On the classification problem for the genera of quotients of the Hermitian curve

In this article, we characterize the genera of those quotient curves H q /G of the F q 2-maximal Hermitian curve H q for which either G is contained in the maximal subgroup M 1 of (H q ) fixing a self-polar triangle, or q is even and G is contained in the maximal subgroup M 2 of (H q ) fixing a pole-polar pair (P,l) with respect to the unitary polarity associated to H q (F q 2) In this way, several new values for the genus of a maximal curve over a finite field are obtained. Our results leave just two open cases to provide the complete list of genera of Galois subcovers of the Hermitian curve; namely, the open cases in [4] when G fixes a point P∈H q (F q 2) and q is even, and the open cases in [33] when G≤M 2 and q is odd