Arithmetic of arithmetic Coxeter groups.
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- Record sourced from PubMed, PMID 30587588.
- Also identified by DOI 10.1073/pnas.1809537115 and PMC identifier 6329956.
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Abstract
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 [Formula: see text] and the Coxeter group of type [Formula: see text] 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.