A database of Belyi maps

Belyi maps are covers of the projective line ramified only above
0, 1, and the point at infinity. But that is merely their
algebraic-geometric guise. They can be interpreted in many alternative and
surprising ways, be it combinatorially, as triples of finite permutations
sigma_0, sigma_1, sigma_inf such that sigma_0 sigma_1 sigma_inf = 1, or
topologically, as drawings on compact orientable surfaces. In his Esquisse
d'un Programme, Grothendieck had high hopes to use these correspondences to
better understand the absolute Galois group of QQ.

In this talk, we describe the aforementioned correspondences in some
detail. After this, we discuss how Belyi maps can be computed explicitly by
using the theory of modular forms. Recently, the implementation of these
techniques in a collaboration with its creators Musty, Schiavone and Voight
has led to a database that contains 1024 Belyi maps of degree up to 9. This
database is freely available online in the L-Functions and Modular Forms
Database.