We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field Fq2 whose number of Fq2 -rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are Fq2 -isomorphic to y q + y = x for some m ∈ Z.

We classify, up to isomorphism, maximal curves covered by the Hermit-ian curve H by a prime degree Galois covering. We also compute the genus of maximal curves obtained by the quotient of H by several automorphisms groups. Finally we discuss the value for the third largest genus that a maximal curve can have.

A family of maximal curves is investigated that are all quotients of the Hermitian curve. These curves provide examples of curves with the same genus, the same automorphism group and in some cases the same order sequence of the linear series associated to maximal curves, but that are not isomorphic. Dedicated with affection to Zhe-Xian Wan on the occasion of his 80-th birthday

We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F q 2 whose number of F q 2-rational points reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a rational point, we prove that maximal curves are F q 2-isomorphic to y q + y = x m , for some m ∈ Z +. As a consequence we show that a maximal curve of genus g = (q − ...