We prove a new formula for the number of integral points on an elliptic curve over a function field without assuming that the coefficient field is algebraically closed. This is an improvement on the standard results of Hindry-Silverman.

Two projective nonsingular complex algebraic curves X and Y defined over the field R of real numbers can be isomorphic while their sets X(R) and Y(R) of R-rational points could be even non homeomorphic. This leads to the count of the number of real forms of a complex algebraic curve X, that is, those nonisomorphic real algebraic curves whose complexifications are isomorphic to X. In this paper we compute, as a function of genus, the maximum number of such real forms that a complex algebraic curve admits.

In a finite Desarguesian plane of odd order, it was shown by Segre thirty years ago that a set of maximum size with at most two points on a line is a conic. Here, in a plane of odd or even order, sufficient conditions are given for a set with at most three points on a line to be a cubic curve. The case of an elliptic curve is of particular interest.

Let K be a number field and E/K an elliptic curve. As is well known [3, 4,[ if K has class number 1, then there exists a global minimal Weierstrass equation for E. Our main goal in this paper is to prove the following converse to this statement.