Point counts, automorphisms, and gonalities of Shimura curves

Abstract: We implement an algorithm to compute the number of points over finite fields for the Shimura curves $X_0^D(N)$ and their Atkin--Lehner quotients. Our computations result in many examples of curves which attain the largest known point counts among curves of specified genus over a finite field of given cardinality. To illustrate the utility of our point counts algorithm in addressing arithmetic questions, we prove that all automorphisms are Atkin--Lehner for many curves $X_0^D(N)$ with $DN\leq 10000$, and we determine all tetragonal curves $X_0^D(N)$ up to a small number of possible exceptions.