Refinements on higher order Weil-Oesterlé bounds via a Serre type argument
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910989755088896 |
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| author | Hallouin, Emmanuel Moustrou, Philippe Perret, Marc |
| author_facet | Hallouin, Emmanuel Moustrou, Philippe Perret, Marc |
| contents | Weil's theorem gives the most standard bound on the number of points of a curve over a finite field. This bound was improved by Ihara and Oesterlé for larger genus. Recently, Hallouin and Perret gave a new point of view on these bounds, that can be obtained by solving a sequence of semi-definite programs, and the two first steps of this hierarchy recover Weil's and Ihara's bounds. On the other hand, by taking into account arithmetic constraints, Serre obtained a refinement on Weil's bound. In this article, we combine these two approaches and propose a strengthening of Ihara's bound, based on an argument similar to Serre's refinement. We show that this generically improves upon Ihara's bound, even in the range where it was the best bound so far. Finally we discuss possible extensions to higher order Weil-Oesterlé bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05212 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Refinements on higher order Weil-Oesterlé bounds via a Serre type argument Hallouin, Emmanuel Moustrou, Philippe Perret, Marc Number Theory Algebraic Geometry 11G20, 14G05 Weil's theorem gives the most standard bound on the number of points of a curve over a finite field. This bound was improved by Ihara and Oesterlé for larger genus. Recently, Hallouin and Perret gave a new point of view on these bounds, that can be obtained by solving a sequence of semi-definite programs, and the two first steps of this hierarchy recover Weil's and Ihara's bounds. On the other hand, by taking into account arithmetic constraints, Serre obtained a refinement on Weil's bound. In this article, we combine these two approaches and propose a strengthening of Ihara's bound, based on an argument similar to Serre's refinement. We show that this generically improves upon Ihara's bound, even in the range where it was the best bound so far. Finally we discuss possible extensions to higher order Weil-Oesterlé bounds. |
| title | Refinements on higher order Weil-Oesterlé bounds via a Serre type argument |
| topic | Number Theory Algebraic Geometry 11G20, 14G05 |
| url | https://arxiv.org/abs/2506.05212 |