On maximal curves that are not quotients of the Hermitian curve

For each prime power ℓ, the plane curve Xℓ with equation $Y^{ℓ^2−ℓ+1}=X^{ℓ^2}−X$ is maximal over F_{ℓ^6}. 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 F_{ℓ^{2n}} 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 F_{2^{2n}} for n≥5.