Small Points and Free Abelian Groups

Abstract

Let F be an algebraic extension of the rational numbers and E an elliptic curve defined over some number field contained in F . The absolute logarithmic Weil height, respectively the Néron-Tate height, induces a norm on F * modulo torsion, respectively on E(F ) modulo torsion. The groups F * and E(F ) are free abelian modulo torsion if the height function does not attain arbitrarily small positive values. In this paper we prove the failure of the converse to this statement by explicitly constructing counterexamples.2010 Mathematics Subject Classification. 11G50, 11G05 (primary); 20K20 (secondary).

Fields Generated by Finite Rank Subgroups of $\overline{\mathbb{Q}}^*$

“…This is mainly an application of a classification result of Pontryagin. The proof follows very closely the proofs of [19], Lemma 1, and [12], Proposition 2.3.…”

“…In particular, for all σ ∈ Gal(K(A tors , 1 would be equal to ℓ m . Then (16) implies that ℓ m ≤ 4M ℓ ord ℓ (f ) (12) < ℓ m ℓ in contradiction to the fact m ≥ m ℓ . Therefore, A[ℓ] r+1 is isomorphic to a subgroup of Gal(K(A tors , 1 ℓ m ℓ Γ ′ α )/K(A tors )).…”

“…Up to this restriction, the fields generated by torsion points are almost as large as possible. Namely, (12) (…”