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

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.