A Minkowski-type theorem on distances to cusps: the general case

TL;DR

This paper extends Minkowski-type theorems relating heights and distances to cusps in hyperbolic spaces to all totally real number fields, removing previous restrictive assumptions.

Contribution

It generalizes previous results by removing the class number one restriction, establishing a Minkowski-type theorem for distances to cusps in the broader setting of totally real fields.

Findings

Established a Minkowski-type inequality for distances to cusps in hyperbolic spaces.

Extended the framework to all totally real number fields, regardless of class number.

Linked heights and cusp distances without prior restrictions.

Abstract

In a previous paper, we studied the connection between points in $\mathbb{H}^n$ and $2$-dimensional rigid adelic spaces on a totally real number field $K$ with class number $h_K = 1$. This last assumption was needed to link heights and distances to cusps. In this paper, we remove this hypothesis to obtain, without restriction on $K$ totally real, an analogue of Minkowski's second theorem on the Roy--Thunder minima of a $2$-dimensional rigid adelic space in the framework of distances between a point $\tau \in \mathbb{H}^n$ and its two closest cusps.