Drinfeld singular moduli, hyperbolas, units
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909166442905600 |
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| author | Anglès, Bruno Armana, Cécile Bosser, Vincent Pazuki, Fabien |
| author_facet | Anglès, Bruno Armana, Cécile Bosser, Vincent Pazuki, Fabien |
| contents | Let $q\geq2$ be a prime power and consider Drinfeld modules of rank 2 over $\mathbb{F}_q[T]$. We prove that there are no points with coordinates being Drinfeld singular moduli, on a family of hyperbolas $XY=γ$, where $γ$ is a polynomial of small degree. This is an effective André-Oort theorem for these curves. We also prove that there are at most finitely many Drinfeld singular moduli that are algebraic units, for every fixed $q\geq2$, and we give an effective bound on the discriminant of such singular moduli. We give in an appendix an inseparability criterion for values of some classical modular forms, generalising an argument used in the proof of our first result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01075 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Drinfeld singular moduli, hyperbolas, units Anglès, Bruno Armana, Cécile Bosser, Vincent Pazuki, Fabien Number Theory 11G09, 11J93 Let $q\geq2$ be a prime power and consider Drinfeld modules of rank 2 over $\mathbb{F}_q[T]$. We prove that there are no points with coordinates being Drinfeld singular moduli, on a family of hyperbolas $XY=γ$, where $γ$ is a polynomial of small degree. This is an effective André-Oort theorem for these curves. We also prove that there are at most finitely many Drinfeld singular moduli that are algebraic units, for every fixed $q\geq2$, and we give an effective bound on the discriminant of such singular moduli. We give in an appendix an inseparability criterion for values of some classical modular forms, generalising an argument used in the proof of our first result. |
| title | Drinfeld singular moduli, hyperbolas, units |
| topic | Number Theory 11G09, 11J93 |
| url | https://arxiv.org/abs/2404.01075 |