  Using original ideas from J.-B. Bost and S. David, we provide an explicit
comparison between the Theta height and the stable Faltings height of a
principally polarized abelian variety. We also give as an application an
explicit upper bound on the number of K-rational points of a curve of genus g>1
over a number filed K under a conjecture of S. Lang and J. Silverman. We
complete the study with a comparison between differential lattice structures.

Number Theory