joan.carles.lario@upc.edu
Is there any distinguished elliptic curve over $\mathbf{Q}$ in its isogeny class? First, Mazur and Swinnerton-Dyer proposed the so-called {\it strong} curve which is an optimal quotient of the Jacobian of the modular curve $X_0(M)$, where $M$ is the conductor of the isogeny class. Later, Stevens suggested that it is better to consider the elliptic curve which is an optimal quotient of the Jacobian of the modular curve $X_1(M)$. In both cases the Manin constant plays a role, and the Stevens proposal seems to be more intrinsically arithmetic due to the intervention of Néron models, étale isogenies, and Parshin-Faltings heights. We define the Faltings curve as the one with minimal height in the isogeny class. Let $G$ be the natural graph attached to an isogeny class: a vertex for every elliptic curve in the class, and edges correspond to isogenies of prime degree among them. For every square-free integer $d$, we can consider the graph $G^d$ attached to the twisted elliptic curves in $G$ by the quadratic character of $\mathbf{Q}(\sqrt{d})$. 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 talk we shall discuss the probability distribution of a vertex in $G^d$ to be a Faltings elliptic curve as $|d|$ grows to infinity. This is work in progress in collaboration with Enrique González-Jiménez, UAM.