We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the $k$th moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the $p^m$t
h Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.
The subject of the present paper is an application of quantum probability to $p$-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for ${rm PGL}_2(F)$, where $F$ is a $p$-adic field. As a byproduct, we obtain
a new proof of the Fourier inversion formula for ${rm PGL}_2(F)$.
Elkies proposed a procedure for constructing explicit towers of curves, and gave two towers of Shimura curves as relevant examples. In this paper, we present a new explicit tower of Shimura curves constructed by using this procedure.
Rank-2 Drinfeld modules are a function-field analogue of elliptic curves, and the purpose of this paper is to investigate similarities and differences between rank-2 Drinfeld modules and elliptic curves in terms of supersingularity. Specifically, we
provide an explicit formula of a supersingular polynomial for rank-2 Drinfeld modules and prove several basic properties. As an application, we give a numerical example of an asymptotically optimal tower of Drinfeld modular curves.
In this paper, we study graph-theoretic analogies of the Mertens theorems by using basic properties of the Ihara zeta-function. One of our results is a refinement of a special case of the dynamical system Mertens second theorem due to Sharp and Pollicott.
