Residual Galois representations of elliptic curves with image in the normaliser of a non-split Cartan
Pedro Lemos (University College London)
Abstract: Due to the work of several mathematicians, 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 (at least for large enough primes) 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
Heilbronn number theory seminar
Series comments: This is part of the University of Bristol's Heilbronn number theory seminar. If you wish to attend the talk (and are not a Bristolian), please register using this form or email us at bristolhnts@gmail.com with your name and affiliation (if any).
We will email out the link to all registered participants the day before.
Organizers: | Min Lee*, Dan Fretwell, Oleksiy Klurman |
*contact for this listing |