Richard Griffon

A parallelogram inequality for abelian varieties over function fields.

Given an abelian variety $A$ over a number field, and two finite subgroups $G$,$H$ of $A$, Rémond has recently proved a so-called ``parallelogram inequality’’ which relates the Faltings heights of the quotients of $A$ by $G$, $H$, $G \cap H$, and $G + H$.
I will talk about a very recent work — joint with Samuel Le Fourn and Fabien Pazuki — in which we prove a perfect analogue of Rémond’s inequality in the context of abelian varieties over function fields, where the role of the Faltings height is played by the differential height. I will introduce the relevant notions and sketch our proof. Among other tools, we need to control the variation of the differential height through isogenies of various types.
Time permitting, I will also discuss an application of this inequality in the function field setting to a bound on the height of an abelian subvariety of a given abelian variety.