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Assume γ = 0. Then the Moebius transformation deﬁned by M −1 is the
translation z → z + β and hence takes z out of the domain − 1 ≤ Re z ≤ 1
δ
2
2
unless β = 0 or β = ±1 and Re z = ± 1 . In the ﬁrst case M = ±I and f = f . In
2
1 ±1
the second case M = ±
, f = ax2 ± axy + cy 2 and f = ax2 axy + cy 2 .
01
Assume γ = ±1. If γ = 1, then z + δ  ≤ 1 implies
(i) δ = 0, z  = 1, or
(ii) z = ρ := √
−1+ −3
2 and δ = 1. In case (i) we have M = ± −1
0 α
1 1
and M · z = α − z . This easily implies 01
and
−1 0
¯
M · f = cx2 − 2bxy + ay 2 . Since (c, b, a) ∈ Ω, we get a = c. Again f is of the
form ax2 + 2bxy + ay 2 and is properly equivalent to ax2 − 2bxy + ay 2 .
In the second case f = a(x2 + xy + y 2 ) and M f = a(x2 − xy + y 2 ).
α α−1
Now, in case (ii), we get M =
and M ·ρ = (αρ+(α−1))/(ρ+1) =
1
1
α + ρ. This implies α = 0, f = a(x2 + xy + y 2 ), M f = f .
Finally, the case γ = −1 is reduced to the case γ = 1 by replacing M with
−M .
This analysis proves the following:
α = 0 or (α, z ) = (−1, ρ), (1, −ρ2 ). So, in the ﬁrst case, M = Theorem 1.4. Let f = ax2 + 2bxy + cy 2 and f = a x2 + 2b xy + c y 2 be two
properly reduced positive deﬁnite binary forms. Then f is properly equivalent
to f if and only if f = f or f = ax2 ± axy + cy 2 , f = ax2 axy + cy 2 , or
f = ax2 + 2bxy + ay 2 , f = ax2 − 2bxy + ay 2 . Moreover, M f = f for some
M = ±I if and only if one of the following cases occurs:
(i) f = a(x2 + y 2 ) and M = ± 0 −1
;
10 (ii) f = a(x2 ± xy + y 2 ) and M = ± 1 −1
0 −1
,±
.
1
0
11 Deﬁnition. Let G be a group acting on a set X . A subset S of X is called a
fundamental domain for the action of G on X if each orbit of G intersects S at
exactly one element.
¯
The proof of Theorem 1.4 shows this enlarged set Ω contains a representative
¯
of each orbit of SL(2, Z). Moreover, two points (a, b, c) and (a , b , c ) in Ω belong
to the same orbit of SL(2, Z) if and only if either a = c = a = c , b = −b or
a = a, b = −b = a/2. Clearly
¯
Ω = R+ × D . ...
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 Fall '09
 ONTONKONG

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