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In mathematics, the Weber modular functions are a family of three modular functions f , f 1 , and f 2 , studied by Heinrich Martin Weber .
Definition
Let
q
=
e
2
π
i
τ
{\displaystyle q=e^{2\pi i\tau }}
τ is an element of the upper half-plane .
f
(
τ
)
=
q
−
1
/
48
∏
n
>
0
(
1
+
q
n
−
1
/
2
)
=
e
−
2
π
i
/
48
η
(
1
2
(
τ
+
1
)
)
η
(
τ
)
=
η
2
(
τ
)
η
(
1
2
τ
)
η
(
2
τ
)
f
1
(
τ
)
=
q
−
1
/
48
∏
n
>
0
(
1
−
q
n
−
1
/
2
)
=
η
(
1
2
τ
)
η
(
τ
)
f
2
(
τ
)
=
2
q
1
/
24
∏
n
>
0
(
1
+
q
n
)
=
2
η
(
2
τ
)
η
(
τ
)
{\displaystyle {\begin{aligned}{\mathfrak {f}}(\tau )&=q^{-1/48}\prod _{n>0}(1+q^{n-1/2})=e^{-2\pi i/48}{\frac {\eta {\big (}{\tfrac {1}{2}}(\tau +1){\big )}}{\eta (\tau )}}={\frac {\eta ^{2}(\tau )}{\eta {\big (}{\tfrac {1}{2}}\tau {\big )}\eta (2\tau )}}\\{\mathfrak {f}}_{1}(\tau )&=q^{-1/48}\prod _{n>0}(1-q^{n-1/2})={\frac {\eta {\big (}{\tfrac {1}{2}}\tau {\big )}}{\eta (\tau )}}\\{\mathfrak {f}}_{2}(\tau )&={\sqrt {2}}\,q^{1/24}\prod _{n>0}(1+q^{n})={\sqrt {2}}\,{\frac {\eta (2\tau )}{\eta (\tau )}}\end{aligned}}}
where η (τ ) is the Dedekind eta function . Note the eta quotients immediately imply that,
f
(
τ
)
f
1
(
τ
)
f
2
(
τ
)
=
2
{\displaystyle {\mathfrak {f}}(\tau ){\mathfrak {f}}_{1}(\tau ){\mathfrak {f}}_{2}(\tau )={\sqrt {2}}}
The transformation τ → –1/τ fixes f and exchanges f 1 and f 2 . So the 3-dimensional complex vector space with basis f , f 1 and f 2 is acted on by the group SL2 (Z ).
Relation to theta functions
Let the argument of the Jacobi theta function be the nome
q
=
e
π
i
τ
{\displaystyle q=e^{\pi i\tau }}
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
)
{\displaystyle {\begin{aligned}{\mathfrak {f}}(\tau )&={\sqrt {\frac {\theta _{3}(0,q)}{\eta (\tau )}}}\\{\mathfrak {f}}_{1}(\tau )&={\sqrt {\frac {\theta _{4}(0,q)}{\eta (\tau )}}}\\{\mathfrak {f}}_{2}(\tau )&={\sqrt {\frac {\theta _{2}(0,q)}{\eta (\tau )}}}\\\end{aligned}}}
Thus,
f
1
(
τ
)
8
+
f
2
(
τ
)
8
=
f
(
τ
)
8
{\displaystyle {\mathfrak {f}}_{1}(\tau )^{8}+{\mathfrak {f}}_{2}(\tau )^{8}={\mathfrak {f}}(\tau )^{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
{\displaystyle \theta _{2}(0,q)^{4}+\theta _{4}(0,q)^{4}=\theta _{3}(0,q)^{4}}
Relation to j-function
The three roots of the cubic equation ,
j
(
τ
)
=
(
x
+
16
)
3
x
{\displaystyle j(\tau )={\frac {(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
{\displaystyle x_{i}={\mathfrak {f}}(\tau )^{24},{\mathfrak {f}}_{1}(\tau )^{24},{\mathfrak {f}}_{2}(\tau )^{24}}
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
{\displaystyle j(\tau )=32{\frac {{\Big (}\theta _{2}(0,q)^{8}+\theta _{3}(0,q)^{8}+\theta _{4}(0,q)^{8}{\Big )}^{3}}{{\Big (}\theta _{2}(0,q)\theta _{3}(0,q)\theta _{4}(0,q){\Big )}^{8}}}}
then,
j
(
τ
)
=
(
f
(
τ
)
16
+
f
1
(
τ
)
16
+
f
2
(
τ
)
16
2
)
3
{\displaystyle j(\tau )=\left({\frac {{\mathfrak {f}}(\tau )^{16}+{\mathfrak {f}}_{1}(\tau )^{16}+{\mathfrak {f}}_{2}(\tau )^{16}}{2}}\right)^{3}}
References
Weber, Heinrich Martin (1981) [1898], Lehrbuch der Algebra ISBN 978-0-8218-2971-4 Yui, Noriko; Zagier, Don (1997), "On the singular values of Weber modular functions", Mathematics of Computation , 66 : 1645– 1662, doi :10.1090/S0025-5718-97-00854-5 , MR 1415803 
