Mika Seppälä

Computation of period matrices of real algebraic curves

In this paper we derive a numerical method which allows us to compute periods of differentials on areal algebraic curve with real points. This leads to an algorithm which can be implemented on a computer and can be used to study the Torelli mapping numerically.

Triangulations and moduli spaces of Riemann surfaces with group actions

SummaryWe study that subset of the moduli space

Computing on Riemann surfaces

\bar M^g

Short homology bases and partitions of Riemann surfaces

of stable genusg,g>1, Riemann surfaces which consists of such stable Riemann surfaces on which a given finite groupF acts. We show first that this subset is compact. It turns out that, for general finite groupsF, the above subset is not connected. We show, however, that for ℤ2 actions this subsetis connected. Finally, we show that even in the moduli space ofsmooth genusg Riemann surfaces, the subset of those Riemann surfaces on which ℤ2 actsis connected. In view of deliberations of Klein ([8]), this was somewhat surprising.These results are based on new coordinates for moduli spaces. These coordinates are obtained by certainregular triangulations of Riemann surfaces. These triangulations play an important role also elsewhere, for instance in approximating eigenfunctions of the Laplace operator numerically.

Real structures of Teichmüller spaces, Dehn twists, and moduli spaces of real curves

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 C ars (Semmler et al., 1996) that can be used to define Riemann surfaces for computations. C ars allows one also to perform the Fenchel?Nielsen twist and other deformations on Riemann surfaces.Almost all theoretical results presented here are well known in classical complex analysis and algebraic geometry. The contribution of the present paper is the design of an algorithm which is based on the classical results and computes first an approximation of a polynomial representing a given compact Riemann surface as a plane algebraic curve and further computes an approximation for a period matrix of this curve. This algorithm thus solves an important problem in the general case. This problem was first solved, in the case of symmetric Riemann surfaces, in Seppala (1994).

Novel Aspects of the Use of ICT in Mathematics Education

These notes are a review on computational methods that allow us to use computers as a tool in the research of Riemann surfaces, algebraic curves and Jacobian varieties. It is well known that compact Riemann surfaces, projective algebraiccurves and Jacobian varieties are only diierent views to the same object, i.e., these categories are equivalent. We want to be able to put our hands on this equivalence of categories. If a Riemann surface is given, we want to compute an equation representing it as a plane algebraic curve, and we want to compute a period matrix for it. Vice versa, we want to be able to compute the uniformization for a given algebraic plane curve, or a Riemann surface corresponding to a given Jacobian variety. In another direction we consider tools that allow us to compute eigenval-ues and eigenfunctions of the Laplace operator for Riemann surfaces. The correspondence between the Laplace spectrum of a Riemann surface and the geometry of the surface in general is intriguing. The programs to be described later give us a possibility to explore this correspondence in an explicit manner. The above mentioned computational problems are hard and most of them are open in the general case. In certain particular cases, like that of hyper-elliptic algebraic curves, interesting results are known We will review some of these results and consider implementations of programs needed to make practical use of these results. For the readers convenience, we also review some of the basic underlying mathematicalconcepts. Our basic referencesto the theory of Riemann surfaces are 4], 6], 8] and 12]. 1. Preliminaries In this section we describe a method for deening MM obius transformations for computations by a computer. We are interested, in particular, in MM obius transformations that map either the upper half{plane or the unit disk onto itself. Such a transformation can be expressed in the form z 7 ! az + b cz + d ; ad ? bc = 1: (1) Transformation (1) maps the upper{half plane U onto itself if and only if all the coeecients a; b; c; and d are real, and it maps the unit disk D onto itself if and only if c = b and d = a, as is well known. For computational purposes, we should deene a MM obius transformation by simply the formula (1), i.e., by giving either exact values (rational or algebraic numbers) The second …

Triangulations and homology of Riemann surfaces

The Teichmuller space T(Σ) of a compact C ∞-surface Σ can be parametrized by geodesic length functions. More precisely, we can find a set {α1... ,α n} of closed curves α j on Σ such that the isotopy class of a hyperbolic metric d on Σ (i.e. the point [d] ∊ T(Σ)) is determined by the lengths of geodesic curves homotopic to the curves α j on (Σ, d). However, since the fundamental group of Σ is not freely generated there is a quite complicated relation among these geodesic length function.

Numerical Schottky Uniformizations: Myrberg’s Opening Process

Abstract The homological systole of a compact Riemann surface X is the minimal length of a simple closed non-separating goedesic curve. Since any homology basis of X must contain curves that intersect any non-separating closed curve, surfaces having small homological systoles cannot have short homology basis. It turns out that this basically the only obstruction to finding short homology basis. We show, in fact, that a compact hyperbolic genus g Riemann surface X with homological systole e has always a canonical homology basis which consists of curves γ satisfying the length bound l (γ)⩽(g−1) 105g+4 arcsin ( 4 e )