Maximal curves over finite fields and related objects

The thesis deals with maximal curves over finite fields, that is, algebraic curves X of genus g over a finite field GF(q^2) of cardinality q^2 attaining the Hasse-Weil upper bound q^2+1+2gq on the number of GF(q^2)-rational places.We construct many quotient curves of the GK maximal curves and give explicit equations; in this way, we obtain many new genera for maximal curves, and new maximal curves which are not covered or Galois covered by the Hermitian curve. We show that another important maximal curves are not Galois covered by the Hermitian curve maximal over their field of maximality: the curves by Garcia-Guneri-Stichtenoth and by Garcia-Stichtenoth, one Suzuki curve and one Ree curve, the covers of the Suzuki and Ree curves introducted by Skabelund. We give applications of (maximal) curves over finite fields in several areas. In Finite Geometry, we construct new small families of complete (k,3)- and (k,4)-arcs in the Galois planes; in Coding Theory, we construct Goppa codes from separable Kummer covers of the projective line and from the GK curve; Permutation Polynomials, we classify explicitely a large class of complete permutation polynomials of monomial type.