Canonical models for triangular modular curves

Contact person for this talk and zoom details: Harry Smit (smit@mpim-bonn.mpg.de)

The study of the classical modular curves has rewarded
mathematicians for perhaps a century.  Triangular modular curves are a
certain generalization of modular curves that arise from quotients of
the upper half-plane by congruence subgroups of hyperbolic triangle
groups.  Despite being nonarithmetic in almost all cases, they
nevertheless carry several appealing features in common with the
classical case: for example, they are defined over explicitly given
number fields, and they have a moduli interpretation over the complex
numbers (by work of Cohen-Wolfart).  We report on progress to extend
this moduli interpretation, exhibiting a canonical model for triangular
modular curves and their modular embeddings into quaternionic and
unitary Shimura varieties.  This is joint work with Robert A. Kucharcyzk.