  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 a family of certain Picard curves, using the action
of the Cartier operator on $H^0(\mathcal{X},\Omega^1)$.
