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 algebraic curves and Jacobian varieties are only different 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 eigenvalues 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 hyperelliptic algebraic curves, interesting results are known (see Seppälä, Mika: Computation of period matrices of real algebraic curves.-Discrete Comput Geom 11:65-81 (1994), or Semmler, K.-D., and M. Seppälä: Numerical uniformization of hyperelliptic curves and references given there). We will review some of these results and consider implementations of programs needed to make practical use of these results. These implementations make use of a larger program, Cars, currently under development, of which we will describe some features in this paper. Cars stands for "Computer Algebra Riemann Surfaces" and offers a convenient way of defining Möbius transformations and Riemann surfaces for computations.
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