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 $\GL_2(\mathbb Z/n\mathbb Z)$ 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.

References in corpus (6)

• $\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$
• The automorphism group of the non-split Cartan modular curve of level 11
• On the automorphisms of the non-split Cartan modular curves of prime level
• Automorphisms of Cartan modular curves of prime and composite level
• Double Covers of Cartan Modular Curves
• Quadratic Points on Non-Split Cartan Modular Curves

Cited by in corpus (1)

• Spectral Theory of Isogeny Graphs