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

Pedro Lemos (University College London)

11-Nov-2020, 16:00-17:00 (3 years ago)

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

Export talk to