Ideals of degree one contribute most of the height

Levin, Aaron; McKinnon, David

doi:10.2140/ant.2012.6.1223

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Ideals of degree one contribute most of the height

Aaron Levin and David McKinnon

Vol. 6 (2012), No. 6, 1223–1238

DOI: 10.2140/ant.2012.6.1223

Abstract

Let $k$ be a number field, $f (x) \in k [x]$ a polynomial over $k$ with $f (0) \neq 0$ , and $O_{k, S}^{*}$ the group of $S$ -units of $k$ , where $S$ is an appropriate finite set of places of $k$ . In this note, we prove that outside of some natural exceptional set $T \subset O_{k, S}^{*}$ , the prime ideals of $O_{k}$ dividing $f (u)$ , $u \in O_{k, S}^{*} ∖ T$ , mostly have degree one over $ℚ$ ; that is, the corresponding residue fields have degree one over the prime field. We also formulate a conjectural analogue of this result for rational points on an elliptic curve over a number field, and deduce our conjecture from Vojta’s conjecture. We prove this conjectural analogue in certain cases when the elliptic curve has complex multiplication.

Keywords

heights, Diophantine approximation, polynomial values, elliptic curves, Vojta's conjecture

Mathematical Subject Classification 2010

Primary: 11G50

Secondary: 11J25

Milestones

Received: 2 June 2011

Revised: 18 October 2011

Accepted: 10 December 2011

Published: 12 August 2012

Authors

Aaron Levin
Department of Mathematics Michigan State University East Lansing, MI 48824 United States

David McKinnon
Department of Pure Mathematics University of Waterloo Waterloo, ON, N2T 2M2 Canada

Vol. 6, No. 6, 2012