Curves over number fields

Speaker: Anthony Weaver
Title: Child's Drawings, map subgroups, and curves over number fields.

Abstract:
In 1984, Grothendieck observed that any freely-drawn connected graph composed of vertices and edges (he called them "child's drawings" or dessins d'enfants), together with a freely-chosen cyclic ordering of the edges incident at each vertex, determines a unique algebraic curve defined over an algebraic number field. This extraordinary fact is a consequence of work by Jones and Singerman in 1978, who completed a discrete version of classical uniformization theory, and G. Bely˘ı in 1979, who proved that a curve can be defined over a number field if and only if it admits a meromorphic function with at most three critical values.

Dessins d'enfants (also known as maps, or, more generally, hypermaps) have a purely algebraic description in terms of finite-index subgroups of Fuchsian triangle groups. There is an exact dictionary between algebraic properties of these subgroups and geometric properties of the maps. I'll describe an abridged version of this dictionary, and then specialize to maps with just one vertex. I'll sketch a proof that even this highly restricted class of maps is sufficient to recover several interesting and well-known families of curves, as well as Klein's famous quartic curve of genus 3, admitting 168 automorphisms

Category:

Education

Tags:

• anthony weaver
• map
• subgroup
• curves
• number
• fields
• Grothendieck
• connected
• graph
• algebraic
• classical
• uniformization
• theory
• yt:stretch=16:9

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The speaker looks like Grothendieck