Drinfeld模的分点与判别式的分布性质
Distributive properties of division points and discriminants of Drinfeld modules
AI总结：
该研究针对带指定位的整体函数域提出Drinfeld模的秩r分布新概念，引入三类基本分布推导分点与判别式的乘积公式，涵盖已知结果并得到典范判别式定义等新结论，还给出经典情形下的显式值。
摘要：
我们针对带有一个指定位∞的整体函数域K，提出了秩r∈ℕ的分布与导出分布的新概念。借助这一概念，我们既能描述对应上述数据、秩为r的固定Drinfeld模的分点、同源与判别式之间的关系，也能刻画相应模形式的这类关系。\n我们引入并研究了三类基本分布，它们的取值分别在有理数域ℚ、K的代数闭包K̄中的单位根群μ(K̄)，以及K∞的完备代数闭包C∞的1-单位群U^(1)(C∞)中。\n由此我们得到了分点与判别式的乘积公式，这些公式涵盖了已知结果（例如秩1情形下(2πi)²的Wallis公式的类似物，秩2情形下Jacobi公式Δ=(2πi)^12 q ∏(1-qⁿ)^24的类似物，以及r>2时的类似边界展开），还包含若干新结果：最一般情形下Drinfeld模的典范判别式的定义，以及分点形式与判别式形式的大小刻画。\n在经典情形即(K,∞)=(𝔽_q(T),∞)且r=1、2或3时，我们给出了这类形式的对数的显式值。
英文摘要：
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.
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