On the spectrum of genera of quotients of the Hermitian curve

ABSTRACT

We investigate the genera of quotient curves ℋ_q∕G of the ${𝔽}_{q^2}$-maximal Hermitian curve ℋ_q, where G is contained in the maximal subgroup ${\mathcal{ℳ}}_q\le Aut\left({\mathcal{\mathscr{H}}}_q\right)$ fixing a pole-polar pair (P,ℓ) with respect to the unitary polarity associated with ℋ_q. To this aim, a geometric and group-theoretical description of ℳ_q is given. The genera of some other quotients ℋ_q∕G with G≰ℳ_q are also computed. In this way we obtain new values in the spectrum of genera of ${𝔽}_{q^2}$-maximal curves. The complete list of genera g>1 of quotients of ℋ_q is given for q≤29, as well as the genera g of quotients of ℋ_q with $g>\frac{q^2-q+30}{32}$ for any q. As a direct application, we exhibit examples of ${𝔽}_{q^2}$-maximal curves which are not Galois covered by ℋ_q when q is not a cube. Finally, a plane model for ℋ_q∕G is obtained when G is cyclic of order p⋅d, with d a divisor of q+1.

KEYWORDS:

• Hermitian curve
• maximal curves
• quotient curves
• unitary groups

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 11G20