Theta height and Faltings height

TL;DR

This paper compares the Theta height and Faltings height of abelian varieties, providing explicit bounds and applications to rational points on curves, based on original ideas from Bost and David.

Contribution

It offers an explicit comparison between two important heights and derives bounds on rational points, advancing understanding in arithmetic geometry.

Findings

Explicit comparison between Theta height and Faltings height

Upper bounds on K-rational points of curves of genus g>1

Comparison of differential lattice structures

Abstract

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.