Caracterizações algébrica e geométrica das regiões de uniformização de curvas hiperelípticas via equação diferencial fuchsiana para a construção de constelações de sinais hiperbólicas

Abstract

In this work, we call attention to the importance of the hyperbolic geometry, of the Fuchsian differential equations, of the Fuchsian groups associated with the corresponding fundamental regions as well as of the hyperbolic tessellations in the construction of new signal constellations, and consequently, in proposing more efficient, reliable and less complex communication systems than the previous known systems. In this direction, we present the mathematical concepts and tools which provide a new approach to the design of new digital communication systems. From the embedding of a discrete memoryless channel in a proper surface, the genus of such surface is known. Knowing the genus, we associate it to a hyperelliptic curve with distinct roots in the Poincaré disk which satisfies Whittaker’s conjecture associated with the Fuchsian differential equation. Thus, the objectives are described as follows: 1) to identify the generators of the Fuchsian group whose fundamental region uniformizes the hyperelliptic curve; 2) to establish a relationship between the degree of the hyperelliptic curve and the tessellation generated by the Fuchsian group associated with the hyperelliptic curve and, consequently, to show a relationship between the hyperelliptic curve and the corresponding arithmetic Fuchsian group; 3) to show the existence or not of an isomorphism by conjugation and/or by linear combination between the Fuchsian groups associated with the hyperelliptic curves; and 4) to show the importance of the fact that the Ricatti and Schwarz differential equations have a linear component, a Fuchsian linear differential equation, and a nonlinear transformation. Certainly, this important characteristic will provide new research directions to be followed.