Weber modular function — latest news, discussion & facts — IJR
About Weber modular function Also known as Weber modular functions
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In mathematics, the Weber modular functions are a family of three functions f, f1, and f2, studied by Heinrich Martin Weber.
Article In mathematics, the Weber modular functions are a family of three functions f, f1, and f2, studied by Heinrich Martin Weber.
Contents Definition Let
q
=
e
2
π
i
τ
{\displaystyle q=e^{2\pi i\tau }}
where τ is an element of the upper half-plane. Then the Weber functions are
f
(
τ
)
=
q
−
1
48
∏
n
>
0
(
1
+
q
n
−
1
/
2
)
=
η
2
(
τ
)
η
(
τ
2
)
η
(
2
τ
)
=
e
−
π
i
24
η
(
τ
+
1
2
)
η
(
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
)
.
{\displaystyle {\begin{aligned}{\mathfrak {f}}(\tau )&=q^{-{\frac {1}{48}}}\prod _{n>0}(1+q^{n-1/2})={\frac {\eta ^{2}(\tau )}{\eta {\big (}{\tfrac {\tau }{2}}{\big )}\eta (2\tau )}}=e^{-{\frac {\pi i}{24}}}{\frac {\eta {\big (}{\frac {\tau +1}{2}}{\big )}}{\eta (\tau )}},\\{\mathfrak {f}}_{1}(\tau )&=q^{-{\frac {1}{48}}}\prod _{n>0}(1-q^{n-1/2})={\frac {\eta {\big (}{\tfrac {\tau }{2}}{\big )}}{\eta (\tau )}},\\{\mathfrak {f}}_{2}(\tau )&={\sqrt {2}}\,q^{\frac {1}{24}}\prod _{n>0}(1+q^{n})={\frac {{\sqrt {2}}\,\eta (2\tau )}{\eta (\tau )}}.\end{aligned}}}
These are also the definitions in Duke's paper "Continued Fractions and Modular Functions". The function
η
(
τ
)
{\displaystyle \eta (\tau )}
is the Dedekind eta function and
(
e
2
π
i
τ
)
α
{\displaystyle (e^{2\pi i\tau })^{\alpha }}
should be interpreted as
e
2
π
i
τ
α
{\displaystyle e^{2\pi i\tau \alpha }}
. The descriptions as
η
{\displaystyle \eta }
quotients immediately imply
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 f1 and f2. So the 3-dimensional complex vector space with basis f, f1 and f2 is acted on by the group SL2(Z).
Alternative infinite product Alternatively, let
q
=
e
π
i
τ
{\displaystyle q=e^{\pi i\tau }}
be the nome,
f
(
q
)
=
q
−
1
24
∏
n
>
0
(
1
+
q
2
n
−
1
)
=
η
2
(
τ
)
Relation to the Ramanujan G and g functions Still employing the nome
q
=
e
π
i
τ
{\displaystyle q=e^{\pi i\tau }}
, define the Ramanujan G- and g-functions as
2
1
/
4
G
n
=
q
−
1
24
∏
n
>
0
(
1
+
q
2
n
−
1
)
=
η
Relation to Jacobi theta functions The argument of the classical Jacobi theta functions is traditionally the nome
q
=
e
π
i
τ
,
{\displaystyle q=e^{\pi i\tau },}
ϑ
10
(
0
;
τ
)
=
θ
2
(
q
)
=
∑
n
=
−
∞
∞
q
(
n
+
1
/
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
Article from Wikipedia (CC BY-SA 4.0), where it is maintained by volunteer editors.
{\displaystyle {\begin{aligned}{\mathfrak {f}}(q)&=q^{-{\frac {1}{24}}}\prod _{n>0}(1+q^{2n-1})={\frac {\eta ^{2}(\tau )}{\eta {\big (}{\tfrac {\tau }{2}}{\big )}\eta (2\tau )}},\\{\mathfrak {f}}_{1}(q)&=q^{-{\frac {1}{24}}}\prod _{n>0}(1-q^{2n-1})={\frac {\eta {\big (}{\tfrac {\tau }{2}}{\big )}}{\eta (\tau )}},\\{\mathfrak {f}}_{2}(q)&={\sqrt {2}}\,q^{\frac {1}{12}}\prod _{n>0}(1+q^{2n})={\frac {{\sqrt {2}}\,\eta (2\tau )}{\eta (\tau )}}.\end{aligned}}}
The form of the infinite product has slightly changed. But since the eta quotients remain the same, then
{\displaystyle {\mathfrak {f}}_{i}(\tau )={\mathfrak {f}}_{i}(q)}
as long as the second uses the nome
{\displaystyle q=e^{\pi i\tau }}
. The utility of the second form is to show connections and consistent notation with the Ramanujan G- and g-functions and the Jacobi theta functions, both of which conventionally uses the nome.
{\displaystyle {\begin{aligned}2^{1/4}G_{n}&=q^{-{\frac {1}{24}}}\prod _{n>0}(1+q^{2n-1})={\frac {\eta ^{2}(\tau )}{\eta {\big (}{\tfrac {\tau }{2}}{\big )}\eta (2\tau )}},\\2^{1/4}g_{n}&=q^{-{\frac {1}{24}}}\prod _{n>0}(1-q^{2n-1})={\frac {\eta {\big (}{\tfrac {\tau }{2}}{\big )}}{\eta (\tau )}}.\end{aligned}}}
The eta quotients make their connection to the first two Weber functions immediately apparent. In the nome, assume
{\displaystyle \tau ={\sqrt {-n}}.}
{\displaystyle {\begin{aligned}2^{1/4}G_{n}&={\mathfrak {f}}(q)={\mathfrak {f}}(\tau ),\\2^{1/4}g_{n}&={\mathfrak {f}}_{1}(q)={\mathfrak {f}}_{1}(\tau ).\end{aligned}}}
Ramanujan found many relations between
which implies similar relations between
{\displaystyle {\mathfrak {f}}(q)}
{\displaystyle {\mathfrak {f}}_{1}(q)}
. For example, his identity,
{\displaystyle (G_{n}^{8}-g_{n}^{8})(G_{n}\,g_{n})^{8}={\tfrac {1}{4}},}
{\displaystyle {\big [}{\mathfrak {f}}^{8}(q)-{\mathfrak {f}}_{1}^{8}(q){\big ]}{\big [}{\mathfrak {f}}(q)\,{\mathfrak {f}}_{1}(q){\big ]}^{8}={\big [}{\sqrt {2}}{\big ]}^{8}.}
For many values of n, Ramanujan also tabulated
for even n. This automatically gives many explicit evaluations of
{\displaystyle {\mathfrak {f}}(q)}
{\displaystyle {\mathfrak {f}}_{1}(q)}
{\displaystyle \tau ={\sqrt {-5}},\,{\sqrt {-13}},\,{\sqrt {-37}}}
, which are some of the square-free discriminants with class number 2,
{\displaystyle {\begin{aligned}G_{5}&=\left({\frac {1+{\sqrt {5}}}{2}}\right)^{1/4},\\G_{13}&=\left({\frac {3+{\sqrt {13}}}{2}}\right)^{1/4},\\G_{37}&=\left(6+{\sqrt {37}}\right)^{1/4},\end{aligned}}}
{\displaystyle {\mathfrak {f}}(\tau )=2^{1/4}G_{n}}
from these, as well as the more complicated examples found in Ramanujan's Notebooks.
{\displaystyle {\begin{aligned}\vartheta _{10}(0;\tau )&=\theta _{2}(q)=\sum _{n=-\infty }^{\infty }q^{(n+1/2)^{2}}={\frac {2\eta ^{2}(2\tau )}{\eta (\tau )}},\\[2pt]\vartheta _{00}(0;\tau )&=\theta _{3}(q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\;=\;{\frac {\eta ^{5}(\tau )}{\eta ^{2}\left({\frac {\tau }{2}}\right)\eta ^{2}(2\tau )}}={\frac {\eta ^{2}\left({\frac {\tau +1}{2}}\right)}{\eta (\tau +1)}},\\[3pt]\vartheta _{01}(0;\tau )&=\theta _{4}(q)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}={\frac {\eta ^{2}\left({\frac {\tau }{2}}\right)}{\eta (\tau )}}.\end{aligned}}}
{\displaystyle \eta (\tau )}
{\displaystyle \eta (\tau )=e^{\frac {-\pi i}{\,12}}\eta (\tau +1)}
, then they are just squares of the Weber functions
{\displaystyle {\mathfrak {f}}_{i}(q)}
{\displaystyle {\begin{aligned}{\frac {\theta _{2}(q)}{\eta (\tau )}}&={\mathfrak {f}}_{2}(q)^{2},\\[4pt]{\frac {\theta _{4}(q)}{\eta (\tau )}}&={\mathfrak {f}}_{1}(q)^{2},\\[4pt]{\frac {\theta _{3}(q)}{\eta (\tau )}}&={\mathfrak {f}}(q)^{2},\end{aligned}}}
with even-subscript theta functions purposely listed first. Using the well-known Jacobi identity with even subscripts on the LHS,
{\displaystyle \theta _{2}(q)^{4}+\theta _{4}(q)^{4}=\theta _{3}(q)^{4};}
{\displaystyle {\mathfrak {f}}_{2}(q)^{8}+{\mathfrak {f}}_{1}(q)^{8}={\mathfrak {f}}(q)^{8}.}
{\displaystyle x_{i}={\mathfrak {f}}(\tau )^{24},-{\mathfrak {f}}_{1}(\tau )^{24},-{\mathfrak {f}}_{2}(\tau )^{24}}
{\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, plus the fact that
{\displaystyle {\mathfrak {f}}_{2}(q)^{2}\,{\mathfrak {f}}_{1}(q)^{2}\,{\mathfrak {f}}(q)^{2}={\frac {\theta _{2}(q)}{\eta (\tau )}}{\frac {\theta _{4}(q)}{\eta (\tau )}}{\frac {\theta _{3}(q)}{\eta (\tau )}}=2}
{\displaystyle j(\tau )=\left({\frac {{\mathfrak {f}}(\tau )^{16}+{\mathfrak {f}}_{1}(\tau )^{16}+{\mathfrak {f}}_{2}(\tau )^{16}}{2}}\right)^{3}=\left({\frac {{\mathfrak {f}}(q)^{16}+{\mathfrak {f}}_{1}(q)^{16}+{\mathfrak {f}}_{2}(q)^{16}}{2}}\right)^{3}}
{\displaystyle {\mathfrak {f}}_{i}(\tau )={\mathfrak {f}}_{i}(q)}
and have the same formulas in terms of the Dedekind eta function
{\displaystyle \eta (\tau )}
