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On the spectrum of genera of quotients of the Hermitian curve

ABSTRACT We investigate the genera of quotient curves ℋq∕G of the -maximal Hermitian curve ℋq, where G is contained in the maximal subgroup fixing a pole-polar pair (P,ℓ) with respect to the unitary polarity associated with ℋq. To this aim, a geometric and group-theoretical description of ℳq is given. The genera of some other quotients ℋq∕G with G≰ℳq are also computed. In this way we obtain new values in the spectrum of genera of -maximal curves. The complete list of genera g>1 of quotients of ℋq is given for q≤29, as well as the genera g of quotients of ℋq with for any q. As a direct application, we exhibit examples of -maximal curves which are not Galois covered by ℋq when q is not a cube. Finally, a plane model for ℋq∕G is obtained when G is cyclic of order p⋅d, with d a divisor of q+1.