Belyi functions: multiplicities and parasitic solutions
by
Elena Kreines

In his classical work [1.] A. Grothendieck put forward a theory of Dessins d'enfants which provides a plenty of new and non-trivial interrelations between different structures in category theory, algebra, algebraic geometry, complex analysis, topology, etc. The equivalence between categories of dessins d'enfants and Belyi pairs is among the key directions of this theory. In particular, this equivalence establishes an approach to a visualization of algebraic curves over number fields and to an interpretation of the action of Galois group Aut([`(Q)]) on the set of their isomorphism classes.

The notion of multiple and parasitic solutions of the equation system on Belyi pair for a given graph is introduced. Series of examples of graphs possessing with geometrical parasitical solutions are considered. It is proved that there are some classes of trees that have no parasitical and multiple solutions. Some examples of non-geometrical parasitical solutions are provided.

References

[1.] A. Grothendieck, Esquisse d'un programme, London Math. Soc. Lecture Notes Series 243, Cambridge Univ. Press, 1997, 3 - 43.

Date received: January 28, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagi-11.