The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations
The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations
复制标题
具有有限维酉表示的余有限 Kleinian 群的 Selberg 迹公式和 Selberg zeta 函数
DOI：
10.1007/s00209-005-0806-9
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发表时间：
2004
影响因子：
0.8
通讯作者：
J. S. Friedman
中科院分区：
文献类型：
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作者：
J. S. Friedman
对于有限维酉表示的余有限克莱因群，我们推导了 Selberg 迹公式。作为一个应用，我们定义相应的 Selberg zeta 函数并计算其除数，从而将 Elstrodt、Grunewald 和 Mennicke 的结果推广到非平凡的酉表示。我们表明，尖状椭圆单元的存在有时会为 zeta 函数添加分支点。事实上，如果 是爱森斯坦整数环，则 的 Selberg zeta 函数包含分枝点，并且是亚纯函数的六次方根。
For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of Elstrodt, Grunewald and Mennicke to non-trivial unitary representations. We show that the presence of cuspidal elliptic elements sometimes adds ramification points to the zeta-function. In fact, if is the ring of Eisenstein integers, then the Selberg zeta-function of contains ramification points and is the sixth-root of a meromorphic function.