Curves of large genus covered by the Hermitian curve

For the Hermitian curve H defined over the finite field F-q2, we give a com plete classification of Galois coverings of H of prime degree. The correspo nding quotient curves turn out to be special cases of wider families of cur ves F-q2-covered by H arising from subgroups of the special linear group SL (2,F-q) or from subgroups in the normaliser of the Singer group of the proj ective unitary group PGU(3, F-q2) Since curves F-q2-covered by H are maxima l over F-q2, our results provide some classification and existence theorems for maximal curves having large genus, as well as several values for the s pectrum of the genera of maximal curves. For every q(2), both the upper lim it and the second largest genus in the spectrum are known, but the determin ation of the third largest value is still in progress. A discussion on the "third largest genus problem" including some new results and a detailed acc ount of current work is given.