Explicit p-adic Hodge theory for elliptic curves and non-split Cartan images

Abstract

Let $E/\mathbb{Q}_p$ be an elliptic curve whose mod $p$ Galois image is contained in the normaliser of a non-split Cartan. We classify the possible $p$-adic images of $E$ using tools from $p$-adic Hodge theory via a careful analysis of the local Galois structure of the $p$-power torsion of $E$. We pay special attention to the case where $E$ has potentially supersingular reduction, where we give an algorithm to determine the corresponding filtered $(\varphi,\operatorname{Gal}(K/\mathbb{Q}_p))$-module from a Weierstrass model (which appears to be novel), and introduce alternative division polynomials that may be of independent interest. We deduce global consequences for elliptic curves $E/\mathbb{Q}$: when the mod $p$ representation of $E$ has non-split Cartan image and $E$ doesn't have CM, the $p$-adic image must be the full preimage of the normaliser of a mod $p^n$ non-split Cartan for some $n \geq 1$. As an application, we sharpen existing bounds on the adelic image in terms of the Weil height of the $j$-invariant.