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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.02187 |
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| _version_ | 1866911186763644928 |
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| author | de Courcy-Ireland, Matthew Litman, Matthew Mizuno, Yuma |
| author_facet | de Courcy-Ireland, Matthew Litman, Matthew Mizuno, Yuma |
| contents | We study orbits in a family of Markoff-like surfaces with extra off-diagonal terms over prime fields $\mathbb{F}_p$. It is shown that, for a typical surface of this form, every non-trivial orbit has size divisible by $p$. This extends a theorem of W.Y. Chen from the Markoff surface itself to others in this family. The proof closely follows and elaborates on a recent argument of D.E. Martin. We expect that there is just one orbit generically. For some special parameters, we prove that there are at least two or four orbits. Cayley's cubic surface plays a role in parametrising the exceptional cases and dictating the number of solutions mod $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02187 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Divisibility by $p$ for Markoff-like Surfaces de Courcy-Ireland, Matthew Litman, Matthew Mizuno, Yuma Number Theory Dynamical Systems Rings and Algebras 11D25, 37P25, 13F60, 11T06, 11T24 We study orbits in a family of Markoff-like surfaces with extra off-diagonal terms over prime fields $\mathbb{F}_p$. It is shown that, for a typical surface of this form, every non-trivial orbit has size divisible by $p$. This extends a theorem of W.Y. Chen from the Markoff surface itself to others in this family. The proof closely follows and elaborates on a recent argument of D.E. Martin. We expect that there is just one orbit generically. For some special parameters, we prove that there are at least two or four orbits. Cayley's cubic surface plays a role in parametrising the exceptional cases and dictating the number of solutions mod $p$. |
| title | Divisibility by $p$ for Markoff-like Surfaces |
| topic | Number Theory Dynamical Systems Rings and Algebras 11D25, 37P25, 13F60, 11T06, 11T24 |
| url | https://arxiv.org/abs/2509.02187 |