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A New Northcott Property for the Faltings Height

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Speaker: 
Lucia Mocz
Affiliation: 
Universität Bonn
Date: 
Wed, 2018-11-21 14:30 - 15:30
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

The Faltings height is a useful invariant for addressing questions in
arithmetic geometry. In his celebrated proof of the Mordell and
Shafarevich conjectures, Faltings shows the Faltings height satisfies a
certain Northcott property, which allows him to deduce his finiteness
statements. In this work we prove a new Northcott property for the
Faltings height. Namely we show, assuming the Colmez Conjecture and the
Artin Conjecture, that there are finitely many CM abelian varieties of a
fixed dimension which have bounded Faltings height. The technique
developed uses new tools from integral p-adic Hodge theory to study the
variation of Faltings height within an isogeny class of CM abelian
varieties. In special cases, we are able to use these techniques to
moreover develop new Colmez-type formulas for the Faltings height.

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