Cartan images and?(ell )-torsion points of elliptic curves with rational?j-invariant

摘要

Let $$ell $$ ? 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 $$ell $$ ? , 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- $$ell $$ ? 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 .