RT Journal Article
SR Electronic
T1 Arithmetic of arithmetic Coxeter groups
JF Proceedings of the National Academy of Sciences
JO Proc Natl Acad Sci USA
FD National Academy of Sciences
SP 442
OP 449
DO 10.1073/pnas.1809537115
VO 116
IS 2
A1 Milea, Suzana
A1 Shelley, Christopher D.
A1 Weissman, Martin H.
YR 2019
UL http://www.pnas.org/content/116/2/442.abstract
AB Conway’s topograph provided a combinatorial-geometric perspective on integer binary quadratic forms—quadratic functions of two variables with integer coefficients. This perspective is practical for solving equations and easily bounds the minima of binary quadratic forms. It appears that Conway’s topograph is just the first in a series of applications of arithmetic Coxeter groups to arithmetic. Four other applications are described in this article, and dozens more may be possible.In the 1990s, J. H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the “topograph,” Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pell’s equation. It appears that the crux of his method is the coincidence between the arithmetic group PGL2(Z) and the Coxeter group of type (3,∞). There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conway’s topograph and generalizations to other arithmetic Coxeter groups. This includes a study of “arithmetic flags” and variants of binary quadratic forms.