Grupos ortogonales sobre cuerpos de caracter\'{i}stica positiva

TL;DR

This paper classifies orthogonal groups over finite fields of positive characteristic using bilinear and quadratic forms, and explores special subgroups based on determinant, spinor norm, and Dickson invariant, relevant to automorphic forms and Langlands program.

Contribution

It provides a classification of orthogonal groups over finite fields in positive characteristic and analyzes their special subgroups using characteristic-specific invariants.

Findings

Classification of orthogonal groups over finite fields.

Identification of special subgroups via determinant, spinor norm, and Dickson invariant.

Application to automorphic forms and Langlands theory.

Abstract

This exposition examines the theory of orthogonal groups and their subgroups over fields of positive characteristic, which has recently been used as an important tool in the study of automorphic forms and Langlands functionality. We present the classification of orthogonal groups over a finite field using the theory of bilinear forms and quadratic forms in positive characteristic. Using the determinant and spinor norm when the characteristic of $F$ is odd and using the Dickson invariant when the characteristic of $F$ is even, we also look at special subgroups of the orthogonal group. -- -- Esta exposici\'{o}n examina la teor\'{i}a de los grupos ortogonales y sus subgrupos sobre cuerpos de caracter\'{i}stica positiva, que recientemente se han utilizado como una herramienta importante en el estudio de las formas autom\'{o}rficas y la funcionalidad de Langlands. Presentamos la clasificaci\'{o}n de grupos ortogonales sobre un cuerpo finito $F$ utilizando la teor\'{i}a de formas bilineales y formas cuadr\'{a}ticas en caracter\'{i}stica positiva. Usando el determinante y la norma del espinor cuando la caracter\'{i}stica de $F$ es impar y usando la invariante de Dickson cuando la caracter\'{i}stica de $F$ es par, tambi\'{e}n encontramos subgrupos especiales del grupo ortogonal.