On computing Belyi maps

Jeroen Sijsling

Date(s) : 13/02/2014   iCal
11 h 00 min - 12 h 00 min

A Belyi map is a finite morphism to the complex projective line that is ramified above at most three points. Surprisingly, the algebraic curves that admit a Belyi map are exactly those that are defined over the algebraic closure of QQ . Due to the simple combinatorial description of covers as finite sets with an action of the fundamental group, the theory of Belyi maps therefore gives a way to study the absolute Galois group of QQ, one of Grothendieck’s dreams.
This talk focuses on the efforts so far to compute Belyi maps explicitly.
The currently available techniques are described, including a recent one due to Voight et al. that uses modular functions. Many asides are explored, such as Galois Belyi maps and questions concerning field of definition. http://arxiv.org/abs/1311.2529

Jeroen Sijsling, Institut für Reine Mathematik at the Universität Ulm

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