arXiv subjects

Marie Frank vom Braucke

Publications and source records attributed to Marie Frank vom Braucke.

2 recordsLinked to original sources

arXiv2026

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

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.

math.AG↗

arXiv2026

On Weierstrass semigroups of maximal Fermat function fields

In this article we explicitly determine the Weierstrass semigroup at any place of some $\mathbb{F}_{q^2}$-maximal Fermat function fields $\mathcal{F}_m$, namely for $m=(q+1)/2$ and $m=(q+1)/3$. These famous function fields arise as Galois subfields of the Hermitian function field, and even though they have been intensively studied in the literature, the Weierstrass semigroup at every place is still not fully known. Surprisingly enough this problem is in fact quite involved and $\mathcal{F}_m$ has many different types of Weierstrass semigroups. Moreover, its set of Weierstrass places is much richer than its set of rational places.

math.AG↗