Smallness of Faltings heights of CM abelian varieties
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".
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2111.00617
 Bibcode:
 2021arXiv211100617W
 Keywords:

 Mathematics  Number Theory