Point counts, automorphisms, and gonalities of Shimura curves

We implement an algorithm to compute the number of points over finite fields for the Shimura curves X₀D(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 X₀D(N) with DN≤ 10000, and we determine all tetragonal curves X₀D(N) up to a small number of possible exceptions.