A numerical survey of the floors of various Hilbert fundamental domains

Harvey Cohn
1965 Mathematics of Computation  
1. Introduction. From the purely computational point of view we are considering a real positive function S2 = f(Rx, Rî , Sx) defined (in (3.15) below) for values Rx , R2, and Si varying on a parallelopiped. The function / is composed of a large and undetermined number (possibly thousands!) of analytic pieces. The object is to find the minimum of / and to estimate ¿he number of pieces which constitute /. What we do is probably the easiest thing: We subdivide the parallelopiped by a regular
more » ... dimensional grid and scan for min S2 as well as the number of pieces in the function /. The function / arises here in an interesting context, however, since it represents part of the boundary (called the "floor") of the fundamental domain R for Hubert's modular group for certain quadratic fields of unique factorization. (We assume some knowledge of factorization theory [1] but we summarize Siegel's theory of these fundamental domains [2], [5], for easy reference.) It is important to know the minimum of S2 (and the optimal "low-point") because of applications to relative-quadratic fields [2, §6], [4] and it is important to know the number of pieces constituting / as a clue to the topological structure of R under boundary identifications.
doi:10.1090/s0025-5718-1965-0195818-4 fatcat:ftvddawepfg3zkvul5pyyx2wcq