In mathematics, the Weber modular functions are a family of three modular functions f, f1, and f2, studied by Heinrich Martin Weber.
Let
q
=
e
2
π
i
τ
where τ is an element of the upper half-plane.
f
(
τ
)
=
q
−
1
48
∏
n
>
0
(
1
+
q
n
−
1
2
)
=
e
−
π
i
24
η
(
τ
+
1
2
)
η
(
τ
)
=
η
2
(
τ
)
η
(
τ
2
)
η
(
2
τ
)
f
1
(
τ
)
=
q
−
1
48
∏
n
>
0
(
1
−
q
n
−
1
2
)
=
η
(
τ
2
)
η
(
τ
)
f
2
(
τ
)
=
2
q
1
24
∏
n
>
0
(
1
+
q
n
)
=
2
η
(
2
τ
)
η
(
τ
)
where
η
(
τ
)
is the Dedekind eta function. Note the
η
quotients immediately imply that
f
(
τ
)
f
1
(
τ
)
f
2
(
τ
)
=
2
.
The transformation τ → –1/τ fixes f and exchanges f1 and f2. So the 3-dimensional complex vector space with basis f, f1 and f2 is acted on by the group SL2(Z).
Let the argument of the Jacobi theta function be the nome
q
=
e
π
i
τ
. Then,
f
(
τ
)
=
θ
3
(
0
,
q
)
η
(
τ
)
f
1
(
τ
)
=
θ
4
(
0
,
q
)
η
(
τ
)
f
2
(
τ
)
=
θ
2
(
0
,
q
)
η
(
τ
)
Thus,
f
1
(
τ
)
8
+
f
2
(
τ
)
8
=
f
(
τ
)
8
which is simply a consequence of the well known identity,
θ
2
(
0
,
q
)
4
+
θ
4
(
0
,
q
)
4
=
θ
3
(
0
,
q
)
4
The three roots of the cubic equation,
j
(
τ
)
=
(
x
+
16
)
3
x
where j(τ) is the j-function are given by
x
i
=
f
(
τ
)
24
,
f
1
(
τ
)
24
,
f
2
(
τ
)
24
. Also, since,
j
(
τ
)
=
32
(
θ
2
(
0
,
q
)
8
+
θ
3
(
0
,
q
)
8
+
θ
4
(
0
,
q
)
8
)
3
(
θ
2
(
0
,
q
)
θ
3
(
0
,
q
)
θ
4
(
0
,
q
)
)
8
then,
j
(
τ
)
=
(
f
(
τ
)
16
+
f
1
(
τ
)
16
+
f
2
(
τ
)
16
2
)
3
