A new proof of Chen's theorem for Markoff graphs
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866929726245830656 |
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| author | Martin, Daniel E. |
| author_facet | Martin, Daniel E. |
| contents | In 2021, Chen proved a congruence for the degree of a certain map on the space of covers of elliptic curves. He concluded as a corollary that the size of any connected component of the Markoff mod $p$ graph is divisible by $p$. In combination with the work of Bourgain, Gamburd, and Sarnak, Chen's result proves a conjecture of Baragar for all but finitely many primes: the Markoff mod $p$ graph is connected. In this note, we provide an alternative proof for the Markoff corollary of Chen's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15960 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new proof of Chen's theorem for Markoff graphs Martin, Daniel E. Number Theory 11D25 In 2021, Chen proved a congruence for the degree of a certain map on the space of covers of elliptic curves. He concluded as a corollary that the size of any connected component of the Markoff mod $p$ graph is divisible by $p$. In combination with the work of Bourgain, Gamburd, and Sarnak, Chen's result proves a conjecture of Baragar for all but finitely many primes: the Markoff mod $p$ graph is connected. In this note, we provide an alternative proof for the Markoff corollary of Chen's theorem. |
| title | A new proof of Chen's theorem for Markoff graphs |
| topic | Number Theory 11D25 |
| url | https://arxiv.org/abs/2502.15960 |