Point counts, automorphisms, and gonalities of Shimura curves

2025/07/21 by Mercuri, Pietro, Padurariu, Oana, Saia, Frederick +1
#11G15 #11G18 #11G20 #11G30 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.15992

Abstract

We implement an algorithm to compute the number of points over finite fields for the Shimura curves X₀^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 X₀^D(N) with DN≤ 10000, and we determine all tetragonal curves X₀^D(N) up to a small number of possible exceptions.

Citations

• Shimura curve Atkin--Lehner quotients of genus at most two
• Maximal curves over finite fields and a modular isogeny
• Bielliptic Shimura curves X₀^D(N) with nontrivial level
• Gonality of the modular curve X₀(N)
• Constraints on the automorphism group of a curve
• On the group of automorphisms of Shimura curves and applications

Related

• Shimura curve Atkin--Lehner quotients of genus at most two
• Counting curves over finite fields
• Point counting on curves using a gonality preserving lift
• Curves with many points over finite fields: the class field theory approach
• Number of points on a family of curves over a finite field