Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow
Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow
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赫克三角形群的奇数和偶数马斯尖点形式以及台球流
DOI：
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发表时间：
2013
影响因子：
0.9
通讯作者：
A. Pohl
中科院分区：
文献类型：
--
作者：
A. Pohl
By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, Möller and the present author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families: $$egin{eqnarray}Z(s)=det (1-{mathcal{L}}_{s}^{+})det (1-{mathcal{L}}_{s}^{-}).end{eqnarray}$$ In this article we show that the operator families ${mathcal{L}}_{s}^{pm }$ arise as families of transfer operators for the triangle groups underlying the Hecke triangle groups, and that for $sin mathbb{C}$, $ ext{Re}s={ extstyle frac{1}{2}}$, the operator ${mathcal{L}}_{s}^{+}$ (respectively ${mathcal{L}}_{s}^{-}$) has a 1-eigenfunction if and only if there exists an even (respectively odd) Maass cusp form with eigenvalue $s(1-s)$. For non-arithmetic Hecke triangle groups, this result provides a new formulation of the Phillips–Sarnak conjecture on non-existence of even Maass cusp forms.
By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, Möller and the present author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families: $$egin{eqnarray}Z(s)=det (1-{mathcal{L}}_{s}^{+})det (1-{mathcal{L}}_{s}^{-}).end{eqnarray}$$ In this article we show that the operator families ${mathcal{L}}_{s}^{pm }$ arise as families of transfer operators for the triangle groups underlying the Hecke triangle groups, and that for $sin mathbb{C}$, $ ext{Re}s={ extstyle frac{1}{2}}$, the operator ${mathcal{L}}_{s}^{+}$ (respectively ${mathcal{L}}_{s}^{-}$) has a 1-eigenfunction if and only if there exists an even (respectively odd) Maass cusp form with eigenvalue $s(1-s)$. For non-arithmetic Hecke triangle groups, this result provides a new formulation of the Phillips–Sarnak conjecture on non-existence of even Maass cusp forms.