On maximal curves that are not quotients of the Hermitian curve

الأصل · EN

For each prime power ℓ the plane curve Xℓ with equation Yℓ²⁻ℓ⁺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.