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The maximal abelian extension contained in a division field of an elliptic curve over ℚ with complex multiplication
Asimina S. Hamakiotes
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Let K be an imaginary quadratic field, and let 𝒪_(K,f) be an order in K of conductor f≥1. Let E be an elliptic curve with CM by 𝒪_(K,f), such that E is defined by a model over ℚ(j_(K,f)), where j_(K,f)=j(E). It has been shown by the author and Lozano-Robledo that Gal(ℚ(j_(K,f),E[N])/ℚ(j_(K,f))) is only abelian for N=2,3, and 4. Let p be a prime and let n≥1 be an integer. In this article, we bound the commutator subgroups of Gal(ℚ(E[pⁿ])/ℚ) and classify the maximal abelian extensions contained in ℚ(E[pⁿ])/ℚ.
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