United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS CURVES OF LARGE GENUS COVERED BY THE HERMITIAN CURVE

For the Hermitian curve H defined over the finite field Fq2, 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 Fq2-covered by H arising from subgroups of the special linear group SL(2, Fq) or from subgroups in the normaliser of the Singer group of the projective unitary group PGU(3, Fq2). Since curves Fq2-covered by H are maximal over Fq2, 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, 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. MIRAMARE TRIESTE July 1999 1 E-mail: [email protected] E-mail: [email protected] Regular Associate of the Abdus Salam ICTP. E-mail: [email protected]