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arXiv · 2609.08565

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

Marie Frank vom Braucke
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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.

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Marie Frank vom Braucke. 2026-09-08. On the $a$-number of Fermat type function fields and some of their subfields. https://arxiv.org/abs/2609.08565

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