Trace formula for \(p\)-adics (Q1599294)

The famous Selberg trace formula relates the trace of the kernel of the Brownian motion, i.e., the heat kernel, of a compact Riemann surface with hyperbolic metric to geometric data parametrized by closed geodesics. In this paper, the upper half plane is replaced by the field of \(p\)-adic numbers and the Riemannian surface by the ball of radius 1, which can be interpreted as a quotient of the \(p\)-adic field, and the Brownian motion is replaced by a random walk. A formula similar to the Selberg trace formula is obtained.