arXiv math/03032512003-03-20math.NTmath.DS

$Γ_0(n)$的转移算子和$PSL(2,Z)$的周期函数的Hecke算子

Transfer operators for $Γ_0(n)$ and the Hecke operators for the period functions for $PSL(2,Z)$

J. Hilgert, D. Mayer, H. Movasati

备注：: 33 pages

摘要：

在这篇文章中，我们报告了一个令人惊讶的关系之间的转移运营商的同余子群$Γ_{0}(n)$和Hecke运营商的空间周期函数的模群$\PSL (2,\mathbb{Z})$。为此，我们研究了具有本征值$\mp 1$的转移算子的特殊本征函数，这些本征函数也是群$Γ_{0}(n)$的Lewis方程的解，并且由模群$\PSL (2,\mathbb{Z})$的转移算子的本征函数确定。因此，在阿特金-莱纳关于新旧形式的理论中，我们应该称它们为刘易斯方程的旧本征函数或旧解。原来，这些旧的解决方案的组成部分的总和为集团$Γ_{0}(n)$确定任何$n$的一个解决方案的刘易斯方程的模群，因此也是一个本征函数的转移运营商为这个群体。

英文摘要：

In this article we report on a surprising relation between the transfer operators for the congruence subgroups $Γ_{0}(n)$ and the Hecke operators on the space of period functions for the modular group $\PSL (2,\mathbb{Z})$. For this we study special eigenfunctions of the transfer operators with eigenvalues $\mp 1$, which are also solutions of the Lewis equations for the groups $Γ_{0}(n)$ and which are determined by eigenfunctions of the transfer operator for the modular group $\PSL (2,\mathbb{Z})$. In the language of the Atkin-Lehner theory of old and new forms one should hence call them old eigenfunctions or old solutions of Lewis equation. It turns out that the sum of the components of these old solutions for the group $Γ_{0}(n)$ determine for any $n$ a solution of the Lewis equation for the modular group and hence also an eigenfunction of the transfer operator for this group.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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