Symmetries of the transfer operator for Γ0(N) and a character deformation of the Selberg zeta function for Γ0(4)
Symmetries of the transfer operator for Γ0(N) and a character deformation of the Selberg zeta function for Γ0(4)
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Г0(N) 的传递算子的对称性和 Г0(4) 的 Selberg zeta 函数的特征变形
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
D. Mayer
中科院分区：
文献类型：
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作者：
M. Fraczek;D. Mayer;D. Mayer
关于Γ0（N）和平凡特征标χ0的转移算子具有一个由置换矩阵P生成的有限对称群，其中P2 = id.每一个这样的对称都导致Selberg zeta函数关于一个约化转移算子的Fredholm行列式的一个因子分解.这些对称性与Γ0（N）的Maass波型在GL（2，Z）中的自同构群有关。对于群Γ0（4）和Selberg特征标χα，只存在一个非平凡对称算子P .相应的本征值λ = ±1的约化转移算子的本征函数在相应的自同构下分别对应于偶数和奇数的Maass形式。从Sarnak和菲利普斯的结果可以得出，由约化转移算子的本征值λ = −1确定的Selberg函数的零点在特征标变形下停留在临界线上。从数值结果我们可以预期，另一方面，对应于本征值λ = +1的所有零点都离开这条线，使α远离零点。
The transfer operator for Γ0(N) and trivial character χ0 possesses a finite group of symmetries generated by permutation matrices P with P 2 = id. Every such symmetry leads to a factorization of the Selberg zeta function in terms of Fredholm determinants of a reduced transfer operator. These symmetries are related to the group of automorphisms in GL(2,Z) of the Maass wave forms of Γ0(N) . For the group Γ0(4) and Selberg’s character χα there exists just one non-trivial symmetry operator P . The eigenfunctions of the corresponding reduced transfer operator with eigenvalue λ = ±1 are related to Maass forms even respectively odd under a corresponding automorphism. It then follows from a result of Sarnak and Phillips that the zeros of the Selberg function determined by the eigenvalues λ = −1 of the reduced transfer operator stay on the critical line under the deformation of the character. From numerical results we expect that on the other hand all the zeros corresponding to the eigenvalue λ = +1 leave this line for α turning away from zero.