A Fundamental Domain for the General Linear Group by Means of Successive Minima

We present a variant of the method of construction of Watanabe of a fundamental domain for the translation action by the subgroup of rational points on the whole group of adlic points of the general linear group over an arbitrary number field.The obtained domain is a generalization of Grenier's one, not Minkowski's.For several fields, we profit our presentation to show that the fundamental domain is in a so-called Siegel set.As an example, inequalities bounding our fundamental domain are explicitly determined in the case of degree 3 over the imaginary quadratic field with discriminant -3 , which will be interesting to some people.