Computing on Riemann Surfaces

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 reader's 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 …