Ideals of degree one contribute most of the height

TL;DR

This paper shows that for most $S$-units in a number field, the prime ideals dividing their values under a polynomial mostly have degree one, and extends this idea conjecturally to rational points on elliptic curves, proving some cases with complex multiplication.

Contribution

It establishes that, outside a natural exceptional set, prime ideals dividing polynomial values at $S$-units are mostly of degree one, and formulates and proves a conjectural analogue for elliptic curves with complex multiplication.

Findings

Most prime ideals dividing polynomial values at $S$-units have degree one.

The conjecture for elliptic curves is proved in cases with complex multiplication.

The result is connected to Vojta's Conjecture.

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 $f(u)$, $u\in \O_{k,S}^*\setminus T$, 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.