Curves of large genus covered by the hermitian curve

Abstract

For the Hermitian curve H defined over the finite field , we give a complete classification of Galois coverings of H of prime degree. The corresponding quotient curves turn out to be special cases of wider families of curves -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 . Since curves -covered by H are maximal over , our results provide some classification 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 work is given.