Cyclic Trigonal Riemann Surfaces of Genus 4 (original) (raw)

Equisymmetric Strata of the Moduli Space of Cyclic Trigonal Riemann Surfaces of Genus 4

Glasgow Mathematical Journal, 2008

A closed Riemann surface which can be realized as a three-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. If the trigonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic trigonal Riemann surface. Using the characterization of cyclic trigonality by Fuchsian groups, we find the structure of the space of cyclic trigonal Riemann surfaces of genus 4.

On real trigonal Riemann surfaces

MATHEMATICA SCANDINAVICA, 2006

A closed Riemann surface X which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering will be called a trigonal morphism. A trigonal Riemann surface X is called real trigonal if there is an anticonformal involution (symmetry) σ of X commuting with the trigonal morphism. If the trigonal morphism is a cyclic regular covering the Riemann surface is called real cyclic trigonal. The species of the symmetry σ is the number of connected components of the fixed point set Fix(σ) and the orientability of the Klein surface X/ σ. We characterize real trigonality by means of Fuchsian and NEC groups. Using this approach we obtain all possible species for the symmetry of real cyclic trigonal and real non-cyclic trigonal Riemann surfaces.

Maximal order of automorphisms of trigonal Riemann surfaces

Journal of Algebra, 2010

In this paper we find the maximal order of an automorphism of a trigonal Riemann surface of genus g, g ≥ 5. We find that this order is smaller for generic than for cyclic trigonal Riemann surfaces, showing that generic trigonal surfaces have "less symmetry" than cyclic trigonal surfaces. Finally we prove that the maximal order is attained for infinitely many genera in both the cyclic and the generic case.

Basis of homology adapted to the trigonal automorphism of a Riemann surface

2007

A closed (compact without boundary) Riemann surface S of genus g is said to be trigonal if there is a three sheeted covering (a trigonal morphism) from S to the Riemann sphere, f : S −→Ĉ. If there is an automorphism of period three, φ, on S permuting the sheets of the covering, we shall call S cyclic trigonal and φ will be called trigonal automorphism.

One-dimensional families of Riemann surfaces of genus g with 4\text {g}+4$$ 4 g + 4 automorphims

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2017

We prove that the maximal number ag + b of automorphisms of equisymmetric and complex-uniparametric families of Riemann surfaces appearing in all genera is 4g + 4. For each integer g ≥ 2 we find an equisymmetric complex-uniparametric family A g of Riemann surfaces of genus g having automorphism group of order 4g + 4. For g ≡ −1mod4 we present another uniparametric family K g with automorphism group of order 4g + 4. The family A g contains the Accola-Maclachlan surface and the family K g contains the Kulkarni surface.

On Riemann surfaces of genus g with 4g–4 automorphisms

Israel Journal of Mathematics

In this article we study compact Riemann surfaces with a nonlarge 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.

Riemann surface with cyclic automorphisms group

Proyecciones (Antofagasta), 1997

In t his paper. we present tllC' uniformization of y 2 = .rP-l, with p > 5 aurl prime. i. e .. the only hyperelliptic Riemann surface of gt'nus (/-7. \\"hich admit Z j2pZ as automorphism group. This 1111ifonnization is fouud by using a fuc:hsian group which rcflects the actiou of Z/2pZ aud is coustructed starting of a triangle group of !YJW (0:¡>.p.p). I\loreover. we describe completely the action of the automorphism group in hmnology. so that we can describe the invariant subvariety for Z /2pZ in A 9 (principally polarized abelian varieties of dimension y). which is detPrmiued bv the real Abe! aplication from M 9 in A 9 .

On cyclic p-gonal Riemann surfaces with several p-gonal morphisms

Geometriae Dedicata, 2009

Let p be a prime number, p > 2. A closed Riemann surface which can be realized as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. If the p-gonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic p-gonal Riemann surface. Accola showed that if the genus is greater than (p − 1) 2 the p-gonal morphism is unique. Using the characterization of p-gonality by means of Fuchsian groups we show that there exists a uniparametric family of cyclic p-gonal Riemann surfaces of genus (p−1) 2 which admit two p-gonal morphisms. In this work we show that these uniparametric families are connected spaces and that each of them is the Riemann sphere without three points. We study the Hurwitz space of pairs (X, f), where X is a Riemann surface in one of the above families and f is a p-gonal morphism, and we obtain that each of these Hurwitz spaces is a Riemann sphere without four points.

On compact Riemann surfaces with dihedral groups of automorphisms

Mathematical Proceedings of the Cambridge Philosophical Society, 2003

We study compact Riemann surfaces of genus g 2 having a dihedral group of automorphisms. We find necessary and sufficient conditions on the signature of a Fuchsian group for it to admit a surface kernel epimorphism onto the dihedral group D N. The question of extendability of the action of D N is considered. We also give an explicit parametrization of the moduli space of Riemann surfaces with maximal dihedral symmetry, showing that it is a one-dimensional complex manifold. Defining equations of all such surfaces and the formulae of their automorphisms are calculated. The locus of this moduli space consisting of those surfaces admitting some real structure is determined.

On the one-dimensional family of Riemann surfaces of genus q with 4q automorphisms

Journal of Pure and Applied Algebra

Bujalance, Costa and Izquierdo have recently proved that all those Riemann surfaces of genus g ≥ 2 different from 3, 6, 12, 15 and 30, with exactly 4g automorphisms form an equisymmetric one-dimensional family, denoted by Fg. In this paper, for every prime number q ≥ 5, we explore further properties of each Riemann surface S in Fq as well as of its Jacobian variety JS.

Analytic computation of some automorphism groups of Riemann surfaces

Kodai Mathematical Journal, 2007

Equations for the locus of Riemann Surfaces of genus three with a nonabelian automorphism group generated by involutions are determined from vanishings of Riemann's theta function. Torelli's Theorem implies that all of the properties of a non-hyperelliptic compact Riemann Surface (complex algebraic curve) X are determined by its period matrix W. This paper shows how to compute the group Aut X of conformal automorphisms of a surface X of genus three using W, in the case when the group is nonabelian and generated by its involutions. The connection between W and X is Riemann's theta function yðz; WÞ. Accola ([1], [2], [3]), building on classical results about hyperelliptic surfaces, found relationships between the theta divisor Y ¼ fz A JacðX Þ : yðz; WÞ ¼ 0g and Aut X. In the case of genus three, certain vanishings of y at quarter-periods of JacðX Þ imply that X has an automorphism s of degree two (or involution) such that X =hsi has genus one (making s an elliptic-hyperelliptic involution). This work derives equations in the moduli space of surfaces of genus three for many of the loci consisting of surfaces with a given automorphism group. It is a two-step process. First, topological arguments determine the order of the dihedral group generated by two non-commuting involutions. Then, combinatorial arguments about larger groups generated by involutions determine the theta vanishings corresponding to each. Much of the work here is based on the author's 1981 PhD dissertation [7] at Brown University. It appears now because of renewed interest in these questions, some of which is inspired by questions in coding theory: See [3], [5]. The research was directed by R. D. M. Accola, and Joe Harris was also a valuable resource. The author extends his (belated) thanks to them. 1. Preliminaries and notation In all that follows, X is a compact Riemann Surface (or complex algebraic curve) of genus three with automorphism group Aut X , period matrix W, jacobian 394

Riemann Surfaces, Plane Algebraic Curves and Their Period Matrices

Journal of Symbolic Computation, 1998

The aim of this paper is to present theoretical basis for computing a representation of a compact Riemann surface as an algebraic plane curve and to compute a numerical approximation for its period matrix. We will describe a program Cars ) that can be used to de ne Riemann surfaces for computations. Cars allows one also to perform the Fenchel{Nielsen twist and other deformations on Riemann surfaces.

On branched covering of compact Riemann surfaces with automorphisms

1997

In this work, we give an algor'ithm to count the different conforma/ equivalence classes of compact Riemann surfaces that admit a group of automorphisms isomorphic to Z/nZ, n E N, and that are branched coverings ofthe Riemann sphere, with signature ((n,O);m 1 ,m 2 ,m 3), m 1 ,mz,m3 E N. By using the previous result, we count the different conforma/ equivalence classes of compact Riemann surfaces in the cases of coverings with signature ((p,O);p,p,p), p 2:: 5 and prime, and signature ((p2,0);p 2 ,p 2 ,p), p 2::3 and prime.

Unicellular Dessins and a Uniqueness Theorem for Klein's Riemann Surface of Genus 3

Bulletin of The London Mathematical Society, 2001

If we consider the 14-sided hyperbolic polygon of Felix Klein that defines his famous surface of genus 3, then we observe that we have a uniform, unifacial dessin whose automorphism group is transitive on the edges, but not on the directed edges of the dessin. We show that Klein's surface is the unique platonic surface with this property.

On the Connectedness of the Branch Locus of the Moduli Space of Riemann Surfaces of Genus 4

Glasgow Mathematical Journal, 2010

Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space of surfaces of genus 4, we prove its connectedness. As a consequence, one can deform a surface of genus 4 with automorphisms, i.e. symmetric, to any other symmetric genus 4 surface through a path consisting entirely of symmetric surfaces.

On Genus Two Riemann Surfaces Formed from Sewn Tori

Communications in Mathematical Physics, 2007

We describe the period matrix and other data on a higher genus Riemann surface in terms of data coming from lower genus surfaces via an explicit sewing procedure. We consider in detail the construction of a genus two Riemann surface by either sewing two punctured tori together or by sewing a twice-punctured torus to itself. In each case the genus two period matrix is explicitly described as a holomorphic map from a suitable domain (parameterized by genus one moduli and sewing parameters) to the Siegel upper half plane H 2 . Equivariance of these maps under certain subgroups of Sp(4, Z) is shown. The invertibility of both maps in a particular domain of H 2 is also shown.

Lectures notes on compact Riemann surfaces

arXiv: Mathematical Physics, 2018

This is an introduction to the geometry of compact Riemann surfaces, largely following the books Farkas-Kra, Fay, Mumford Tata lectures. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations. 2) Space of meromorphic functions and forms, we classify them with the Newton polygon. 3) Abel map, the Jacobian and Theta functions. 4) The Riemann--Roch theorem that computes the dimension of spaces of functions and forms with given orders of poles and zeros. 5) The moduli space of Riemann surfaces, with its combinatorial representation as Strebel graphs, and also with the uniformization theorem that maps Riemann surfaces to hyperbolic surfaces. 6) An application of Riemann surfaces to integrable systems, more precisely finding sections of an eigenvector bundle over a Riemann surface, which is known as the "algebraic reconstruction" method in integrable systems, and we mention how it is related to Baker-Akhiezer functions and Tau funct...