On maximal curves that are not quotients of the Hermitian curve

For each prime power ℓ the plane curve Xℓ with equation Yℓ2ℓ+1=Xℓ2-X is maximal over Fℓ6. Garcia and Stichtenoth in 2006 proved that X3 is not Galois covered by the Hermitian curve and raised the same question for Xℓ with ℓ>3; in this paper we show that Xℓ is not Galois covered by the Hermitian curve for any ℓ>3. Analogously, Duursma and Mak proved that the generalized GK curve Cℓn over Fℓ2n is not a quotient of the Hermitian curve for ℓ>2 and n≥5, leaving the case ℓ=2 open; here we show that C2n is not Galois covered by the Hermitian curve over F22n for n≥5.