𝔽 p 2 -maximal curves with many automorphisms are Galois-covered by the Hermitian curve

Let 𝔽 be the finite field of order q 2. It is sometimes attributed to Serre that any curve 𝔽-covered by the Hermitian curve H q + 1 : y q + 1 = x q + x ${{\mathcal{H}}_{q+1}}:{{y}^{q+1}}={{x }^{q}}+x$ is also 𝔽-maximal. For prime numbers q we show that every 𝔽-maximal curve x $\mathcal{x}$ of genus g ≥ 2 with | Aut(𝒳) | > 84(g − 1) is Galois-covered by H q + 1 . ${{\mathcal{H}}_{q+1}}.$ The hypothesis on | Aut(𝒳) | is sharp, since there exists an 𝔽-maximal curve x $\mathcal{x}$ for q = 71 of genus g = 7 with | Aut(𝒳) | = 84(7 − 1) which is not Galois-covered by the Hermitian curve H 72 . ${{\mathcal{H}}_{72}}.$

The curve Yn = Xℓ (Xm + 1) over finite fields II

Let F be the finite field of order q 2. In this paper we continue the study in [24], [23], [22] of F-maximal curves defined by equations of type y n = x ℓ ( x m + 1 ) . ${{y}^{n}}={{x}^{\ell }}\left( {{x}^{m}}+1 \right).$ New results are obtained via certain subcovers of the nonsingular model of v N = u t 2 − u ${{v}^{N}}={{u}^{{{t}^{2}}}}-u$ where q = tα , α ≥ 3 is odd and N = (tα + 1)/(t + 1). We observe that the case α = 3 is closely related to the Giulietti–Korchmáros curve.

On space maximal curves

Any maximal curve X is equipped with an intrinsic embedding π: X → Pr which reveal outstanding properties of the curve. By dealing with the contact divisors of the curve π(X) and tangent lines, in this paper we investigate the first positive element that the Weierstrass semigroup at rational points can have whenever r = 3 and π(X) is contained in a cubic surface.