The p-rank of curves of Fermat type

Abstract

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 $\mathrm{Aut}\left(X\right)$ 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 $K={\overline{F}}_p$. 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.