Fp2 - Maximal Curves with Many Automorphisms are Galois-Covered by the Hermitian Curve

Let F be the finite field of order q2, q = ph with p prime. It is commonly
atribute to J.P. Serre the fact that any curve F-covered by the Hermitian curve Hq+1 :
yq+1 = xq + x is also F-maximal. Nevertheless, the converse is not true as the Giulietti-
Korchm´aros example shows provided that q > 8 and h ≡ 0 (mod 3). In this paper, we
show that if an F-maximal curve X of genus g ≥ 2 where q = p is such that |Aut(X)| >
84(g − 1) then X is Galois-covered by Hp+1. Also, we show that the hypothesis on the
order of Aut(X) is sharp, since there exists an F-maximal curve X for q = 71 of genus
g = 7 with |Aut(X)| = 84(7 − 1) which is not Galois-covered by the Hermitian curve
H72.