Summer Learning/Research Seminar

A parallelogram height inequality relates the "arithmetic complexities'' (heights) of four arithmetic objects linked by morphisms arranged in a parallelogram-shaped diagram. For instance, given an elliptic curve $E$ over a number field, and two finite subgroups $G$, $H$ of $E$, Rémond's inequality relates the Faltings heights of the quotients of $E$ by $G$, $H$, $G \cap H$, and $G + H$. This "parallelogram inequality’’ is but a special case of a theorem of Rémond, valid for abelian varieties of arbitrary dimension over any number field. This inequality has interesting Diophantine consequences. More broadly, it complements a classical theorem of Faltings which relates the heights of $E$ and $E/G$, and contributes to our understanding of how heights vary under isogenies. In recent work with Le Fourn and Pazuki, we prove a parallelogram inequality for elliptic curves over function fields, where the rôle of the Faltings height is played by the differential height (our proof is actually written in the context of higher-dimensional abelian varieties). In another project with Baker and Pazuki, we prove a parallelogram inequality in the analogous setting of Drinfeld modules over a function field of positive characteristic.

In this talk, I will introduce the relevant notions, explain the general questions around understanding how maps interact with heights. I will also sketch the key parts of the proof of the parallelogram inequality. Time permitting, I will mention applications of the inequality.