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Residual Galois representations of elliptic curves with image contained in the normaliser of a nonsplit Cartan
Algebra & Number Theory  (IF0.938),  Pub Date : 2021-05-20, DOI: 10.2140/ant.2021.15.747
Samuel Le Fourn, Pedro Lemos

It is known that if p > 37 is a prime number and E is an elliptic curve without complex multiplication, then the image of the mod p Galois representation

ρ̄E,p : Gal(¯) GL(E[p])

of E is either the whole of GL(E[p]), or is contained in the normaliser of a nonsplit Cartan subgroup of GL(E[p]). In this paper, we show that when p > 1.4 × 107, the image of ρ̄E,p is either GL(E[p]), or the full normaliser of a nonsplit Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For d 1, let I(d) denote the set of primes p for which there exists an elliptic curve defined over and without complex multiplication admitting a degree p isogeny defined over a number field of degree d. We show that, for d 1.4 × 107, we have

I(d) = {p prime : p d 1}.