Trace formula on the p-adic upper half-plane

This article aims at showing a p-adic analogue of Selbergʹs 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.