scholarly journals Arithmetic of arithmetic Coxeter groups

2018 ◽  
Vol 116 (2) ◽  
pp. 442-449 ◽  
Author(s):  
Suzana Milea ◽  
Christopher D. Shelley ◽  
Martin H. Weissman
Keyword(s):  
Coxeter Group ◽  
Quadratic Forms ◽  
Coxeter Groups ◽  
Diophantine Equations ◽  
Geometric Method ◽  
Arithmetic Group ◽  
Binary Quadratic Forms ◽  
Pell’S Equation

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.

scholarly journals On S-class number relations of algebraic tori in Galois extensions of global fields

1991 ◽  
Vol 124 ◽  
pp. 133-144 ◽  
Author(s):  
Masanori Morishita
Keyword(s):  
Class Number ◽  
Quadratic Forms ◽  
Euler Number ◽  
Number Fields ◽  
Binary Quadratic Forms ◽  
Algebraic Number Fields ◽  
Algebraic Tori ◽  
Genus Theory ◽  
Class Number Formula ◽  
Number Formula

As an interpretation and a generalization of Gauss’ genus theory on binary quadratic forms in the language of arithmetic of algebraic tori, Ono [02] established an equality between a kind of “Euler number E(K/k)” for a finite Galois extension K/k of algebraic number fields and other arithmetical invariants associated to K/k. His proof depended on his Tamagawa number formula [01] and Shyr’s formula [Sh] which follows from the analytic class number formula of a torus. Later, two direct proofs were given by Katayama [K] and Sasaki [Sa].

scholarly journals The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space

2014 ◽  
Vol 66 (2) ◽  
pp. 354-372 ◽  
Author(s):  
Ruth Kellerhals ◽  
Alexander Kolpakov
Keyword(s):  
Growth Rate ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Triangle Group ◽  
Minimal Growth ◽  
Salem Number

AbstractDue to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on H3 is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has the smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is, as such, unique. Our approach provides a different proof for the analog situation in H2 where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).

Representing Primes by Binary Quadratic Forms

1991 ◽  
Vol 64 (1) ◽  
pp. 34
Author(s):  
Steven Galovich ◽  
Jeremy Resnick
Keyword(s):  
Quadratic Forms ◽  
Binary Quadratic Forms

scholarly journals Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions

2011 ◽  
Vol 16 ◽  
pp. 82 ◽  
Author(s):  
Mehmet Koca ◽  
Nazife Ozdes Koca ◽  
Muna Al-Shueili
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Linear Combinations ◽  
Octahedral Group ◽  
Snub Cube ◽  
Proper Rotation

There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. In this paper we construct the chiral polyhedra and their dual solids in a systematic way. We use the proper rotational subgroups of the Coxeter groups and to derive the orbits representing the solids of interest. They lead to the polyhedra tetrahedron, icosahedron, snub cube, and snub dodecahedron respectively. We prove that the tetrahedron and icosahedron can be transformed to their mirror images by the proper rotational octahedral group so they are not classified in the class of chiral polyhedra. It is noted that vertices of the snub cube and snub dodecahedron can be derived from the vectors, which are linear combinations of the simple roots, by the actions of the proper rotation groupsand  respectively. Their duals are constructed as the unions of three orbits of the groups of concern. We also construct the polyhedra, quasiregular in general, by combining chiral polyhedra with their mirror images. As a by-product we obtain the pyritohedral group as the subgroup the Coxeter group and discuss the constructions of pyritohedrons. We employ a method which describes the Coxeter groups and their orbits in terms of quaternions.  

scholarly journals Congruences for representations of primes by binary quadratic forms

1982 ◽  
Vol 41 (4) ◽  
pp. 311-322
Author(s):  
Richard Hudson ◽  
Kenneth Williams
Keyword(s):  
Quadratic Forms ◽  
Binary Quadratic Forms
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