论文标题
Rademacher符号的双曲线类似物
A Hyperbolic Analogue of the Rademacher Symbol
论文作者
论文摘要
Dedekind最著名的结果之一是$ \logδ（z）$的转换定律。半个世纪后，Rademacher修改了Dedekind的结果，并引入了$ \ mathrm {sl} _2（\ Mathbb {Z}）$ - 连接类别不变（Integer-valued）函数$ψ（γ）$称为Rademacher符号。受Ghys在模块化结上的工作的启发，Duke-Mimamogluth（2017）构建了该符号的双曲线类似物。 在本文中，我们研究了它们对Rademacher符号$ψ_γ（σ）$的双曲线类似物，并通过将其与经典的Rademacher符号进行比较来提供两种类型的显式公式。与之相关的是，我们对抛物线，椭圆和双曲双曲线的Kronecker极限类型公式进行了相反。这些极限可提供谐波，极性谐波和局部谐波的重量形式2。
One of the most famous results of Dedekind is the transformation law of $\log Δ(z)$. After a half-century, Rademacher modified Dedekind's result and introduced an $\mathrm{SL}_2(\mathbb{Z})$-conjugacy class invariant (integer-valued) function $Ψ(γ)$ called the Rademacher symbol. Inspired by Ghys' work on modular knots, Duke-Imamoglu-Tóth (2017) constructed a hyperbolic analogue of the symbol. In this article, we study their hyperbolic analogue of the Rademacher symbol $Ψ_γ(σ)$ and provide its two types of explicit formulas by comparing it with the classical Rademacher symbol. In association with it, we contrastively show Kronecker limit type formulas of the parabolic, elliptic, and hyperbolic Eisenstein series. These limits give harmonic, polar harmonic, and locally harmonic Maass forms of weight 2.
