On maximal curves that are not quotients of the Hermitian curve

For each prime power ℓ   the plane curve XℓXℓ with equation Yℓ2−ℓ+1=Xℓ2−XYℓ2−ℓ+1=Xℓ2−X is maximal over Fℓ6Fℓ6. Garcia and Stichtenoth in 2006 proved that X3X3 is not Galois covered by the Hermitian curve and raised the same question for XℓXℓ with ℓ>3ℓ>3; in this paper we show that XℓXℓ is not Galois covered by the Hermitian curve for any ℓ>3ℓ>3. Analogously, Duursma and Mak proved that the generalized GK curve CℓnCℓn over Fℓ2nFℓ2n is not a quotient of the Hermitian curve for ℓ>2ℓ>2 and n≥5n≥5, leaving the case ℓ=2ℓ=2 open; here we show that C2nC2n is not Galois covered by the Hermitian curve over F22nF22n for n≥5n≥5.