The p-rank of curves of Fermat type

Autor(es)

Borges, Herivelto ; Goncalves, Cirilo

• Tipo de documento: Artigo Científico
• Fonte: FINITE FIELDS AND THEIR APPLICATIONS; v. 97, p. 36-pg., 2024-04-16.

Resumo

Let K be an algebraically closed field of characteristic p > 0. A pressing problem in the theory of algebraic curves is the determination of the p -rank of a (nonsingular, projective, irreducible) curve X over K. This birational invariant affects arithmetic and geometric properties of X, and its fundamental role in the study of the automorphism group Aut(X) has been noted by many authors in the past few decades. In this paper, we provide an extensive study of the p -rank of curves of Fermat type y(m) = x(n) + 1 over K = F<overline>(p). We determine a combinatorial formula for this invariant in the general case and show how this leads to explicit formulas of the p -rank of several such curves. By way of illustration, we present explicit formulas for more than twenty subfamilies of such curves, where m and n are generally given in terms of p. We also show how the approach can be used to compute the p -rank of other types of curves. (AU)

• Processo FAPESP: 22/15301-5 - Curvas algébricas sobre corpos finitos e aplicações
• Beneficiário: Herivelto Martins Borges Filho
• Modalidade de apoio: Auxílio à Pesquisa - Regular