Modular curves with many points over finite fields

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 GL2(Z/nZ) such that for each prime p dividing n, the subgroup H at p is either a Borel subgroup, a Cartan subgroup, or the normalizer of a Cartan subgroup of GL2(Z/peZ), and for W any subgroup of the Atkin-Lehner involutions of X_H . We applied our algorithm to more than ten thousand curves of genus up to 50, finding more than one hundred record-breaking curves, namely curves X/Fq with genus g that improve the previously known lower bound for the maximum number of points over Fq 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.