Faltings elliptic curves in twisted $\mathbb Q$-isogeny classes

Abstract

Let $G$ be the graph attached to the $\mathbb Q$-isogeny class of an elliptic curve defined over $\mathbb Q$: that is, a vertex for every elliptic curve defined over $\mathbb Q$ in the isogeny class, and edges in correspondence with the prime degree rational isogenies between them. Stevens shows that there is a unique elliptic curve in $G$ with minimal Faltings height. We call this curve the Faltings elliptic curve in $G$. For every square-free integer $d$, we consider the graph~$G^d$ attached to the twisted elliptic curves in $G$ by the quadratic character of $\mathbb Q(\sqrt{d})$. It turns out that $G$ and $G^d$ are canonically isomorphic as abstract graphs (the isomorphism identifies the vertices with equal $j$-invariant). In this paper we determine which vertex is the Faltings elliptic curve in $G^d$. We also obtain the probability of a vertex in $G$ to be the Faltings elliptic curve in~$G^d$. It turns out that this probability depends on the $p$-adic valuations of rational values of certain modular functions.