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.
Giulietti, M., Montanucci, M., Zini, G. (2016). On maximal curves that are not quotients of the Hermitian curve. FINITE FIELDS AND THEIR APPLICATIONS, 41, 72-88 [10.1016/j.ffa.2016.05.005].
On maximal curves that are not quotients of the Hermitian curve
Zini, G
2016
Abstract
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.
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