Cartan images and $$\ell $$ ℓ -torsion points of elliptic curves with rational j-invariant

Abstract

Abstract Let

$$\ell $$

ℓ

be an odd prime and d a positive integer. We determine when there exists a degree-d number field K and an elliptic curve E / K with

$$j(E)\in \mathbb {Q}\setminus \{0,1728\}$$

j ( E ) ∈ Q \ { 0 , 1728 }

for which

$$E(K)_\mathrm {tors}$$

E

( K )

tors

contains a point of order

$$\ell $$

ℓ

, conditionally on a conjecture of Sutherland. We likewise determine when there exists such a pair (K, E) for which the image of the associated mod-

$$\ell $$

ℓ

Galois representation is contained in a Cartan subgroup or its normalizer. We do the same under the stronger assumption that E is defined over

$$\mathbb {Q}$$

Q

.