A uniform relative Dobrowolski's lower bound over abelian extensions

A uniform relative Dobrowolskis lower bound over abelian extensions

Let L/K be an abelian extension of number fields. We prove an uniform lower bound for the height in L∗ outside roots of unity. This lower bound depends only on the degree [L : K].

A Lower Bound for the Canonical Height on Abelian Varieties over Abelian Extensions

Let A be an abelian variety defined over a number field K and let ĥ be the canonical height function on A(K̄) attached to a symmetric ample line bundle L. We prove that there exists a constant C = C(A, K,L) > 0 such that ĥ(P ) ≥ C for all nontorsion points P ∈ A(K), where K is the maximal abelian extension of K. Introduction The purpose of this paper is to provide a proof of the following result...

A Lower Bound for the Canonical Height on Elliptic Curves over Abelian Extensions

Let E/K be an elliptic curve defined over a number field, let ĥ be the canonical height on E, and let K/K be the maximal abelian extension of K. Extending work of Baker [4], we prove that there is a constant C(E/K) > 0 so that every nontorsion point P ∈ E(K) satisfies ĥ(P ) > C(E/K).

A Lower Bound for the Canoni al Height on Ellipti Curves over Abelian Extensions

Lower Bounds for the Canonical Height on Elliptic Curves over Abelian Extensions

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 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 t...