No abstract

Let 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) ∈ Q \ {0, 1728} for which E(K ) tors contains a point of order , 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-Galois representation is contained in a Cartan subgroup or its normalizer. We do the same under the stronger assumption that E is defined over Q.

Let 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) ∈ Q \ {0, 1728} for which E(K)tors contains a point of order ℓ. We also determine when there exists such a pair (K, E) for which the image of the associated mod-ℓ Galois representation is contained in a Cartan subgroup or its normalizer, conditionally on a conjecture of Sutherland. We do the same under the stronger assumption that E is defined over Q.