Symmetries of the transfer operator for \(\Gamma_0(N)\) and a character deformation of the Selberg zeta function for \(\Gamma_0(4)\)
factorization of the Selberg zeta function
Let \(\Gamma=\Gamma_0(N) \subset \mathrm{SL}(2,\mathbb{Z})\) denote the Hecke congruence subgroup of level \(N\) containing all matrices of the form \({ a \;\;\; b \choose Nc \;\; d }\) with integer entries and determinant \(1\). Its index is \(\mu_N= N \prod_{p\mid N}(1+1/p)\) with \(p\) running through the prime-divisors of \(N\). \(\chi\) denotes a unitary character of \(\Gamma\).NEWLINENEWLINEWe consider the geodesic flow \(\Phi_t: SM_\Gamma \to SM_\Gamma\) on the unit tangent bundle \(SM_\Gamma\) of the corresponding modular surface \(M_\Gamma = \Gamma \backslash \mathbb{H}\) of constant negative curvature and \(\mathbb{H}\) denoting the upper complex half-plane.NEWLINENEWLINELet \(\rho:\Gamma \to \text{end}(\mathbb{C}^d)\) denote a unitary representation of \(\Gamma\). The Selberg zeta function \(Z_{\Gamma,\rho}\) is defined as NEWLINE\[NEWLINE Z_{\Gamma,\rho}(\beta) = \prod_\gamma \prod_{k=0}^\infty \det\bigg(1-\rho(g_\gamma) \, \text{exp}\big(-(k+\beta)l_\gamma) \big) \bigg) NEWLINE\]NEWLINE where \(l_\gamma\) denotes the period (i.e.\ length) of the prime periodic orbit \( \gamma\) of \(\Phi_t\) and \(g_\gamma \in \Gamma\) is hyperbolic with \(g_\gamma(\gamma)=\gamma\).NEWLINENEWLINEThe dynamical approach is to express the Selberg zeta function in terms of the so-called transfer operator, known from Ruelle's thermodynamic formalism approach to dynamical systems. The transfer operator \(L_{\beta,\chi}: B \to B\) was determined in [\textit{C.-H.\ Chang} and \textit{D.\ Mayer}, in: Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, 523--562 (2001; Zbl 1211.37030)] and [\textit{C.-H.\ Chang} and \textit{D.\ Mayer}, Regul.\ Chaotic Dyn.\ 5, No.\ 3, 281--312 (2000; Zbl 0979.37014)] as NEWLINE\[NEWLINE L_{\beta,\chi} = { 0 \;\; \mathcal{L}^+_{\beta, \rho_\chi} \choose \mathcal{L}^-_{\beta, \rho_\chi} \;\; 0 } NEWLINE\]NEWLINE where \(\chi\) is a unitary character of \(\Gamma\), \(B=B(D,\mathbb{C}^\mu) \oplus B(D,\mathbb{C}^\mu)\) is the Banach space of holomorphic functions on the disc \(D=\{z;\; |z-1| < 3/2 \}\) and \(\rho_\chi\) denotes the representation of \(\mathrm{SL}(2,\mathbb{Z})\) induced from the character \(\chi\) of \(\Gamma\). For \(\mathrm{Re}(\beta)>1/2\), the operator \(\mathcal{L}^\pm_{\beta, \rho_\chi}\) is given by NEWLINE\[NEWLINE \mathcal{L}^\pm_{\beta, \rho_\chi} = \sum_{n=1}^\infty \frac{1}{(z+n)^{2\beta}} \, \rho_\chi\big( ST^{\pm n} \big)\, \underline{f}\left( \frac{1}{z+n}\right) NEWLINE\]NEWLINE with \(S={ 0 \; -1 \choose 1 \;\; 0 }\), \(T={ 1 \;\; 1 \choose 0 \;\; 1 }\) and \(\underline{f} \in B(D,\mathbb{C}^\mu)\). It was proven in [\textit{C.-H.\ Chang} and \textit{D.\ Mayer}, in: B.\ Fiedler, Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer.\ 523--562 (2001; Zbl 1211.37030)] that \(L_{\beta,\chi}\) and \(\mathcal{L}^\pm_{\beta, \rho_\chi}\) extend to a family of nuclear operators of order zero on the entire complex plane \(\beta\) with possible poles at \(\beta_k=(1-k)/2\), \(k=0,1,2,\dots\). The Selberg zeta function \(Z_{\Gamma,\chi}\) can be expressed as NEWLINE\[NEWLINE Z_{\Gamma,\chi}(\beta) = \det( 1- L_{\beta,\chi}) = \det (1 - \mathcal{L}^+_{\beta, \rho_\chi}\mathcal{L}^-_{\beta, \rho_\chi}) = \det(1-\mathcal{L}^-_{\beta, \rho_\chi}\mathcal{L}^+_{\beta, \rho_\chi}). NEWLINE\]NEWLINE One of the main results of the present paper is the following new result:NEWLINENEWLINE\textbf{Theorem.} For Hecke congruence subgroups \(\Gamma=\Gamma_0(N)\) and trivial character \(\chi_0 \equiv 1\) there exists \(h_N\) matrices \(\tilde{P}= { 0 \;\; P \choose P \;\; 0 }\) commuting with the transfer operator \(L_{\beta,\chi_0}\) where \(P\) is a \(\mu_N \times \mu_N\) permutation matrix satisfying \(P^2=1_{\mu_N}\), NEWLINE\[NEWLINE P \, \rho_{\chi_0}(S) = \rho_{\chi_0}(S) \, P \quad \text{and} \quad P \, \rho_{\chi_0}(T) = \rho_{\chi_0}\big(T^{-1}\big) \, P, NEWLINE\]NEWLINE and hence NEWLINE\[NEWLINE P \, \mathcal{L}^+_{\beta,\chi_0} = \mathcal{L}^-_{\beta,\chi_0} \, P. NEWLINE\]NEWLINE Thereby \(h_N = \max \{ k; \; k | 24 \text{ and } k^2|N\}\). The permutation matrices \(P\) are determined by the \(h_N\) generators \(j\) of the normalizer group \(\mathcal{N}_N\) of \(\Gamma_0(N)\) in \(\mathrm{GL}(2,\mathbb{Z})\). The Selberg zeta function \(Z_{\Gamma_0(N),\chi_0}\) can be written as NEWLINE\[NEWLINE Z_{\Gamma_0(N),\chi_0} = \det \big(1- P \, \mathcal{L}^+_{\beta,\chi_0} \big) \, \det \big(1+ P \, \mathcal{L}^-_{\beta,\chi_0} \big). NEWLINE\]NEWLINENEWLINENEWLINEThe authors also relate eigenfunctions of the transfer operator \(\mathcal{L}^+_{\beta,\chi_0} \) with eigenvalue \(\pm 1\) to solutions of two functional equations, one parity equation depending on the eigenvalue \(\pm 1\) and a vector valued version of the ``three-term equation'' introduced in [\textit{J.\ Lewis} and \textit{D. Zagier}, Ann.\ Math.\ (2) 153, No.\ 1, 191--258 (2001; Zbl 1061.11021)] for period functions. The authors also relate these \(\pm 1\)-eigenfunctions to the Maaß\ cusp forms associated to the Fuchsian group \(\Gamma\) with character \(\chi\).
scientific article; zbMATH DE number 6064723