Parametric geometry of numbers over a number field and extension of scalars

TL;DR

This paper extends the parametric geometry of numbers to number fields and their completions, revealing that approximation exponents over $\

Contribution

It generalizes the theory of rational approximation to number fields and relates approximation over $K$ to that over $\\mathbb{Q}$ via scalar extension, enabling new constructions of algebraic curves with singular points.

Findings

Exponents of approximation over $\

Same spectrum for approximation exponents over $\

Construction of algebraic curves with very singular points

Abstract

The parametric geometry of numbers of Schmidt and Summerer deals with rational approximation to points in $\mathbb{R}^n$. We extend this theory to a number field $K$ and its completion $K_w$ at a place $w$ in order to treat approximation over $K$ to points in $K_w^n$. As a consequence, we find that exponents of approximation over $\mathbb{Q}$ in $\mathbb{R}^n$ have the same spectrum as their generalizations over $K$ in $K_w^n$. When $w$ has relative degree one over a place $\ell$ of $\mathbb{Q}$, we further relate approximation over $K$ to a point $\boldsymbol{\xi}$ in $K_w^n$, to approximation over $\mathbb{Q}$ to a point $\Xi$ in $\mathbb{Q}_\ell^{nd}$, obtained by extension of scalars, where $d$ is the degree of $K$ over $\mathbb{Q}$. By combination with a result of Bel, this allows us to construct algebraic curves in $\mathbb{R}^{3d}$ defined over $\mathbb{Q}$, of degree $2d$, containing points that are very singular with respect to rational approximation.