Let A be an abelian variety defined over a number field K and let Kab be the maximal abelian extension of K. We show that there only finitely many torsion points of A which are defined over Kab iff A has no abelian subvariety with complex multiplication over K. We use this to give another proof of Ribet's result that A has only finitely many torsion points which are defined over the cyclotomic extension of K.
On $p$-adic $L$-functions for symplectic representations of GL(N) over number fields
The maximal abelian extension contained in a division field of an elliptic curve over $\mathbb{Q}$ with complex multiplication
Additive Complexity and the Roots of Polynomials Over Number Fields and p-adic Fields
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