  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.
