Hermite's Constant for Quadratic Number Fields

Abstract

We develop a method to compute the Hermite-Humbert constants $\gam_{K,n}$ of a real quadratic number field $K$, the analogue of the classical Hermite constant $\gam_n$ when $\funnyQ$ is replaced by a quadratic extension. In the case $n=2$, the problem is equivalent to the determination of lowest points of fundamental domains in $\H^2$ for the Hilbert modular group over $K$, that had been studied experimentally by H. Cohn. We establish the results he conjectured for the fields $ \funnyQ@(\sqrt{2})$, $\funnyQ@(\sqrt{3})$ and $\funnyQ@(\sqrt{5})$. The method relies on the characterization of extreme forms in terms of perfection and eutaxy given by the second author in an earlier paper.