On the essential minimum of Faltings' height

TL;DR

This paper investigates the minimal values of Faltings' height on elliptic curves, providing bounds, numerical evidence of multiple small values, and approximations of related Green functions using complex analysis techniques.

Contribution

It offers the first detailed bounds and numerical evidence for the essential minimum of Faltings' height, contrasting it with other heights and employing advanced approximation methods.

Findings

Faltings' height has at least two values below its essential minimum.

Bounds for the essential minimum are computed up to five decimal places.

Numerical evidence suggests at least four isolated small values before the minimum.

Abstract

We study the essential minimum of the (stable) Faltings height on the moduli space of elliptic curves. We prove that, in contrast to the Weil height on a projective space and the N{\'e}ron-Tate height of an abelian variety, Faltings' height takes at least two values that are smaller than its essential minimum. We also provide upper and lower bounds for this quantity that allow us to compute it up to five decimal places. In addition, we give numerical evidence that there are at least four isolated values before the essential minimum. One of the main ingredients in our analysis is a good approximation of the hyperbolic Green function associated to the cusp of the modular curve of level one. To establish this approximation, we make an intensive use of distortion theorems for univalent functions. Our results have been motivated and guided by numerical experiments that are described in detail in the companion files.