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Let $\mathbb{F}_{q^2}$ be the finite field with $q^2$ elements. Most of theknown $\mathbb{F}_{q^2}$-maximal curves arise as quotient curves of the$\mathbb{F}_{q^2}$-maximal Hermitian curve $\mathcal{H}_{q}$. After a seminalpaper by Garcia, Stichtenoth and Xing, many papers have provided genera ofquotients of $\mathcal{H}_q$, but their complete determination is a challengingopen problem. In this paper we determine completely the spectrum of genera ofquotients of $\mathcal{H}_q$ for any $q\equiv1\pmod4$.

The complete list of genera of quotients of the $\mathbb{F}_{q^2}$-maximal Hermitian curve for $q\equiv1\pmod{4}$