Heights and regulators of number fields and elliptic curves

TL;DR

This paper compares invariants of number fields and abelian varieties, showing finiteness results and properties of regulators under certain conjectures, highlighting similarities in their arithmetic structures.

Contribution

It establishes finiteness of non-CM number fields with bounded regulator and, assuming the height conjecture, proves a Northcott property for regulators of abelian varieties.

Findings

Finite number of non-CM fields with bounded regulator.

Northcott property for regulators of abelian varieties under conjecture.

Arithmetic similarities between CM fields and abelian varieties with non Zariski dense Mordell-Weil groups.

Abstract

We compare general inequalities between invariants of number fields and invariants of abelian varieties over number fields. On the number field side, we remark that there is only a finite number of non-CM number fields with bounded regulator. On the abelian side, assuming the height conjecture of Lang and Silverman, we obtain a Northcott property for the regulator on the set of abelian varieties with dense rational points over a number field. This amounts to say that the arithmetic of CM fields is similar, with respect to the invariants considered here, to the arithmetic of abelian varieties over a number field having a non Zariski dense Mordell-Weil group.