For the Hermitian curve H dened over the nite eld F q 2 , we give a complete classication of Galois coverings of H of prime degree. The corresponding quotient curves turn out to be special cases of wider families of curves F q 2 -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 projective unitary group P G U (3; F q 2 ). Since curves F q 2 -covered by H are maximal over F q 2 , our results provide some classication and existence theorems for maximal curves having large genus, as well as several values for the spectrum of the genera of maximal curves. For every q 2 , both the upper limit and the second largest genus in the spectrum are known, but the determination of the third largest value is still in progress. A discussion on the \third largest genus problem" including some new results and a detailed account of current w ork is given.
MIRAMARE { TRIESTE
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