A Gap in the Spectrum of the Faltings Height

TL;DR

This paper proves the existence of a positive gap above the minimal stable Faltings height for elliptic curves with semistable reduction, providing an explicit constant, but shows no such gap exists for curves with unstable reduction.

Contribution

It establishes a quantitative gap in the spectrum of the stable Faltings height for semistable elliptic curves and explicitly determines the constant, contrasting with the unstable case.

Findings

Existence of a positive gap above the minimal height for semistable elliptic curves.

Explicit determination of the gap constant.

No gap exists for elliptic curves with unstable reduction.

Abstract

We show that the minimum $h_{\text{min}}$ of the stable Faltings height on elliptic curves found by Deligne is followed by a gap. This means that there is a constant $C >0$ such that for every elliptic curve $E/K$ with everywhere semistable reduction over a number field $K$, we either have $h(E/K)=h_{\text{min}}$ or $h(E/K)\geq h_{\text{min}} +C$. We determine such an absolute constant explicitly. On the contrary, we show that there is no such gap for elliptic curves with unstable reduction.