Curves over finite fields (whose cardinality is a square) attaining the Hasse-Weil upper bound for the number of rational points are called maximal curves. Here we deal with three problems on maximal curves: 1. Determination of the possible genera of maximal curves. 2. Determination of explicit equations for maximal curves. 3. Classification of maximal curves having a fixed genus.

Using an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field Fq. This bound enables us to provide explicit conditions on q, g and π for the nonexistence of absolutely irreducible projective algebraic curves defined over Fq of geometric genus g, arithmetic genus π an...

A lower bound on the minimum degree of the plane algebraic curves containing every point in a large point-set K of the Desarguesian plane PG(2, q) is obtained. The case where K is a maximal (k, n)-arc is considered in greater depth.

In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.