of an element in a group $G$ of conformal automorphisms of a closed Riemann surface $S$ of genus at least two. Such a formula was initially used to obtain the character of the representation associated to the induced action of $G$ on the first homology group of $S$, and later turned out to be extremely useful in many other contexts. By using a simple counting procedure, we provide a similar formula for the number of connected components of an element in a finite group of isometries of a hyperbolic manifold.

**(with M. Izquierdo)**On the connectedness of the branch locus of Schottky space.

Albanian Journal of Mathematics 12 (1) (2018), 235-246 . ISSN: 1930-1235

Schottky space of rank $g$ is the space ${\mathcal S}_{g}$that parametrizes conjugacy classes of Schottky groups of rank $g$. Its branch locu consists of the classes of Schottky groups which are proper finite index normal subgroups of a Kleinian group. In this paper it is proved the the connectivity of the branch locus.

**2019**

**Automorphisms of dessins d'enfants.**Archiv der Mathematik 112 (2019), 13-18. ISSN: 0003-889X

In this paper we observe that given any finite group $G$, together a fixed topological action of it of some genus $g \geq 2$, there is a dessin d'enfant having it as its full groiup of automorphisms. In particular, the symmetric strong genus of $G$ equals to the minimal genus (at least two) action on a dessin d'enfant.

**An explicit descent of real algebraic varieties.**In: Algebraic Curves and Their Applications.

Ed. Lubjana Beshaj and Tony Shaska.

Contemporary Math. Amer. Math. Soc. 724 (2019), 235-246. ISBN: 978-1-4704-4247-7

Let $X$ be an smooth complex affine algebraic variety admitting a symmetry $L$, that is, an antiholomorphic automorphism of order two. If both, $X$ and $L$ are defined over $\overline{\mathbb Q}$, then Koeck, Lau and Singerman showed the existence of a complex smooth algebraic variety $Z$ admitting a symmetry $T$, both defined over ${\mathbb R} \cap \overline{\mathbb Q}$, and of an isomorphism $R:X \to Z$ so that $R \circ L \circ R^{-1}=T$. The provided proof is existential and, if explicit equations for $X$ and $L$ are given over $\overline{\mathbb Q}$, then it is not described how to get the explicit equations for $Z$ and $T$ over ${\mathbb R} \cap \overline{\mathbb Q}$. In this paper we provide an explicit rational map $R$ defined over $\overline{\mathbb Q}$ so that $Z=R(X)$ is defined over ${\mathbb R} \cap \overline{\mathbb Q}$, $R:X \to Z$ is an isomorphisms and $T=R \circ L \circ R^{-1}$ being the usual conjugation map.

**(with G. Honorato and F. Valenzuela-Henriquez )**On the dynamics on n-circle inversion.

Nonlinearity 32 (2019), 1242-1274. ISSN: 0951-7715

The article deals with singular perturbation of polynomial maps \[R_{\lambda,\,n}(z)=\frac{z^{n}+\lambda}{z},\] where $\lambda$ is a complex parameter and $n$ is the degree, which is a particular case of the family of rational maps known as McMullen maps. Our main result shows that even when the geometric limit of Julia set converges to the unit circle or the annulus for a.e. Lebesgue $\lambda \in \mathbb{C}^*$, as $n$ tends to infinity, the measure of maximal entropy always converges to the Lebesgue measure supported on the unit circle. Additionally we describe the dynamics on the Julia set and show that is related to a quotient of a shift of $n$ symbols by an equivalence relation. Finally we prove that the Thurston's pull--back map associated to a particular $4$--circle inversion is a ramified Galois covering. From the arithmetical point of view we prove that each $n$--circle inversion can be defined over its field of moduli.

**(with Maximiliano Leyton-Alvarez)**Weierstrass weight of the hyperosculating points of generalized Fermat curves.

Journal of Pure and Applied Algebra 227 (7) (2019), 3057-3070. ISSN: 0022-4049

Let $(S,H)$ be a generalized Fermat pair of the type $(k,n)$. If $F\subset S$ is the set of fixed points of the non-trivial elements of the group $H$, then $F$ is exactly the set of hyperosculating points of the standard embedding $S\hookrightarrow {\mathbb{P}}^{n}$. We provide an optimal lower bound (this being sharp in a dense open set of the moduli space of the generalized Fermat curves) for the Weierstrass weight of these points.

**(with Y. Atarihuana )**On the connectivity of the branch and real locus of ${\mathcal M}_{0,[n+1]}$.

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales (RACSAM). Serie A. Matematicas 113 (2019), 2981-2998. ISSN: 1578-7303

If $n \geq 3$, then moduli space ${\mathcal M}_{0,[n+1]}$, of isomorphisms classes of $(n+1)$-marked spheres, is a complex orbifold of dimension $n-2$.

Its branch locus ${\mathcal B}_{0,[n+1]}$ consists of the isomorphism classes of those $(n+1)$-marked spheres with non-trivial group of conformal automorphisms. We prove that

${\mathcal B}_{0,[n+1]}$ is connected if either $n \geq 4$ is even or if $n \geq 6$ is divisible by $3$, and that it has exactly two connected components otherwise. The orbifold ${\mathcal M}_{0,[n+1]}$ also admits a natural real structure, this being induced by the complex conjugation on the Riemann sphere. The locus ${\mathcal M}_{0,[n+1]}({\mathbb R})$ of its fixed points, the real points, consists of the isomorphism classes of those marked spheres admitting an anticonformal automorphism. Inside this locus is the real locus ${\mathcal M}_{0,[n+1]}^{\mathbb R}$, consisting of those classes of marked spheres admitting an anticonformal involution. We prove that ${\mathcal M}_{0,[n+1]}^{\mathbb R}$ is connected for $n \geq 5$ odd, and that it is disconnected for $n=2r$ with $r \geq 5$ is odd.

**(with J. C. Garcia)**On square roots of meromorphic maps.

Results in Mathematics 74:118 (2019), ISSN: 1422-6383

Let $S$ be a connected Riemann surface and let $\varphi:S \to \widehat{\mathbb C}$ be a surjective meromorphic map. A simple geometrical necessary and sufficient condition is provided for the existence of a square root of $\varphi$, that is, a meromorphic map $\psi:S \to \widehat{\mathbb C}$ such that $\varphi=\psi^{2}$.

**2020**

**with Saul Quispe)**Regular dessins d'enfants with dicyclic group of automorphisms.

Journal of Pure and Applied Algebra 224 No. 5 (2020), Article 1062. ISSN: 0022-4049. https://doi.org/10.1016/j.jpaa.2019.106242

Let $G_{n}$ be the dicyclic group of order $4n$. We observe that, up to isomorphisms, (i) for $n \geq 2$ even there is exactly one regular dessin d'enfant with automorphism group $G_{n}$, and (ii) for $n \geq 3$ odd there are exactly two of them. Each of them is produced on well known hyperelliptic Riemann surfaces. We obtain that the minimal genus over which $G_{n}$ acts purely-non-free is $\sigma_{p}(G_{n})=n$ (this coincides with the strong symmetric genus of $G_{n}$ when $n$ is even). For each of the triangular conformal actions, every non-trivial subgroup of $G_{n}$ has genus zero quotient, in particular, that the isotypical decomposition, induced by the action of $G_{n}$, of its jacobian variety has only one component. We also study conformal/anticonformal actions of $G_{n}$, on closed Riemann surfaces, with the property that $G_{n}$ admits anticonformal elements. It is known that $G_{n}$ always acts on a genus one Riemann surface with such a property. We observe that the next genus $\sigma^{hyp}(G_{n})\geq 2$ over which $G_{n}$ acts in that way is $n+1$ for $n \geq 2$ even, and $2n-2$ for $n \geq 3$ odd. We also provide examples of pseudo-real Riemann surfaces admitting $G_{n}$ as the full group of conformal/anticonformal automorphisms.

**(with G. Girondo, G. Gonzalez-Dieaz and G. Jones )**Zapponi-orientable dessins d'enfants.

Revista Matematica Iberoamericana 36 No 2 (2020), 549-570. ISSN: 0213-2230

Almost two decades ago Zapponi introduced a notion of orientability of a clean dessin d'enfant, based on an orientation of the embedded bipartite graph. We extend this concept, which we call Z-orientability to distinguish it from the traditional topological definition, to the wider context of all dessins, and we use it to define a concept of twist orientability, which also takes account of the Z-orientability properties of those dessins obtained by permuting the roles of white and black vertices and face-centres. We observe that these properties are Galois-invariant, and we study the extent to which they are determined by the standard invariants such as the passport and the monodromy and automorphism groups. We find that in general they are independent of

these invariants, but in the case of regular dessins they are determined by the monodromy group.

**A remark on the field of moduli of Riemann surfaces.**Archiv der Mathematik 114 (2020) 515-526., ISSN: 0003-889X https://doi.org/10.1007/s00013-019-01411-9

Let $S$ be a closed Riemann surface of genus $g\geq 2$ and let ${\rm Aut}(S)$ be its group of conformal automorphisms. It is well known that if either: (i) ${\rm Aut}(S)$ is trivial or (ii) $S/{\rm Aut}(S)$ is an orbifold of genus zero with exactly three cone points, then $S$ is definable over its field of moduli ${\mathcal M}(S)$. In the complementary situation, explicit examples for which ${\mathcal M}(S)$ is not a field of definition are known. We provide upper bounds for the minimal degree extension of ${\mathcal M}(S)$ by a field of definition in terms of the quotient orbifold $S/{\rm Aut}(S)$.

**A sufficiently complicated noded Schottky group of rank three.**Cubo, A Mathematical Journal 22 (1) (2020), 39-53. ISSN: 0719-0646.

In 1974, Marden proved the existence of non-classical Schottky groups by a theoretical and non-constructive argument. Explicit examples are only known in rank two; the first one by Yamamoto in 1991 and later by Williams in 2009. In 2006, Maskit and the author provided a theoretical method to construct non-classical Schottky groups in any rank. The method assumes the knowledge of certain algebraic limits of Schottky groups, called sufficiently complicated noded Schottky groups. The aim of this paper is to provide explicitly a sufficiently complicated noded Schottky group of rank three and explain how to use it to construct explicit non-classical Schottky groups.

**Geometric description of Virtual Schottky groups.**Bulletin of the London Math. Soc. 55 (2020), 530-545. ISSN: 1469-2120.

A virtual Schottky group is a Kleinian group $K$ containing a Schottky group $G$ as a finite index normal subgroup. These groups correspond to those groups of automorphisms of closed Riemann surfaces which can be realized at the level of their lowest uniformizations. In this paper we provide a geometrical structural decomposition of $K$. When $K/G$ is an abelian group, an explicit free product decomposition in terms of Klein-Maskit's combination theorems is provided.

**(with J. C. Garcia)**On n-th roots of meromorphic maps.

Revista Colombiana de Matematicas 54 (2020), 65-74.

Let S be a connected Riemann surface and $\varphi:S \to \widehat{\mathbb C}$ be a holomorphic branched covering. We provide a geometrical condition for $\varphi$ to admit n-roots. This extends previous results done for n=2.

**Holomorphic differentials of Generalized Fermat curves.**Journal of Number Theory 217 (2020), 78-101.

A non-singular complete irreducible algebraic curve $F_{k,n}$, defined over an algebraically closed field $K$, is called a generalized Fermat curve of type $(k,n)$, where $n, k \geq 2$ are integers and $k$ is relatively prime to the characteristic $p$ of $K$, if it admits a group $H \cong {\mathbb Z}_{k}^{n}$ of automorphisms such that $F_{k,n}/H$ is isomorphic to ${\mathbb P}_{K}^{1}$ and it has exactly $(n+1)$ cone points, each one of order $k$. By the Riemann-Hurwitz-Hasse formula, $F_{k,n}$ has genus at least one if and only if $(k-1)(n-1) >1$. In such a situation, we construct a basis, called a standard basis, of its space $H^{1,0}(F_{k,n})$ of regular forms, containing a subset of cardinality $n+1$ that provides an embedding of $F_{k,n}$ into ${\mathbb P}_{K}^{n}$ whose image is the fiber product of $(n-1)$ classical Fermat curves of degree $k$. For $p=2$, we obtain

a lower bound (which is sharp for $n=2,3$) for the dimension of the space of exact one-forms, that is, the kernel of the Cartier operator.

We also do this for $(p,k,n)=(3,2,4)$.

**2021**

**(with. A.Carocca and R. E. Rodriguez)**q-etale covers of cyclic p-gonal covers.

Journal of Algebra 573 (2021), 393-409.

In this paper we study the Galois group of the Galois cover of the composition of a $q$-cyclic \'etale cover and a cyclic $p$-gonal cover for any odd prime $p$. Furthermore, we give properties of isogenous decompositions of certain Prym and Jacobian varieties associated to intermediate subcovers given by subgroups.

**Constructing jacobian varieties with many elliptic factors.**Proceedings "Geometry at the Frontier Symmetries and Moduli Spaces of Algebraic Varieties" , Contemporary Mathematics 766 (2021), pp. 201-216. (ISBN: 978-1-4704-6422-6)

Given two elliptic curves, $E_{1}$ and $E_{2}$, Earle provided an explicit genus two Riemann surface $R_{2}$ such that $JR_{2} \cong_{isog.} E_{1} \times E_{2}$.

In this paper, given $s \geq 3$ elliptic curves $E_{1},\ldots, E_{s}$,

we construct an explicit closed Riemann surface $R_{s}$, of genus $g=1+2^{s-2}(s-2)$, such that $JR_{s} \cong_{isog.} E_{1} \times \cdots \times E_{s} \times A$, where $A$ is also a product of at least $s(s-3)/2$ elliptic curves and jacobian varieties of some hyperelliptic Riemann surfaces, all of these curves explicitly given in terms of the given elliptic curves. In particular, for every triple $E_{1}, E_{2}, E_{3}$ of elliptic curves this provides an explicit Riemann surface $R_{3}$ of genus three with $JR_{3} \cong_{isog.} E_{1} \times E_{2} \times E_{3}$.

**(with Sebastian Reyes-Carocca)**Weil's descent theorem from a computational point of view.

Proceedings Geometry at the Frontier Symmetries and Moduli Spaces of Algebraic Varieties" , Contemporary Mathematics 766 (2021), pp. 217-228. (ISBN: 978-1-4704-6422-

Let ${\mathcal L}/{\mathcal K}$ be a finite Galois extension and let $X$ be an affine algebraic variety defined over ${\mathcal L}$. Weil's Galois descent theorem provides necessary and sufficient conditions for $X$ to be definable over ${\mathcal K}$, that is, for the existence of an algebraic variety $Y$ defined over ${\mathcal K}$ together with a birational isomorphism $R:X \to Y$ defined over ${\mathcal L}$. Weil's proof does not provide a method to construct the birational isomorphism $R.$ The aim of this paper is to give an explicit construction of $R$.

**Automorphism groups of origami curves.**Archiv der Mathematik 116 (2021), 385-390. (https://doi.org/10.1007/s00013-020-01559-9) ISSN: 0003-889X.

A closed Riemann surface $S$ (of genus at least one) is called an origami curve if it admits a non-constant holomorphic map $\beta:S \to E$ with at most one branch value, where $E$ is a genus one Riemann surface. In this case, $(S,\beta)$ is called an origami pair and ${\rm Aut}(S,\beta)$ is the group of conformal automorphisms $\phi$ of $S$ such that $\beta=\beta \circ \phi$. Let $G$ be a finite group. It is a known fact that $G$ can be realized as a subgroup of ${\rm Aut}(S,\beta)$ for a suitable origami pair $(S,\beta)$. It is also known that $G$ can be realized as a group of conformal automorphisms of a Riemann surface $X$ of genus $g \geq 2$ and with quotient orbifold $X/G$ of genus $\gamma \geq 1$. Given a conformal action of $G$ on a surface $X$ as before,

we prove that there is an origami pair $(S,\beta)$, where $S$ has genus $g$ and $G \cong {\rm Aut}(S,\beta)$ such that the actions of ${\rm Aut}(S,\beta)$ on $S$ and that of $G$ on $X$ are topologically equivalent.

**(with Raquel Diaz)**Stable Riemann orbifolds of Schottky type.

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales (RACSAM) 115, 111 (2021). https://doi.org/10.1007/s13398-021-01052-0. (ISSN: 1578-7303)

Let ${\mathcal O}$ be a stable Riemann orbifold, that is, a closed $2$-dimensional orbifold with nodes such that each connected component of the complement of the nodes has an analytically finite complex structure of hyperbolic type. We say that ${\mathcal O}$ is of Schottky type if there is a virtual noded Schottky group $K$ such that $\Omega^{ext}/K$ is isomorphic to it, where $\Omega^{ext}$ is the extended domain of discontinuity of $K$.

This is the same as saying that $\mathcal O$ is the conformal boundary at infinity of the hyperbolic $3$-dimensional handlebody orbifold $\mathbb H^3/K$.

In this paper we prove that the stable Riemann orbifolds of certain signature are of Schottky type.

**(with Saul Quispe)**On Real and Pseudo-Real Rational Maps.

Nonlinearity 34 (2021) 6248-6272. https://doi.org/10.1088/1361-6544/ac12ad (ISSN: 0951-7715)

The moduli space ${\rm M}_{d}$, of complex rational maps of degree $d \geq 2$, is a connected complex orbifold which carries a natural real structure, coming from usual complex conjugation. Its real points are the classes of rational maps admitting antiholomorphic automorphisms. The locus of the real points ${\rm M}_{d}({\mathbb R})$ decomposes as a disjoint union of the loci ${\rm M}_{d}^{\mathbb R}$, consisting of the real rational maps, and ${\mathcal P}_{d}$, consisting of the pseudo-real ones. We obtain that, both ${\rm M}_{d}^{\mathbb R}$ and ${\rm M}_{d}({\mathbb R})$, are connected and that ${\mathcal P}_{d}$ is disconnected. We also observe that the group of holomorphic automorphisms of a pseudo-real rational map is either trivial or a cyclic group. For every $n \geq 1$, we construct pseudo-real rational maps whose group of holomorphic automorphisms is cyclic of order $n$. As the field of moduli of a pseudo-real rational map is contained in ${\mathbb R}$, these maps provide examples of rational maps which are not definable over their field of moduli. It seems that these are the only explicit examples in the literature (Silverman) of rational maps which cannot be defined over their field of moduli.

We provide explicit examples of real rational maps which cannot be defined over their field of moduli. Finally, we also observe that every real rational map, which admits a model over the algebraic numbers, can be defined over the real algebraic numbers.

**2022**

**Dessins d'enfants with a given bipartite graph.**Contemporary Mathematics 776 (2022), pp. 249-267 (ISBN: 978-1-4704-6025-9)

An algorithm to produce all those dessins (up to isomorphisms) with a given bipartite graph is provided.

**(with Sebastian Reyes-Caroca and Angelica Vega)**Fiber products of Riemann surfaceds

Contemporary Mathematics 776 (2022), pp. 161-175 (ISBN: 978-1-4704-6025-9)

If $S_{0}, S_{1}, S_{2}$ are connected Riemann surfaces, $\beta_{1}:S_{1} \to S_{0}$ and $\beta_{2}:S_{2} \to S_{0}$ are surjective holomorphic maps, then the associated fiber product $S_{1} \times_{(\beta_{1},\beta_{2})} S_{2}$ has the structure of a one-dimensional complex analytic space, endowed with a canonical map $\beta: S_{1} \times_{(\beta_{1},\beta_{2})} S_{2} \to S_{0}$, such that, for $j=1,2$, $\beta_{j} \circ \pi_{j}=\beta$, where $\pi_{j}: S_{1} \times_{(\beta_{1},\beta_{2})} S_{2} \to S_{j}$ is the natural coordinate projection.The connected components of its singular locus provide its irreducible components. A Fuchsian description of the irreducible components of $S_{1} \times_{(\beta_{1},\beta_{2})} S_{2}$ is provided and, as a consequence, we obtain that if one the maps $\beta_{j}$ is a regular branched covering, then all its irreducible components are isomorphic. Also, if both $\beta_{1}$ and $\beta_{2}$ are of finite degree, then we observe that the number of these irreducible components is bounded above by the greatest common divisor of the two degrees, and that such an upper bound is sharp.

We also provide sufficient conditions for the irreducibility of the connected components of the fiber product. In the case that $S_{0}=\widehat{\mathbb C}$, and $S_{1}$ and $S_{2}$ are compact, we define the strong field of moduli of the pair $(S_{1} \times_{(\beta_{1},\beta_{2})} S_{2},\beta)$ and observe that this field coincides with the minimal field containing the fields of moduli of both pairs $(S_{1},\beta_{1})$ and $(S_{2},\beta_{2})$. Finally, in the case all surfaces $S_{1}$, $S_{2}$ and $S_{0}$ are compact and the fiber product is a connected Riemann surface, we observe that the Jacobian variety $J(S_{1} \times_{(\beta_{1},\beta_{2})}S_{2}) \times JS_{0}$ is isogenous to $JS_{1} \times JS_{2} \times P$, where $P$ is a suitable abelian subvariety of $J (S_{1} \times_{(\beta_{1},\beta_{2})}S_{2})$.

*)*

**(with Y. Atarihuana, J. Garcia, S. Quispe and C. Ramirez M**Dessins d’enfants and some holomorphic structures on the Loch Ness monster.

Quarterly Journal of Mathematics73 (2022), 349-369 (ISSN 0033-5606)

The classical theory of dessin d'enfants, which are bipartite maps on compact orientable surfaces, are combinatorial objects used to study branched covers between compact Riemann surfaces and the absolute Galois group of the field of rational numbers. In this paper, we show how this theory is naturally extended to non-compact orientable surfaces and, in particular, we observe that the Loch Ness monster (the surface of infinite genus with exactly one end) admits infinitely many regular dessins d'enfants (either chiral or reflexive). In addition, we study different holomorphic structures on the Loch Ness monster, which come from homology covers of compact Riemann surfaces, infinite hyperelliptic and infinite superelliptic curves.

**ACCEPTED ARTICLES**

**(with Grzegorz Gromadzki)**Structural description of dihedral extended Schottky groups and application in study of symmetries of handlebodies.

Topology and its Applications (ISSN 0166-8641)

Given a symmetry $\tau$ of a closed Riemann surface $S$, there exists an extended Kleinian group $K$, whose orientation-preserving half is a Schottky group $\Gamma$ uniformizing $S$, such that $K/\Gamma$ induces $\langle \tau \rangle$; the group $K$ is called an extended Schottky group. A geometrical structural description, in terms of the Klein-Maskit combination theorems, of both Schottky and extended Schottky groups is well known.

A dihedral extended Schottky group is a group generated by the elements of two different extended Schottky groups, both with the same orientation-preserving half. Such configuration of groups corresponds to closed Riemann surfaces together with two different symmetries and the aim of this paper is to provide a geometrical structure of them. This result can be used in study of three dimensional manifolds and as an illustration we give the sharp upper bounds for the total number of connected components of the locus of fixed points of two and three different symmetries of a handlebody with a Schottky structure.

**(with A. Cañas, F. Turiel and A. Viruel)**Groups as automorphisms of dessins d'enfants

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales (RACSAM) (ISSN: 1578-7303)

It is known that every finite group can be represented as the full group of automorphisms of a suitable compact dessin d'enfant. In this paper, we give a constructive and easy proof that the same holds for any countable group by considering non-compact dessins. Moreover, we show that any tame action of a countable group is so realisable.

Smooth quotients of generalized Fermat curves

Revista Matemática Complutense (ISSN 1139-1138)

A closed Riemann surface $S$ is called a generalized Fermat curve of type $(p,n)$, where $n,p \geq 2$ are integers such that $(p-1)(n-1)>2$, if it admits a group $H \cong {\mathbb Z}_{p}^{n}$ of conformal automorphisms with quotient orbifold $S/H$ of genus zero with exactly $n+1$ cone points, each one of order $p$; in this case $H$ is called a generalized Fermat group of type $(p,n)$. In this case, it is known that $S$ is non-hyperelliptic and that $H$ is its unique generalized Fermat group of type $(p,n)$. Also, explicit equations for them, as a fiber product of classical Fermat curves of degree $p$, are known. For $p$ a prime integer, we describe those subgroups $K$ of $H$ acting freely on $S$, together with algebraic equations for $S/K$, and determine those $K$ such that $S/K$ is hyperelliptic.

On $p$-gonal fields of definition

Ars Mathematica Contemporanea (ISSN 1855-3966)

Let $S$ be a closed Riemann surface of genus $g \geq 2$ and $\varphi$ be a conformal automorphism of $S$ of prime order $p$ such that $S/\langle \varphi \rangle$ has genus zero.

Let ${\mathbb K} \leq {\mathbb C}$ be a field of definition of $S$.

We provide an argument for the existence of a field extension ${\mathbb F}$ of ${\mathbb K}$, of degree at most $2(p-1)$, for which $S$ is definable by a curve of the form $y^{p}=F(x) \in {\mathbb F}[x]$, in which case $\varphi$ corresponds to $(x,y) \mapsto (x,e^{2 \pi i/p} y)$.

If, moreover, $\varphi$ is also definable over ${\mathbb K}$, then ${\mathbb F}$ can be chosen to be at most a quadratic extension of ${\mathbb K}$.

For $p=2$, that is when $S$ is hyperelliptic and $\varphi$ is its hyperelliptic involution, this fact is due to Mestre (for even genus) and Huggins and Lercier-Ritzenthaler-Sijslingit in the case that ${\rm Aut}(S)/\varphi\rangle$ is non-trivial.