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Cartan images and ℓ-torsion points of elliptic curves with rational j-invariant
Oron Y. Propp
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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.
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