Point counts, automorphisms, and gonalities of Shimura curves

TL;DR

This paper presents an algorithm for counting points on Shimura curves over finite fields, enabling the identification of curves with maximal point counts and analysis of their automorphisms and gonalities.

Contribution

It introduces a new algorithm for point counting on Shimura curves and applies it to determine automorphisms and classify tetragonal curves.

Findings

Many curves attain the largest known point counts for their genus.

All automorphisms are Atkin--Lehner for many curves with DN ≤ 10000.

Complete classification of tetragonal curves with few exceptions.

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.