  We consider an abelian variety defined over a number field. We give
conditional bounds for the order of its Tate-Shafarevich group, as well as
conditional bounds for the N\'eron-Tate height of generators of its
Mordell-Weil group. The bounds are implied by strong but nowadays classical
conjectures, such as the Birch and Swinnerton-Dyer conjecture and the
functional equation of the L-series. In particular, we improve and generalise a
result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group,
and extends a conjecture of S. Lang on the canonical height of a system of
generators of the free part of the Mordell-Weil group. The method is an
extension of the algorithm proposed by Yu. Manin for finding a basis for the
non-torsion rational points of an elliptic curve defined over the rationals.
