We define a set of holomorphic functions in terms the Hauptmodul quotient Riemann surface and prove that these are on upper half-plane. It is also shown automorphic forms weight k with respect to Fuchsian group.
 GANIT J. Bangladesh Math. Soc. 43.1 (2023) 036- 044

A differential-difference operator is used to model the heat equation on a finite graph analogue of Poincaré’s upper half-plane. Finite analogues classical theta functions are shown be solutions in this setting.

In this paper, we study composition operators Cφf = f ◦ φ, induced by a fixed analytic self-map of the of the upper half-plane, acting between Hardy and Bloch-type spaces of the upper half-plane.

This space was first introduced by Mumford, where it plays a key role in the generalization to higher genus of Tate’s theory of p-adic uniformization of elliptic curves with semistable reduction. Slightly later, Drinfeld and Cerednik showed that appropriate quotients of this space by discrete arithmetic subgroups of PGL2(K) coming from quaternion algebras yield Shimura curves. Since that time, ...

The Half-Plane Half-Comb walk is a random on the plane, when we have square lattice upper half-plane and comb structure lower half-plane, i.e., horizontal lines below x-axis are removed. We prove that probability this returns to origin in 2N steps asymptotically equal $$2/(\pi N).$$ 2 / ( ? N ) . As consequence, strong laws limit distribution for local time.