Appears in collection : Conférence à la mémoire de Jean-Pierre Demailly

Consider a homogeneous polynomial P with integer coefficents. Its “naive” height is, in classical Diophantine geometry, defined as the maximum of the absolute values of the coefficients of P. More generally, in the framework of Arakelov geometry, the height is a real number, depending on the choice of a Hermitian metric on the hyperplane line bundle over the complex projective variety X, cut out by P. In a recent joint work with Rolf Andreasson we introduce a canonical height, obtained by taking the metric in question to be Kähler-Einstein - if such a metric exists. This canonical height has several useful properties. In particular, it can be expressed as a limit of periods on the N-fold products XN, as N tends to infinity. In this talk I will explain how this leads to an explicit expression for the canonical height of any diagonal homogeneous polynomial P in three variables. The formula involves the Hurwitz zeta function and its derivative at s=−1. Some connections to Shimura curves and Parshin's conjectural arithmetic Miyaoka–Yau type inequality will also be pointed out. No prior knowledge of Arakelov geometry will be assumed.