Number Theory: An Introduction to Mathematics

5 The Modular Group 205

This is illustrated in Figure 2, where the domaing(F)is represented simply by the
group elementg.
There is an interesting connection between the modular group and binary quadratic
forms. Thediscriminantof a binary quadratic form

f=ax^2 +bxy+cy^2

with coefficientsa,b,c∈RisD:=b^2 − 4 ac. The quadratic form isindefinite(i.e.
assumes both positive and negative values) if and only ifD>0, andpositive definite
(i.e. assumes only positive values unlessx=y=0) if and only ifD<0,a>0,
which implies alsoc>0. (IfD=0, the quadratic form is proportional to the square
of a linear form.)
If we make a linear change of variables

x=αx′+βy′, y=γx′+δy′,

whereα,β,γ,δ∈Zandαδ−βγ=1, the quadratic formfis transformed into the
quadratic form

f′=a′x′^2 +b′x′y′+c′y′^2 ,

where

a′=aα^2 +bαγ+cγ^2 ,

b′= 2 aαβ+b(αδ+βγ)+ 2 cγδ,

c′=aβ^2 +bβδ+cδ^2 ,

and hence

b′^2 − 4 a′c′=b^2 − 4 ac=D.

The quadratic formsfandf′are said to beproperly equivalent.

I T

ST STS

TSTS TST

–1 –1/2 0 1/2 1 3/2 2

Fig. 2.Tiling ofHbyΓ.