𝔽p2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve

Abstract

Abstract Let 𝔽 be the finite field of order q2. It is sometimes attributed to Serre that any curve 𝔽-covered by the Hermitian curveHq+1:yq+1=xq+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 Hq+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 H72. ${{\mathcal{H}}_{72}}.$