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arXiv · 1106.1385

Ideals of degree one contribute most of the height

Aaron Levin·David McKinnon
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Abstract

Let $k$ be a number field, $f(x)\in k[x]$ a polynomial over $k$ with $f(0)\neq 0$, 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 $T\subset Ø_{k,S}^*$, the prime ideals of $Ø_k$ dividing $f(u)$, $u\in Ø_{k,S}^*\setminus T$, 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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Aaron Levin, David McKinnon. 2011-06-07. Ideals of degree one contribute most of the height. https://arxiv.org/abs/1106.1385

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