Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps
Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps
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带尖点的奇维双曲流形的 Eta 不变量和正则化行列式
DOI：
10.1353/ajm.2005.0023
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发表时间：
2001
影响因子：
1.7
通讯作者：
Jinsung Park
中科院分区：
文献类型：
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作者：
Jinsung Park
研究了具有尖点的双曲流形上Dirac算子的eta不变量和Dirac Laplacian算子的正则化行列式及其与Selberg zeta函数的关系。利用Selberg迹公式和对幂幺轨道积分的详细分析，证明了由相对迹定义的eta和zeta函数在原点处是正则的，从而可以定义eta不变量和正则化行列式.证明了奇型Selbergzeta函数在C上有亚纯扩张，证明了奇型Selbergzeta函数的eta不变量与某个值之间的关系，并导出了相应的函数方程.这些结果推广了John Millson的早期工作到具有尖点的双曲流形。证明了偶型Selbergzeta函数在C上有一个亚纯扩张，并将其与正则行列式联系起来，得到了相应的函数方程.
We study eta invariants of Dirac operators and regularized determinants of Dirac Lapla- cians over hyperbolic manifolds with cusps and their relations with Selberg zeta functions. Using the Selberg trace formula and a detailed analysis of the unipotent orbital integral, we show that the eta and zeta functions defined by the relative traces are regular at the origin so that we can define the eta invariant and the regularized determinant. We also show that the Selberg zeta function of odd type has a meromorphic extension over C, prove a relation of the eta invariant and a certain value of the Selberg zeta function of odd type, and derive a corresponding functional equation. These results generalize the earlier work of John Millson to hyperbolic manifolds with cusps. We also prove that the Selberg zeta function of even type has a meromorphic extension over C, relate it to the regularized determinant, and obtain a corresponding functional equation.
S.Koyama：“Selberg zeta 函数的决定式表达（II）”TranS.A.M.S.
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