Modular Curves with many Points over Finite Fields
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2016
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| _version_ | 1866909094523174912 |
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| author | Dose, Valerio Lido, Guido Mercuri, Pietro Stirpe, Claudio |
| author_facet | Dose, Valerio Lido, Guido Mercuri, Pietro Stirpe, Claudio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1603_07489 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Modular Curves with many Points over Finite Fields Dose, Valerio Lido, Guido Mercuri, Pietro Stirpe, Claudio Number Theory 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. |
| title | Modular Curves with many Points over Finite Fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/1603.07489 |