# On the spectrum of genera of quotients of the Hermitian curve

## TL;DR

This paper studies the genera of quotient curves derived from the Hermitian curve over finite fields, providing new genus values and geometric models, especially for quotients by certain subgroups of the automorphism group.

## Contribution

It offers a detailed geometric and group-theoretical analysis of the automorphism subgroup fixing a pole-polar pair, and computes new genus values for quotient curves beyond known cases.

## Key findings

- New genus values for quotient curves of the Hermitian curve.
- A geometric model for quotients when the group is cyclic of order p·d.
- Extension of the spectrum of genera for maximal curves.

## Abstract

We investigate the genera of quotient curves $\mathcal H_q/G$ of the $\mathbb F_{q^2}$-maximal Hermitian curve $\mathcal H_q$, where $G$ is contained in the maximal subgroup $\mathcal M_q\leq{\rm Aut}(\mathcal H_q)$ fixing a pole-polar pair $(P,\ell)$ with respect to the unitary polarity associated with $\mathcal H_q$. To this aim, a geometric and group-theoretical description of $\mathcal M_q$ is given. The genera of some other quotients $\mathcal H_q/G$ with $G\not\leq\mathcal M_q$ are also computed. Thus we obtain new values in the spectrum of genera of $\mathbb F_{q^2}$-maximal curves. A plane model for $\mathcal H_q/G$ is obtained when $G$ is cyclic of order $p\cdot d$, with $d$ a divisor of $q+1$.