Vol. 6, No. 6, 2012

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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

Let k be a number field, f(x) k[x] a polynomial over k with f(0)0, and Ok,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 Ok,S, the prime ideals of Ok dividing f(u), u Ok,S T, mostly have degree one over ; 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.

heights, Diophantine approximation, polynomial values, elliptic curves, Vojta's conjecture
Mathematical Subject Classification 2010
Primary: 11G50
Secondary: 11J25
Received: 2 June 2011
Revised: 18 October 2011
Accepted: 10 December 2011
Published: 12 August 2012
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