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Scholarly Works (2 results)

  • Thesis
  • Peer Reviewed
UC Santa Cruz Electronic Theses and Dissertations (2020)

Conway’s topograph can be used in the study of binary quadratic forms (BQFs)

to replace tedious algebraic computations with straightforward geometric arguments.

The crux of his method is the isomorphism between the arithmetic group PGL2(Z) and

the Coxeter group (3, infinity). We introduce the arithmetic groups called dilinear groups

and construct generalizations of Conway’s topograph called dilinear topographs. Then

we use them to study variants of BQFs, called binary quadratic diforms (BQDs). The

payoff can be seen in the last chapter in our investigation of minimum value bounds for

diforms and pairs of BQFs.

    Cover page: Generalizations of Conway’s Topograph arising from Arithmetic Coxeter Groups
    • Article
    • Peer Reviewed
    UC Santa Cruz Previously Published Works (2019)
    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.
      Cover page: Arithmetic of arithmetic Coxeter groups
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