The complete list of genera of quotients of the ${\mathbb{F}}_{q^2}$-maximal Hermitian curve for $q\equiv 1(\mathrm{mod}4)$

Abstract

Let ${\mathbb{F}}_{q^2}$ be the finite field with $q^2$ elements. Most of the known ${\mathbb{F}}_{q^2}$-maximal curves arise as quotient curves of the ${\mathbb{F}}_{q^2}$-maximal Hermitian curve ${\mathcal{H}}_q$. After a seminal paper by Garcia, Stichtenoth and Xing, many papers have provided genera of quotients of ${\mathcal{H}}_q$, but their complete determination is a challenging open problem. In this paper we determine completely the spectrum of genera of quotients of ${\mathcal{H}}_q$ for any $q\equiv 1(\mathrm{mod}4)$.

arXiv admin note: text overlap with arXiv:1805.09118