On the $a$-number of Fermat type function fields and some of their subfields

Abstract

The $a$-number of a function field $F$ is the dimension of the kernel of the Cartier operator acting on the holomorphic differentials of $F$. We compute the action of the Cartier operator on the holomorphic differentials on the Hermitian function field $\mathcal{H}$, function fields of Fermat type ${\mathcal{F}}_{n,m}$ and the Hurwitz function field ${\mathcal{H}}_n$. From this, we obtain various results about the $a$-number of these function fields, including a generalisation of the results on the Fermat function field from Kodama and Washio.