Heights, ranks and regulators of abelian varieties

Abstract

We lower bound the Faltings height of an abelian variety over a number field by the sum of its injectivity diameter and the norm of its bad reduction primes. It leads to an unconditional bound on the rank of Mordell-Weil groups. Assuming the height conjecture of Lang and Silverman, we then obtain a Northcott property for the regulator on the set of simple abelian varieties defined over a fixed number field, of fixed dimension $g$, bounded rank and with dense rational points over a number field. We remove the simplicity assumption in the principally polarized case by giving a refined version of the Lang-Silverman conjecture.

Several improvements. arXiv:1406.0120v3 and the present text are independent, but both come from the obsolete arXiv:1406.0120

References in corpus (2)

• Polynomial bounds for Arakelov invariants of Belyi curves
• New explicit formulas for Faltings' delta-invariant

Cited by in corpus (2)

• Canonical Kahler metrics and Arithmetics -- Generalising Faltings heights
• Homothéties explicites des représentations $\ell$-adiques