A Gap in the Spectrum of the Faltings Height

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 $h(E/K)\ge h_{\text{min}}+C$. We determine such an absolute constant explicitly. On the contrary, we show that there is no such gap for elliptic curves with unstable reduction.