Computing canonical heights using arithmetic intersection theory

The canonical height ĥ on an abelian variety A defined over a global field k is an object of fundamental importance in the study of the arithmetic of A. For many applications it is required to compute ĥ(P ) for a given point P ∈ A(k). For instance, given generators of a subgroup of the Mordell-Weil group A(k) of finite index, this is necessary for most known approaches to the computation of generators of the Mordell-Weil group A(k). Furthermore, the regulator of A(k), which appears in the statement of the conjecture of Birch and Swinnerton-Dyer, is defined in terms of the canonical height and thus we need the ability to compute canonical heights in order to gather numerical evidence for the conjecture in the case of positive rank.