Fundamental domains for the general linear group

Abstract

Historically the most familiar fundamental domain for P n /GL n (Z) has been that of Minkowski. This paper develops a new fundamental domain more suited to applications in number theory. It is shown that these domains can be determined explicitly for given n and this is done for n = 3,4, 5,6. A reduction algorithm for an arbitrary element of P n is also determined.

Units in some families of algebraic number fields

“…Hence, D[g] = K, the union being taken over all g ∈ Γ ∞ . Note that the fundamental domain described in [16] coincides with the domain found by Korkine and Zolotarev in 1873 (see [18] or [24]). Unless a point X ∈ P is integral, like point I in Example 1 below, in order to prove that X is extremal, we shall show that X[h] is Minkowski reduced for some h ∈ Γ ∞ .…”

An analogue of Siegel’s 𝜙-operator for automorphic forms for 𝐺𝐿_{𝑛}(𝑍)

“…Here the square bracket notation has the same meaning as before. It may seem a little strange at first to decompose Y in this fashion, but as was shown in [3] this generalizes the identification of the upper half-plane to S?P2 to the spaces S?Pn in a convenient fashion. For the matters at hand, W £ S^P"-X means that for fixed v , f can be thought of as a function of W, and so a function on <9?P"-X.…”

Ryshkov domains of reductive algebraic groups

“…Since Q depends on Q, we obtain r G different kinds of fundamental domains of G.‫/ށ‬ 1 with respect to G.k/. The method to construct Q may be viewed as a generalization of the highest point method (see [Grenier 1988] and[Terras 1988, §4,4]). For example, let k D ‫,ޑ‬ G D GL n and Q be a standard maximal ‫-ޑ‬parabolic subgroup such that QnG is a projective space.…”