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\centerline{\large \bf Computations on tori}

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\centerline{Mika Sepp\"al\"a} \centerline{\it University of
Helsinki, Finland, and Florida State University, USA}
\centerline{e-mail {\tt Mika.Seppala@Helsinki.Fi}}


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{\bf Problem 1:} Given two smooth algebraic curves, $C_1$ and
$C_2$, of the same genus, find extremal quasiconformal mappings
$C_1\to C_2$.  If the curves are given as affine plane curves, one
would like to be able to find the Teichm\"uller extremal mapping
in a given homotopy class of mappings $C_1\to C_2$ in some
explicit way.

In the case of tori, this problem has a complete solution.
Teichm\"uller mappings can be expressed in terms of certain
elliptic integrals, and the Teichm\"uller distance between two
given tori can be explicitly computed.  This is work of C. Nolder,
E. Klassen, M. Sepp\"al\"a and T. Sutton.  The methods rely on
explicit numerical uniformizations of tori.

  Let $T^g$ be the Teichm\"uller space of genus $g$
Riemann surfaces, and let $\Gamma_\alpha$ be the family of simple
closed curves homotopic to a given simple closed curve $\alpha$ on
a topological genus $g$ surface. One can define the modulus
function
\[
m_\alpha : T^g\to {\mathbb R},\; [X] \mapsto \text{ modulus of
$\Gamma_\alpha$  on $X$}.
\]
Moduli of path families are defined in analysis, and they form an
important tool in the study of quasiconformal mappings.  The
number $m_\alpha (X)$ is related to the euclidean area of
euclidean cynliders embedded in $X$ and containing the curve
$\alpha$.

{\bf Problem 2:}  For a given algebraic curve $C$ and a given
simple closed curve $\alpha$ on $C$, compute $m_\alpha(C)$ in an
explicit way.

In the case of tori, this problem has an explicit solution again
in terms of certain elliptic integrals.



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