The p-rank of curves of Fermat type

Abstract

Let $\mathbb{K}$ 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 $\mathcal{X}$ over $\mathbb{K}$, This birational invariant affects arithmetic and geometric properties of $\mathcal{X}$, and its fundamental role in the study of the automorphism group $\operatorname{Aut}(\mathcal{X})$ 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 $\mathbb{K}=\bar{\mathbb{F}}_p$. 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.