We compute the number of points over finite fields of some classes of modular
curves, namely $X_0(N)$, $X_0^+(N)$, without using explicit equations. In this
way we could improve many lower bounds for the maximum number of points of a
curve over finite fields

Dose, Valerio

Lido, Guido

Mercuri, Pietro

Stirpe, Claudio

English

arXiv

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

Archivio della ricerca- Università di Roma La Sapienza

Modular curves with many points over finite fields

arXiv.org e-Print Archive

Modular Curves with many Points over Finite Fields

Modular Curves with many Points over Finite Fields

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Archivio della ricerca- Università di Roma La Sapienza

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