## 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 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.

The full text (23 pages)
is available either as a
dvi-file or as a
postscript-file.

The second author wishes to thank the ** Taniguchi foundation **
for generous support.
This work has also been supported by the European Communities
HCM-network "Computational Conformal Geometry",
contract nr ERBCHRX-CT-930408,
by the Academy of Finland and by the Swiss National Science Foundation Grant
20-34099.92.

April 1, 1996
Mika Seppälä