Residual Galois representations of elliptic curves with image in the normaliser of a non-split Cartan

Abstract: It is known that if $p$ is a prime $>37$, then the image of the residual Galois representation $\bar{\rho}_{E,p}: G_{\mathbb{Q}}\rightarrow {\rm GL}_2(\mathbb{F}_p)$ attached to an elliptic curve $E/\mathbb{Q}$ without complex multiplication is either ${\rm GL}_2(\mathbb{F}_p)$, or is contained in the normaliser of a non-split Cartan subgroup of ${\rm GL}_2(\mathbb{F}_p)$. I will report on a recent joint work with Samuel Le Fourn where we improve this result by showing that if $p>1.4\times 10^7$, then $\bar{\rho}_{E,p}$ is either surjective, or its image is the normaliser of a non-split Cartan subgroup of ${\rm GL}_2(\mathbb{F}_p)$.

number theory

Audience: researchers in the topic

London number theory seminar

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