On possibilities of 3-adic Galois images associated to isogeny-torsion graphs

Abstract

Let $E$ be a non CM elliptic curve defined over $\Q$. There is an isogeny-torsion graph associated to $E$ and there is also a Galois representation $\rho_{E,l^{\infty}} \colon \Gal(\Qbar/\Q) \to \GL_2(\ZZ_{\ell})$ associated to $E$ for every prime $\ell.$ In this article, we explore the relation between these two objects when $\ell=3$. More precisely, we give a classification of 3-adic Galois images associated to vertices of the isogeny-torsion graph of $E.$