John Voight (Vermont)
Shimura curves of low genus and totally real fields of small root discriminant
Shimura curves are generalizations of modular curves, where the matrix
ring is replaced by a quaternion algebra over a totally real field.
Recently, Long, Maclachlan, and Reid proved that the number of Shimura
curves of bounded genus is finite. In this talk, we describe a method
to explicitly enumerate all Shimura curves of genus at most 2; along
the way, we tabulate all totally real number fields of root
discriminant at most 14. We examine some of the mathematically and
computationally interesting aspects of each of these tasks in turn.