### Abstract

This article aims at showing a p-adic analogue of Selberg's trace formula, which describes a duality between the spectrum of a Hilbert-Schmidt operator and the length of prime geodesics appearing in the p-adic upper half-plane associated with a hyperbolic discontinuous subgroup of SL(2,Q_{p}). Then we construct Markov processes on the fundamental domain relative to such subgroups, to whose transition operators the trace formula applied and a p-adic analogue of prime geodesic theorem is proved.

Original language | English |
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Pages (from-to) | 422-454 |

Number of pages | 33 |

Journal | Journal of Functional Analysis |

Volume | 216 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2004 Nov 15 |

Externally published | Yes |

### Fingerprint

### Keywords

- Ihara zeta function
- Markov process
- p-Adic field
- Prime geodesic theorem
- Trace formula

### ASJC Scopus subject areas

- Analysis

### Cite this

**Trace formula on the p-adic upper half-plane.** / Yasuda, Kumi.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 216, no. 2, pp. 422-454. https://doi.org/10.1016/j.jfa.2004.03.008

}

TY - JOUR

T1 - Trace formula on the p-adic upper half-plane

AU - Yasuda, Kumi

PY - 2004/11/15

Y1 - 2004/11/15

N2 - This article aims at showing a p-adic analogue of Selberg's trace formula, which describes a duality between the spectrum of a Hilbert-Schmidt operator and the length of prime geodesics appearing in the p-adic upper half-plane associated with a hyperbolic discontinuous subgroup of SL(2,Qp). Then we construct Markov processes on the fundamental domain relative to such subgroups, to whose transition operators the trace formula applied and a p-adic analogue of prime geodesic theorem is proved.

AB - This article aims at showing a p-adic analogue of Selberg's trace formula, which describes a duality between the spectrum of a Hilbert-Schmidt operator and the length of prime geodesics appearing in the p-adic upper half-plane associated with a hyperbolic discontinuous subgroup of SL(2,Qp). Then we construct Markov processes on the fundamental domain relative to such subgroups, to whose transition operators the trace formula applied and a p-adic analogue of prime geodesic theorem is proved.

KW - Ihara zeta function

KW - Markov process

KW - p-Adic field

KW - Prime geodesic theorem

KW - Trace formula

UR - http://www.scopus.com/inward/record.url?scp=4344699754&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4344699754&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2004.03.008

DO - 10.1016/j.jfa.2004.03.008

M3 - Article

AN - SCOPUS:4344699754

VL - 216

SP - 422

EP - 454

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -