Saúl Quispe

Universidad de La Frontera

Conformal/anticonformal actions of the quasi-abelian group $QA_{n}$ of order $2^n$, for $n\geq 4$, on closed Riemann surfaces, pseudo-real Riemann surfaces and closed Klein surfaces are considered. We obtain several consequences, such as the solution of the minimum genus problem for the $QA_n$-actions, and for each of these actions, we study the to...

Dear colleagues: Along with greetings, we are pleased to invite you to participate in the "Workshop on surfaces in the frontier", to be held on February 1, 2, and 3, 2023, in a hybrid way, at the National University of Colombia (Manizales Headquarters). , in the areas of surfaces of finite and infinite type. The website that contains more informa...

Considering non-constant holomorphic maps $\beta_{i}:S_{i}\to S_{0}$, $i\in\{1,2\}$, between non-compact Riemann surfaces for which it is associated its fiber product $S_{1}\times_{(\beta_{1},\beta_{2})}S_{2}$. With this setting, in this paper we relate the ends space of such fiber product to the ends space of its normal fiber product. Moreover, we...

In this paper, we discuss certain types of conformal/anticonformal actions of the generalized quasi-dihedral group $G_{n}$ of order $8n$, for $n\geq 2$, on closed Riemann surfaces, pseudo-real Riemann surfaces and compact Klein surfaces, and in each of these actions we study the uniqueness (up to homeomorphisms) action problem.

The theory of dessins d’enfants on compact Riemann surfaces, 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 ori...

The classical theory of dessin d'enfants, which are bipartite maps on compact 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 surfaces and, in particular, we o...

A conformal automorphism $\tau$, of order $n \geq 2$, of a closed Riemann surface $\X$, of genus $g \geq 2$, which is central in ${\rm Aut}(\X)$ and such that $\X/\langle \tau\rangle$ has genus zero, is called a superelliptic automorphism of level $n$. If $n=2$, then $\tau$ is called hyperelliptic involution and it is known to be unique. In this pa...

Let Gn be the dicyclic group of order 4n. We observe that, up to isomorphisms, (i) for n≥2 even there is exactly one regular dessin d'enfant with automorphism group Gn, and (ii) for n≥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 Gn acts purely...

We build a database of genus 2 curves defined over $\mathbb Q$ which contains all curves with minimal absolute height $h \leq 5$, all curves with moduli height $\mathfrak h \leq 20$, and all curves with extra automorphisms in standard form $y^2=f(x^2)$ defined over $\mathbb Q$ with height $h \leq 101$. For each isomorphism class in the database, an...

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. All of them are produced on very well known hyperelliptic Riemann surfaces. We observe, for each of the...

We build a database of genus 2 curves defined over $\Q$ which contains all curves with minimal absolute height $\h \leq 5$, all curves with moduli height $\mH \leq 20$, and all curves with extra automorphisms in standard form $y^2=f(x^2)$ defined over $\Q$ with height $\h \leq 101$. For each isomorphism class in the database, an equation over its m...

Milnor proved that the moduli space M d of rational maps of degree d ≥ 2 has a complex orbifold structure of dimension 2( d − 1). Let us denote by ${\mathcal S}$d the singular locus of 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 ma...

Herradon has recently provided an example of a regular dessin d'enfant whose field of moduli is the non-abelian extension ℚ(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 ≥ 3 we...

It is well known that every closed Riemann surface S of genus g≥2, admitting a group G of conformal automorphisms so that S/G has triangular signature, can be defined over a finite extension of ℚ. It is interesting to know, in terms of the algebraic structure of G, if S can in fact be defined over ℚ. This is the situation if G is either abelian or...

A smooth complex projective curve is called pseudoreal if it is isomorphic to its conjugate but is not definable over the reals. Such curves, together with real Riemann surfaces, form the real locus of the moduli space $\mathcal M_g$. This paper deals with the classification of pseudoreal curves according to the structure of their automorphism grou...

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 Rodríguez using Kani–Rosen results. In the last, it was seen that all factors in the isogen...

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...

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 curves.

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 examples of real Riemann surfaces which cannot be defined over the field of moduli.

A closed Riemann surface $S$ is called a generalized Fermat curve of type $(p,n)$, where $p,n \geq 2$ are integers, if it admits a group $H \cong {\mathbb Z}_{p}^{n}$ of conformal automorphisms so that $S/H$ is an orbifold of genus zero with exactly $n+1$ cone points, each one of order $p$. It is known that $S$ is a fiber product of $(n-1)$ classic...

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. Kontogeorgis.

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...

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 automorphis...

Let $X$ be a smooth projective algebraic curve of genus $g\geq 2$ defined over a field $K$. We show that $X$ can be defined over its field of moduli if it has odd signature, i.e. if the signature of the covering $X\to X/\Aut(X)$ is of type $(0;c_1,...,c_k)$, where some $c_i$ appears an odd number of times. This result is applied to $q$-gonal curves...