The $a$-numbers of Fermat and Hurwitz 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.