Heights of Drinfeld modular polynomials and Hecke images
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916439727800320 |
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| author | Breuer, Florian Pazuki, Fabien Ran, Zhenlin |
| author_facet | Breuer, Florian Pazuki, Fabien Ran, Zhenlin |
| contents | We obtain explicit upper and lower bounds on the size of the coefficients of the Drinfeld modular polynomials $Φ_N$ for any monic $N\in\mathbb{F}_q[t]$. These polynomials vanish at pairs of $j$-invariants of Drinfeld $\mathbb{F}_q[t]$-modules of rank 2 linked by cyclic isogenies of degree $N$. The main term in both bounds is asymptotically optimal as $\mathrm{deg}(N)$ tends to infinity. We also obtain precise estimates on the Weil height and Taguchi height of Hecke images of Drinfeld modules of rank 2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11132 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Heights of Drinfeld modular polynomials and Hecke images Breuer, Florian Pazuki, Fabien Ran, Zhenlin Number Theory 11F52, 11G09, 11R58, 14H25 We obtain explicit upper and lower bounds on the size of the coefficients of the Drinfeld modular polynomials $Φ_N$ for any monic $N\in\mathbb{F}_q[t]$. These polynomials vanish at pairs of $j$-invariants of Drinfeld $\mathbb{F}_q[t]$-modules of rank 2 linked by cyclic isogenies of degree $N$. The main term in both bounds is asymptotically optimal as $\mathrm{deg}(N)$ tends to infinity. We also obtain precise estimates on the Weil height and Taguchi height of Hecke images of Drinfeld modules of rank 2. |
| title | Heights of Drinfeld modular polynomials and Hecke images |
| topic | Number Theory 11F52, 11G09, 11R58, 14H25 |
| url | https://arxiv.org/abs/2410.11132 |