Modular Curves with many Points over Finite Fields

TL;DR

This paper introduces an algorithm to compute points on a broad class of modular curves over finite fields, leading to the discovery of record-breaking curves with many points for given genus.

Contribution

The paper develops a new algorithm for counting points on specific modular curves and generalizes Chen's isogeny to all Cartan modular curves of composite level.

Findings

Identified over one hundred curves with record-breaking number of points.

Successfully computed points on over ten thousand curves of genus up to 50.

Improved known lower bounds for the maximum number of points on curves of certain genera.

Abstract

We describe an algorithm to compute the number of points over finite fields on a broad class of modular curves: we consider quotients $X_H/W$ for $H$ a subgroup of $\GL_2(\mathbb Z/n\mathbb Z)$ 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 $\GL_2(\mathbb Z/p^e\mathbb Z)$, 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 $X/\FF_q$ with genus $g$ that improve the previously known lower bound for the maximum number of points over $\FF_q$ 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.