On sesquilinear forms over finite fields

TL;DR

This paper develops a comprehensive theory of sesquilinear forms over finite fields, linking them to quadratic forms, character sums, and algebraic curves, with new classification and enumeration results.

Contribution

It introduces a novel framework for sesquilinear forms over finite fields, including their polynomial and matrix representations, and connects them to quadratic forms and algebraic curves.

Findings

Classification results for sesquilinear forms

Explicit enumeration of solutions to equations involving these forms

Characterization of certain Artin-Schreier curves

Abstract

We develop a theory of sesquilinear forms over finite fields, investigating their representations via polynomials and coefficient matrices, along with classification results for these forms. Through their connection to quadratic forms, we calculate certain character sums to resolve enumeration problems for equations defined by sesquilinear forms. This provides a characterization of a class of maximal or minimal Artin-Schreier curves with explicit examples.