Lower bounds for the canonical height on elliptic curves over abelian extensions

Details Lower bounds for the canonical height on elliptic curves over abelian extensions Lower bounds for the canonical height on elliptic curves over abelian extensions

2002-12-10

arxiv.org/abs/math/0212132v2

Mathematics

Number Theory

14 pages. Proof of Theorem 1.4 clarified

Scientific paper

Let K be a number field and let E/K be an elliptic curve. If E has complex multiplication, we show that there is a positive lower bound for the canonical height of non-torsion points on E defined over the maximal abelian extension K^ab of K. This is analogous to results of Amoroso-Dvornicich and Amoroso-Zannier for the multiplicative group. We also show that if E has non-integral j-invariant (so that in particular E does not have complex multiplication), then there exists C > 0 such that there are only finitely many points P in E(K^ab) of canonical height less than C. This strengthens a result of Hindry and Silverman.

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