Classification of quadratic forms over finite fields with maximal and minimal Artin-Schreier curves

Ruikai Chen

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arXiv · 2411.11705

Classification of quadratic forms over finite fields with maximal and minimal Artin-Schreier curves

Ruikai Chen

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Abstract

This paper explores quadratic forms over finite fields with associated Artin-Schreier curves. Specifically, we investigate quadratic forms of $\mathbb F_{q^n}/\mathbb F_q$ represented by polynomials over $\mathbb F_{q^n}$ with $q$ odd, characterizing them using certain matrices defined by coefficients of the polynomials. In particular, a comprehensive treatment will be given for those polynomials whose coefficients all lie in $\mathbb F_q$. Afterwards, the results on quadratic forms will be applied to get maximal and minimal Artin-Schreier curves explicitly.

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Ruikai Chen. 2024-11-18. Classification of quadratic forms over finite fields with maximal and minimal Artin-Schreier curves. https://arxiv.org/abs/2411.11705

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