A parallelogram inequality for elliptic curves over function fields
Given an elliptic curve \(E\) over a number field, and two finite subgroups \(G\), \(H\) of \(E\), one can relate the Faltings heights of the quotients of \(E\) by \(G\), \(H\), \(G \cap H\), and \(G + H\). This relation is called the ``parallelogram inequality’’ and is but a special case of a theorem of Rémond ((his result works for abelian varieties of arbitrary dimension)). In a very recent work, joint with Le Fourn and Pazuki, we prove a perfect analogue of Rémond’s inequality in the context of elliptic curves over function fields, where the rôle of the Faltings height is played by the differential height (Our proof actually works for higher-dimensional abelian varieties too). In this talk, I will introduce the relevant notions, explain the general questions around understanding how isogenies interact with heights, and sketch the key parts of the proof of the parallelogram inequality. Time permitting, I will mention an application of the inequality, as well as a conjectural analogue in another context.