A finiteness property of torsion points

Abstract

Let k be a number field, and let G be either the multiplicative group Gm/k or an elliptic curve E/k. Let S be a finite set of places of k containing the archimedean places. We prove that if α ∈ G(k) is nontorsion, then there are only finitely many torsion points ξ ∈ G(k)tors which are S-integral with respect to α. We also formulate conjectural generalizations for dynamical systems and for abelian varieties. 0. Introduction. Let k be a number field, with ring of integers Ok and algebraic closure k. In this paper we prove finiteness theorems for torsion points which are integral with respect to a given nontorsion point, for the multiplicative group Gm/k and for elliptic curves E/k. We then attempt to place these results in a conceptual framework, and conjecture generalizations to dynamical systems and abelian varieties. Let S be a finite set of places of k containing the archimedean places. Given α, β ∈ P(k), let cl(α), cl(β) be their Zariski closures in P/Spec(Ok). By definition, β is S-integral relative to α if cl(β) does not meet cl(α) outside S. Thus, β is S-integral relative to α if and only if for each place v of k not in S, and each pair of k-embeddings σ : k(β) ↪→ kv, τ : k(α) ↪→ kv, we have ‖σ(β), τ(α)‖v = 1 under the spherical metric on P(kv). Equivalently, for all σ, τ , { |σ(β)− τ(α)|v ≥ 1 if |τ(α)|v ≤ 1 , |σ(β)|v ≤ 1 if |τ(α)|v > 1 . Theorem 0.1. Let k be a number field, and let S be a finite set of places of k containing all the archimedean places. Fix α ∈ P(k) with Weil height h(α) > 0; that is, identifying P(k) with k ∪{∞}, α is not 0 or ∞ or a root of unity. Then there are only finitely many roots of unity in k which are S-integral with respect to α. Similarly, let E/k be an elliptic curve, and let E/Spec(Ok) be a model of E. Theorem 0.2. Let k be a number field, and let S be a finite set of places of k containing all the archimedean places. If α ∈ E(k) is nontorsion (has canonical height ĥ(α) > 0), there are only finitely many torsion points ξ ∈ E(k)tors which are S-integral with respect to α. By S-integrality we mean that the Zariski closures of ξ and α in the model E/Spec(Ok) do not meet outside fibres above S. Since any two models are isomorphic outside a finite set of places, it follows from the theorem that the finiteness property is independent of the choice of the set S and the model E . Date: October 22, 2007. 2000 Mathematics Subject Classification. Primary 11G05, 37F10, Secondary 11J86, 11J71, 11G50.