Modular Curves with many Points over Finite Fields

We describe an algorithm to compute the number of points over finite fields on a broad class of modular curves: we consider quotients $X_H\mathrm{/}W$ for $H$ a subgroup of ${\text{GL}}_2(\mathbb{Z}\mathrm{/}n\mathbb{Z})$ such that for each prime $p$ dividing $n$, the subgroup $H$ at $p$ is either a Borel subroup, a Cartan subgroup, or the normalizer of a Cartan subgroup of ${\text{GL}}_2(\mathbb{Z}\mathrm{/}{p}^e\mathbb{Z})$, and for $W$ any subgroup of the Atkin-Lehner involutions of $X_H$. We applied our algorithm to more than ten thousands curves of genus up to 50, finding more than one hundred record-breaking curves, namely curves $X\mathrm{/}{\text{FF}}_q$ with genus $g$ that improve the previously known lower bound for the maximum number of points over ${\text{FF}}_q$ of a curve with genus $g$. As a key technical tool for our computations, we prove the generalization of Chen’s isogeny to all the Cartan modular curves of composite level.