Smallness of Faltings heights of CM abelian varieties

TL;DR

This paper establishes bounds on the Faltings height of CM abelian varieties in relation to the root discriminant of their fields, assuming certain conjectures, and proves unconditional bounds for specific cases.

Contribution

It proves bounds on the Faltings height of CM abelian varieties assuming conjectures and provides unconditional bounds for cases with no complex quadratic subfields.

Findings

Bound on Faltings height in terms of root discriminant

Unconditional bounds for CM fields without complex quadratic subfields

Smallness of root discriminant for fields of good reduction

Abstract

We prove that assuming the Colmez conjecture and the ``no Siegel zeros" conjecture, the stable Faltings height of a CM abelian variety over a number field is less than or equal to the logarithm of the root discriminant of the field of definition of the abelian variety times an effective constant depending only on the dimension of the abelian variety. In view of the fact that the Colmez conjecture for abelian CM fields, the averaged Colmez conjecture, and the ``no Siegel zeros" conjecture for CM fields with no complex quadratic subfields are already proved, we prove unconditional analogues of the result above. In addition, we also prove that the logarithm of the root discriminant of the field of everywhere good reduction of CM abelian varieties can be ``small".