Volume of truncated fundamental domains

Volume of Truncated Fundamental Domains

Fundamental Domains for Shimura Curves

We describe a process for defining and computing a fundamental domain in the upper half plane H of a Shimura curve X 0 (N) associated to an order in a quaternion algebra A/Q. A fundamental domain for X 0 (N) realizes a finite presentation of the quaternion unit group, modulo units of its center. We give explicit examples of domains for the curves X 0 (1), X 0 (1), and X 35 0 (1). The first exam...

Volume of a doubly truncated hyperbolic tetrahedron

The present paper regards the volume function of a doubly truncated hyperbolic tetrahedron. Starting from the previous results of J. Murakami, U. Yano and A. Ushijima, we have developed a unified approach to expressing the volume in different geometric cases by dilogarithm functions and to treat properly the many analytic strata of the latter. Finally, several numeric examples are given.

Fundamental Domains of Some Drinfeld Modular Curves

We construct fundamental domains for arithmetic subgroups of Γ = GL2(Fq [t]). Given ∆ ⊇ Γ(a) we construct a contracted form T of the Bruhat-Tits tree T and a fundamental domain F of ∆ acting on T. We define a lift of F to F ⊂ T called the “bipartite” lift. We show that F is a fundamental domain of ∆ acting on T precisely when F is “∆-compressed.”

Fundamental Domains of Gamma and Zeta Functions

Branched covering Riemann surfaces C, f are studied, where f is the Euler Gamma function and the Riemann Zeta function. For both of them fundamental domains are found and the group of cover transformations is revealed. In order to find fundamental domains, preimages of the real axis are taken and a thorough study of their geometry is performed. The technique of simultaneous continuation, introd...