A Minkowski-type theorem on distances to cusps: the class number one case

TL;DR

This paper extends Minkowski's second theorem to a setting involving distances to cusps in hyperbolic spaces, using analogies between Euclidean lattices, number fields, and adelic spaces.

Contribution

It introduces a Minkowski-type theorem relating distances to cusps in hyperbolic spaces to Roy--Thunder minima in number theory, bridging geometric and algebraic frameworks.

Findings

Establishes an analogy between rank 2 Euclidean lattices and points in hyperbolic space.

Translates Minkowski's theorem into a statement about distances to cusps in hyperbolic spaces.

Provides bounds on distances to cusps using number-theoretic minima.

Abstract

In the study of Euclidean lattices, the product of the successive minima is bounded from above and below by explicit quantities. This result is known as Minkowski's second theorem, and can be refined to include Hermite's constant in the upper bound, which measures how short a non-zero vector can be in a given lattice. A version of this result exists in the context of number fields, where lattices are replaced with rigid adelic spaces, and successive minima with the Roy--Thunder minima. In this paper, drawing on the analogy between rank $2$ Euclidean lattices and points in $\mathbb{H}$, we will see an analogy between $2$-dimensional rigid adelic spaces and points in $\mathbb{H}^n$, and use that to translate the Minkowski-type theorem on Roy--Thunder minima into a theorem on the distances to cusps in $\mathbb{H}^n$.