In mathematics , the Weber modular functions are a family of three functions f , f 1 , and f 2 ,[ note 1] Heinrich Martin Weber .
Definition
Let
q
=
e
π
i
τ
{\displaystyle q=e^{\pi i\tau }}
nome ) where τ is an element of the upper half-plane .
f
(
τ
)
=
q
−
1
24
∏
n
>
0
(
1
+
q
2
n
−
1
)
=
e
−
π
i
24
η
(
τ
+
1
2
)
η
(
τ
)
=
η
2
(
τ
)
η
(
τ
2
)
η
(
2
τ
)
f
1
(
τ
)
=
q
−
1
24
∏
n
>
0
(
1
−
q
2
n
−
1
)
=
η
(
τ
2
)
η
(
τ
)
f
2
(
τ
)
=
2
q
1
12
∏
n
>
0
(
1
+
q
2
n
)
=
2
η
(
2
τ
)
η
(
τ
)
{\displaystyle {\begin{aligned}{\mathfrak {f}}(\tau )&=q^{-{\frac {1}{24}}}\prod _{n>0}(1+q^{2n-1})=e^{-{\frac {\pi i}{24}}}{\frac {\eta {\big (}{\frac {\tau +1}{2}}{\big )}}{\eta (\tau )}}={\frac {\eta ^{2}(\tau )}{\eta {\big (}{\tfrac {\tau }{2}}{\big )}\eta (2\tau )}}\\{\mathfrak {f}}_{1}(\tau )&=q^{-{\frac {1}{24}}}\prod _{n>0}(1-q^{2n-1})={\frac {\eta {\big (}{\tfrac {\tau }{2}}{\big )}}{\eta (\tau )}}\\{\mathfrak {f}}_{2}(\tau )&={\sqrt {2}}\,q^{\frac {1}{12}}\prod _{n>0}(1+q^{2n})={\frac {{\sqrt {2}}\,\eta (2\tau )}{\eta (\tau )}}\end{aligned}}}
where
η
(
τ
)
{\displaystyle \eta (\tau )}
Dedekind eta function and
(
e
π
i
τ
)
α
{\displaystyle (e^{\pi i\tau })^{\alpha }}
e
π
i
τ
α
{\displaystyle e^{\pi i\tau \alpha }}
η
{\displaystyle \eta }
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
)
2
=
θ
3
(
q
)
η
(
τ
)
f
1
(
τ
)
2
=
θ
4
(
q
)
η
(
τ
)
f
2
(
τ
)
2
=
θ
2
(
q
)
η
(
τ
)
{\displaystyle {\begin{aligned}{\mathfrak {f}}(\tau )^{2}&={\frac {\theta _{3}(q)}{\eta (\tau )}}\\{\mathfrak {f}}_{1}(\tau )^{2}&={\frac {\theta _{4}(q)}{\eta (\tau )}}\\{\mathfrak {f}}_{2}(\tau )^{2}&={\frac {\theta _{2}(q)}{\eta (\tau )}}\\\end{aligned}}}
Using the well-known identity,
θ
2
(
q
)
4
+
θ
4
(
q
)
4
=
θ
3
(
q
)
4
{\displaystyle \theta _{2}(q)^{4}+\theta _{4}(q)^{4}=\theta _{3}(q)^{4}}
thus,
f
2
(
τ
)
8
+
f
1
(
τ
)
8
=
f
(
τ
)
8
{\displaystyle {\mathfrak {f}}_{2}(\tau )^{8}+{\mathfrak {f}}_{1}(\tau )^{8}={\mathfrak {f}}(\tau )^{8}}
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
(
q
)
8
+
θ
3
(
q
)
8
+
θ
4
(
q
)
8
)
3
(
θ
2
(
q
)
θ
3
(
q
)
θ
4
(
q
)
)
8
{\displaystyle j(\tau )=32{\frac {{\Big (}\theta _{2}(q)^{8}+\theta _{3}(q)^{8}+\theta _{4}(q)^{8}{\Big )}^{3}}{{\Big (}\theta _{2}(q)\,\theta _{3}(q)\,\theta _{4}(q){\Big )}^{8}}}}
and using the definitions of the Weber functions in terms of the Jacobi theta functions
f
(
τ
)
2
f
1
(
τ
)
2
f
2
(
τ
)
2
=
θ
3
(
q
)
η
(
τ
)
θ
4
(
q
)
η
(
τ
)
θ
2
(
q
)
η
(
τ
)
=
2
{\displaystyle {\mathfrak {f}}(\tau )^{2}\,{\mathfrak {f}}_{1}(\tau )^{2}\,{\mathfrak {f}}_{2}(\tau )^{2}={\frac {\theta _{3}(q)}{\eta (\tau )}}{\frac {\theta _{4}(q)}{\eta (\tau )}}{\frac {\theta _{2}(q)}{\eta (\tau )}}=2}
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}}
See also
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 (220): 1645– 1662, doi :10.1090/S0025-5718-97-00854-5 MR 1415803
Notes
^ f , f 1 and f 2 are not modular functions (per the Wikipedia definition), but every modular function is a rational function in f , f 1 and f 2 . Some authors use a non-equivalent definition of "modular functions".
