UC Santa Cruz

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.