Distributive properties of division points and discriminants of Drinfeld modules
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916112118054912 |
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| author | Gekeler, Ernst-Ulrich |
| author_facet | Gekeler, Ernst-Ulrich |
| contents | We present a new notion of distribution and derived distribution of rank $r \in \mathbb{N}$ for a global function field $K$ with a distinguished place $\infty$. It allows to describe the relations between division points, isogenies, and discriminants both for a fixed Drinfeld module of rank $r$ for the above data, or for the corresponding modular forms.
We introduce and study three basic distributions with values in $\mathbb{Q}$, in the group $μ(\overline{K})$ of roots of unity in the algebraic closure $\overline{K}$ of $K$, and in the group $U^{(1)}(C_{\infty})$ of $1$-units of the completed algebraic closure $C_{\infty}$ of $K_{\infty}$, respectively.
There result product formulas for division points and discriminants that encompass known results (e.g. analogues of Wallis' formula for $(2πi)^{2}$ in the rank-$1$ case, of Jacobi's formula $Δ= (2πi)^{12} q \prod (1-q^{n})^{24}$ in the rank-$2$ case, and similar boundary expansions for $r > 2$) and several new ones: the definition of a canonical discriminant for the most general case of Drinfeld modules and the description of the sizes of division and discriminant forms.
In the now classical case where $(K, \infty) = (\mathbb{F}_{q}(T), \infty)$ and $r = 1$, $2$ or $3$, we give explicit values for the logarithms of such forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00545 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distributive properties of division points and discriminants of Drinfeld modules Gekeler, Ernst-Ulrich Number Theory 11G09, 11F52, 11R58 We present a new notion of distribution and derived distribution of rank $r \in \mathbb{N}$ for a global function field $K$ with a distinguished place $\infty$. It allows to describe the relations between division points, isogenies, and discriminants both for a fixed Drinfeld module of rank $r$ for the above data, or for the corresponding modular forms. We introduce and study three basic distributions with values in $\mathbb{Q}$, in the group $μ(\overline{K})$ of roots of unity in the algebraic closure $\overline{K}$ of $K$, and in the group $U^{(1)}(C_{\infty})$ of $1$-units of the completed algebraic closure $C_{\infty}$ of $K_{\infty}$, respectively. There result product formulas for division points and discriminants that encompass known results (e.g. analogues of Wallis' formula for $(2πi)^{2}$ in the rank-$1$ case, of Jacobi's formula $Δ= (2πi)^{12} q \prod (1-q^{n})^{24}$ in the rank-$2$ case, and similar boundary expansions for $r > 2$) and several new ones: the definition of a canonical discriminant for the most general case of Drinfeld modules and the description of the sizes of division and discriminant forms. In the now classical case where $(K, \infty) = (\mathbb{F}_{q}(T), \infty)$ and $r = 1$, $2$ or $3$, we give explicit values for the logarithms of such forms. |
| title | Distributive properties of division points and discriminants of Drinfeld modules |
| topic | Number Theory 11G09, 11F52, 11R58 |
| url | https://arxiv.org/abs/2402.00545 |