Topics in Algebraic Curves: Zeta Function and Frobenius nonclassical curves

• Grant number: 24/19443-4
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: July 01, 2025
• End date: June 30, 2028
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Saeed Tafazolian
• Grantee: João Paulo Guardieiro Sousa
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

This project has two main goals, that are related to the candidate's PhD thesis: studying the Zeta function of Artin-Schreier curves and classifying Frobenius nonclassical curves. In \cite{bootsma}, the supervisor and his collaborators provided a method for constructing elliptic curves with arbitrarily large rank over some function fields using maximal curves. In his PhD thesis, the candidate extended this construction to function fields of characteristic $3$ without imposing the maximality condition, but some information on the Zeta function of Artin-Schreier curves is required in order to construct families of examples having the desired property (see Section \ref{zeta-function}).We also want to find necessary and sufficient conditions for a curve to be Frobenius nonclassical. The candidate already classified the Frobenius nonclassical trinomial curves in his PhD studies, and the paper \cite{borges-guardieiro} containing the results is in preparation. We now propose to study quadrinomial curves (see Section \ref{quadrinomial-curves}) and then compare with the results the candidate obtained before. We also propose to study the Weierstrass semigroups at the points of those trinomial curves, since those semigroups are related to several classification results (see Section \ref{weierstrass-semigroup}).We expect that this project will also provide interesting theoretical results for the field of algebraic curves defined over finite fields and its applications, namely: a criterion for the irreducibility of quadrinomial curves, advancements in the theory of Weierstrass semigroups in Kummer extensions and the construction of error-correcting codes with high length.