Rational points of small height on elliptic curves

In this thesis, we are interested in counting problems concerning quadratic twists of a fixed elliptic curve defined over the field of rational numbers.
Inspired by of the analogy that exists between quadratic twists and real quadratic fields, we show an estimate for the number of quadratic twists having a nontorsion rational point whose canonical height is almost minimal. This establishes an analogue of a result of Hooley about the fundamental solution of the Pell equation.
Building upon this result, we then show that the average analytic rank is greater than one in the family of quadratic twists having a nontorsion rational point of almost minimal height.