Ideals of degree one contribute most of the height

2011/06/07 by Aaron Levin, Levin, Aaron, David McKinnon +1
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.1106.1385

Abstract

Let k be a number field, f(x)∈ k[x] a polynomial over k with f(0)≠ 0, and Ø_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⊂ Ø_k,S^*, the prime ideals of Ø_k dividing f(u), u∈ Ø_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.

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