A Minkowski-type theorem on distances to cusps: the general case
Mathieu Dutour
TL;DR
The paper extends a Minkowski-type framework for distances to cusps from the class number one setting to all totally real number fields by associating a 2-dimensional rigid adelic space $E_{\mathfrak{a},\tau}$ to each $\tau\in\mathbb{H}^n$ and fractional ideal $\mathfrak{a}$. It establishes a precise link between the Roy–Thunder minima of $E_{\mathfrak{a},\tau}$ and the $\mathfrak{a}$-distances to the cusps via $\Lambda_1(E_{\mathfrak{a},\tau})=N(\mathfrak{a})^{-1/n}\mu_{\mathfrak{a},1}(\tau)^{-1/2n}$ and $\Lambda_2(E_{\mathfrak{a},\tau})=N(\mathfrak{a})^{-1/n}\mu_{\mathfrak{a},2}(\tau)^{-1/2n}$, and proves a Minkowski-type inequality: $\frac{1}{c_{II}^{\Lambda}(2,K)^{4n}}\cdot\frac{1}{N(\mathfrak{a})^2} \leq \mu_{\mathfrak{a},1}(\tau)\mu_{\mathfrak{a},2}(\tau) \leq \frac{1}{N(\mathfrak{a})^2}$. The results yield effective, uniform cusp-separation statements and optimal lower bounds for $\mu_{\mathfrak{a},1}$, plus asymptotic controls on Hermite-type constants. The paper also analyzes a class of integrals over $\widehat{\Gamma}_K(\mathfrak{a})\backslash\mathbb{H}^n$ and develops a partial-volume framework, linking geometric, arithmetic, and adelic perspectives in the Hilbert modular setting. Together, these contributions broaden the applicability of Minkowski-type distance estimates to broader arithmetic settings and offer tools for height-estimate problems in related contexts.
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 $τ\in \mathbb{H}^n$ and its two closest cusps.