Point counts, automorphisms, and gonalities of Shimura curves
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912496360620032 |
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| author | Mercuri, Pietro Padurariu, Oana Saia, Frederick Stirpe, Claudio |
| author_facet | Mercuri, Pietro Padurariu, Oana Saia, Frederick Stirpe, Claudio |
| contents | We implement an algorithm to compute the number of points over finite fields for the Shimura curves $X_0^D(N)$ and their Atkin--Lehner quotients. Our computations result in many examples of curves which attain the largest known point counts among curves of specified genus over a finite field of given cardinality. To illustrate the utility of our point counts algorithm in addressing arithmetic questions, we prove that all automorphisms are Atkin--Lehner for many curves $X_0^D(N)$ with $DN\leq 10000$, and we determine all tetragonal curves $X_0^D(N)$ up to a small number of possible exceptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15992 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Point counts, automorphisms, and gonalities of Shimura curves Mercuri, Pietro Padurariu, Oana Saia, Frederick Stirpe, Claudio Number Theory 11G18, 11G20, 11G30, 11G15 We implement an algorithm to compute the number of points over finite fields for the Shimura curves $X_0^D(N)$ and their Atkin--Lehner quotients. Our computations result in many examples of curves which attain the largest known point counts among curves of specified genus over a finite field of given cardinality. To illustrate the utility of our point counts algorithm in addressing arithmetic questions, we prove that all automorphisms are Atkin--Lehner for many curves $X_0^D(N)$ with $DN\leq 10000$, and we determine all tetragonal curves $X_0^D(N)$ up to a small number of possible exceptions. |
| title | Point counts, automorphisms, and gonalities of Shimura curves |
| topic | Number Theory 11G18, 11G20, 11G30, 11G15 |
| url | https://arxiv.org/abs/2507.15992 |