The $a$-numbers of Fermat and Hurwitz curves

TL;DR

This paper computes the $a$-numbers, which measure the space of exact holomorphic differentials, for infinite families of Fermat and Hurwitz curves, providing new insights and proofs related to Hermitian curves.

Contribution

It introduces explicit calculations of $a$-numbers for Fermat and Hurwitz curves, extending understanding of their algebraic and geometric properties.

Findings

Computed $a$-numbers for Fermat curves

Computed $a$-numbers for Hurwitz curves

Provided a new proof for the $a$-number of Hermitian curves

Abstract

For an algebraic curve $\mathcal{X}$ defined over an algebraically closed field of characteristic $p >0$, the $a$-number $a(\mathcal{X})$ is the dimension of the space of exact holomorphic differentials on $\mathcal{X}$. We compute the $a$-number for an infinite families of Fermat and Hurwitz curves. Our results apply to Hermitian curves giving a new proof for a previous result of Gross.