### Abstract

We prove that the number variance for the spectrum of an arithmetic surface is highly nonrigid in part of the universal range. In fact it is close to having a Poisson behavior. This fact was discovered numerically by Schmit, Bogomolny, Georgeot and Giannoni. It has its origin in the high degeneracy of the length spectrum, first observed by Selberg.

Original language | English (US) |
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Pages (from-to) | 419-432 |

Number of pages | 14 |

Journal | Communications In Mathematical Physics |

Volume | 161 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 1994 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Luo, W., & Sarnak, P. (1994). Number variance for arithmetic hyperbolic surfaces.

*Communications In Mathematical Physics*,*161*(2), 419-432. https://doi.org/10.1007/BF02099785