Moduli of path families on hyperelliptic algebraic curves

Mika Seppälä

University of Helsinki and Florida State University

 

 

1.   Introduction

 

The concept of moduli of path families as defined by Ahlfors and Beurling in (Ahlfors and Beurling 1952) extends to homotopy classes of simple closed curves on Riemann surfaces as follows.  Let  X be a compact Riemann surface, and α a simple closed curve on  X.   Let Γ(α)  denote the set of simple closed curves on  X  homotopic to the given curve α.   Let M (α) denote the set of all metrics ρdw on  X  such that

 

∫β ρdw≥ 1

for all rectifiable curves  β  in  Γ(α).  Define the modulus, m(α), of the path family Γ(α)    by setting

m(α) = inf ∫∫X  ρ2 du dv,

where the infimum is taken over all metrics ρdw  in  M (α).  

 

Let      

 

(1.1)

 

 

   

 

 be a hyperelliptic affine algebraic plane curve.  Here we assume that the branch points  are all distinct.  Every hyperelliptic curve of genus g can always be represented in this form, and this standard representation for any hyperelliptic curve can always be found algorithmically.  The algcurves package of Maple has  built in routines for testing whether a given curve is hyperelliptic and if it is, then the above standard presentation can always be found automatically.

 

In this paper we define an algorithm that will compute the modulus of a given homotopy class of simple closed curves on the curve C defined by the equation (1.1).  The key tool in this algorithm is the Myrberg algorithm for finding a Schottky uniformization of a hyperelliptic curve.   The algorithm was developed by P. J. Myrberg (1926).  For details of this see (Myrberg 1920)  or for a modern exposition, with detailed proofs and extension to Fuchsian uniformizations, see (Seppälä 1995), (Seppälä 1995), and (Seppälä 2003).

 

 

2.   Myrberg’s uniformization

 

For the convenience of the reader we recall here briefly Myrberg’s algorithm (see (Myrberg 1920), (Seppälä 2003).  To find a Schottky group uniformizing a hyperelliptic curve C given by equation (1.1) one forms  sequences ,  so that , and each  has been obtained from  by the inverse of a rational mapping of the complex plane onto itself.  This is the mapping (5) of Lemma 1 in (Seppälä 2003).  For the purposes of this paper it is not necessary to go into the details of how the sequences  have been formed.  By standard analytical considerations one shows that these sequences converge.  In fact they do converge rather rapidly. 

 

For  let

(2.1)

,

 

and, for , let

(2.2)

 

 

be the elliptic rotation of angle  with fixed-points .  If all the branch points  are real, then also the limits points  are real.   In this case the mappings  map the upper half plane onto the lower half plane.

 

Let

(2.3)

 

 

 

be the group generated by the products of the elliptic rotations  and .  The group   is a Schottky group acting in a domain  so that  

3.   Lifting of path families on algebraic curves

 

Consider a hyperelliptic curve  given by the affine equation (1.1).  The following arguments apply to any simple closed path   in the finite complex plane going around exactly two points , i.e., the bounded complement of the curve   contains exactly two points , and the path  does not go through any point .  For the sake of clarity, let us assume that  goes around  and .  Then  lifts to a simple closed path on the hyperelliptic curve   going around a handle of .   

 

 

Figure (3.1). This figure illustrates the hyperelliptic algebraic curve  together with the points  on the finite complex plane.  The genus of the curve is two, hence there are six points  in the plane.  These points are indicated as dots on the plane.  The path  is indicated by the dotted line in the plane (below).  The dotted line above is a lifting of the path .

 

Myrberg uniformization consists of the following:

  1. A group 
     of Möbius transformations generated by certain products of elliptic Möbius rotations.  This group acts discontinuously in a domain
     so that
    . The domain
     is the complement of the limit set of the group
    .
  2. An onto mapping
    , this mapping is the uniformizing projection.

 

To understand  the geometry of the group   better, let us, for the moment, assume that the branch points  are real and ordered so that .  We do this assumption to simplify considerations.  The results hold in the general case.

 

The corresponding points  are then also real and ordered in the same way as the original branch points .  Then the group

 

 

 

Ahlfors, L. V. and A. Beurling (1952). Conformal Invariants. Construction and application of conformal maps.  Proceedings of a symposium., Washington DC, U.S. Government Printing Office.

Myrberg, P. J. (1920). "Über die Numerische Ausführung der Uniformisierung." Acta Soc. Sci. Fenn. XLVIII(7): 1-53.

Seppälä, K. D. S. a. M. (1995). Numerical uniformization of hyperelliptic curves. ISSAC 1995.

Seppälä, M. (2003). "Myrberg's Numerical Uniformization of Hyperelliptic Curves." Ann. Acad. Sci. Fenn.

Seppälä, P. B. a. M. (1995). Computing on Riemann surfaces. Topology and Teichmüller spaces, Katinkulta, Sotkamo, Finland, World Sci. Publishing.