标题： 到尖点的距离的闵可夫斯基型定理：类数一的情况 显示英文标题

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

摘要： 在欧几里得格的研究中，连续极小的乘积被显式量从上方和下方界定了。 这个结果被称为闵可夫斯基第二定理，并可以改进以包含赫尔米特常数在上界中，该常数衡量给定格中非零向量可以有多短。 在数域的背景下存在这一结果的一个版本，其中格被刚性阿代尔空间代替，连续极小被Roy--Thunder极小代替。 在本文中，基于秩$2$欧几里得格与$\mathbb{H}$中点之间的类比，我们将看到$2$维刚性阿代尔空间与$\mathbb{H}^n$中点之间的类比，并利用这一点将关于Roy--Thunder极小的闵可夫斯基型定理转化为关于$\mathbb{H}^n$中尖点距离的定理。 显示英文摘要

摘要： 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 $\mathbb{H}^n$.