Abstract. In this paper, we consider a complex-valued d-dimensional fractional Brownian motion defined on the closure of the complex upper half-plane, called analytic fractional Brownian motion and denoted by Γ. This process has been introduced in [16], and both its real and imaginary parts, restricted on the real axis, are usual fractional Brownian motions. The current note is devoted to prove...

Φ : H/Γ0(N) −→ E(C), (0–1) where H is the Poincaré upper half-plane and Γ0(N) is Hecke’s congruence group of level N . Fix a quadratic field K; when it is imaginary, the theory of complex multiplication combined with (0–1) yields the construction of a remarkable collection of points on E defined over certain ring class fields of K. These are the Heegner points recalled in Section 1 whose study ...

Motivated by Kesten’s bridge decomposition for two-dimensional self-avoiding walks in the upper half plane, we show that the conjectured scaling limit of the half-plane SAW, the SLE(8/3) process, also has an appropriately defined bridge decomposition. This continuum decomposition turns out to entirely be a consequence of the restriction property of SLE(8/3), and as a result can be generalized t...

A theorem of W. Kohnen states that the generalized Riemann hypothesis (GRH) holds on an average for holomorphic cusp forms on the upper half plane for the full modular group SL2(Z). In this article we prove a couple of generalizations of this theorem of Kohnen that the GRH holds on an average for holomorphic cusp forms on the upper half plane for arbitrary level, weight and primitive nebentypus...

Group factorisation plays a vital part in the inverse scattering procedure [2,10]. For example the Riemann-Hilbert problem is a (not quite exact) factorisation of group valued functions on the real line into functions analytic on the lower half plane times functions analytic on the upper half plane. However there is a problem, a group valued function which is analytic on the lower half plane ne...

We generalize to Hilbert modular varieties of arbitrary dimension the work of W. Duke [14] on the equidistribution of Heegner points and of primitive positively oriented closed geodesics in the Poincaré upper half plane, subject to certain subconvexity results. We also prove vanishing results for limits of cuspidal Weyl sums associated with analogous problems for the Siegel upper half space of ...

Definitions and classifications of reduced surds for non-arithmetical groups on the hyperbolic plane is proposed, as well those some congruence subgroups. The considered are desymmetrized group, its Γ0 subgroup, a generalized Hecke group according choice commutator subgroup group. definitions studied to ordering generators conjugacy subclasses groups. construction tori defined obtained after te...