Mathematical Formulation of Arithmetic Surface (3, 5) Over Q

Abstract

The primary arithmetic surface (3, 5) over Q is the focus of this work, which examines its construction and properties. Attaching the two symbols i,j where i2 = 3, j2 = 5, and ij = -ji yields a quaternion algebra over Q. This "example issue" involves the creation of a hyperbolic surface by linking the points (3, 5) over Q. If you want to demonstrate how to discover a Dirichlet domain, or how to generate a hyperbolic polygon, you may use this issue as an illustration. The list of generators that corresponds to the edge gluings is one possible motivation for seeking for a hyperbolic surface. The generators in this list have applications beyond only pure mathematics and science. Creating a co-compact Dirichlet domain in Hyperbolic 3-space is the focus of this article. Based on the concepts introduced in "Expository Note: An Arithmetic Surface," this article was written. We provide a brief summary of the material presented before constructing the Dirichlet domain in Hyperbolic 2-space associated with the quadratic form Q(x) = x² +y² -7z². It is then shown that the new quadratic, Q(x) = w² + x² + y² -7z², also has a Dirchlet Domain in Hyperbolic 3-space.