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.

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