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

Heilbronn Number Theory Seminar

11th November 2020, 4:00 pm – 5:00 pm

Zoom,

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}: Gal(\overline{Q}/ Q) -> GL_2 (F_p) attached to an elliptic curve E/Q without complex multiplication is either GL_2(F_p), or is contained in the normaliser of a non-split Cartan subgroup of GL_2(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 x 10^7, then \bar{\rho}_{E,p} is either surjective, or its image is the normaliser of a non-split Cartan subgroup of GL_2(F_p).

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