Arithmetically equivalent number fields have approximately the same successive minima

TL;DR

This paper proves that arithmetically equivalent number fields of the same degree have approximately the same successive minima, differing only by a constant depending on the degree, with examples showing exact equality is not always possible.

Contribution

It establishes a bound on the difference of successive minima for arithmetically equivalent number fields, advancing understanding of their geometric properties.

Findings

Successive minima are approximately equal for arithmetically equivalent fields.

The difference in successive minima is bounded by a constant depending only on the degree.

Examples demonstrate that exact equality of successive minima does not always hold.

Abstract

Let $K$ and $K'$ be arithmetically equivalent number fields, both of degree $d$. We prove that $K$ and $K'$ have the same successive minima, up to a constant depending only on $d$. We give examples showing that one cannot expect equality.