Weber modular function

In mathematics, the Weber modular functions are a family of three functions f, f1, and f2, studied by Heinrich Martin Weber.

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 ) η ( τ 2 ) η ( 2 τ ) , f 1 ( q ) = q − 1 24 ∏ n > 0 ( 1 − q 2 n − 1 ) = η ( τ 2 ) η ( τ ) , f 2 ( q ) = 2 q 1 12 ∏ n > 0 ( 1 + q 2 n ) = 2 η ( 2 τ ) η ( τ ) . {\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 f i ( τ ) = f i ( q ) {\displaystyle {\mathfrak {f}}_{i}(\tau )={\mathfrak {f}}_{i}(q)} as long as the second uses the nome q = e π i τ {\displaystyle q=e^{\pi i\tau }}. The utility of the second form is to show connections and consistent notation with the and the Jacobi theta functions, both of which conventionally uses the nome.

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 as

2 1 / 4 G n = q − 1 24 ∏ n > 0 ( 1 + q 2 n − 1 ) = η 2 ( τ ) η ( τ 2 ) η ( 2 τ ) , 2 1 / 4 g n = q − 1 24 ∏ n > 0 ( 1 − q 2 n − 1 ) = η ( τ 2 ) η ( τ ) . {\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 τ = − n . {\displaystyle \tau ={\sqrt {-n}}.} Then,

2 1 / 4 G n = f ( q ) = f ( τ ) , 2 1 / 4 g n = f 1 ( q ) = f 1 ( τ ) . {\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 G n {\displaystyle G_{n}} and g n {\displaystyle g_{n}} which implies similar relations between f ( q ) {\displaystyle {\mathfrak {f}}(q)} and f 1 ( q ) {\displaystyle {\mathfrak {f}}_{1}(q)}. For example, his identity,

( G n 8 − g n 8 ) ( G n g n ) 8 = 1 4 , {\displaystyle (G_{n}^{8}-g_{n}^{8})(G_{n}\,g_{n})^{8}={\tfrac {1}{4}},}

leads to

[ f 8 ( q ) − f 1 8 ( q ) ] [ f ( q ) f 1 ( q ) ] 8 = [ 2 ] 8 . {\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 G n {\displaystyle G_{n}} for odd n, and g n {\displaystyle g_{n}} for even n. This automatically gives many explicit evaluations of f ( q ) {\displaystyle {\mathfrak {f}}(q)} and f 1 ( q ) {\displaystyle {\mathfrak {f}}_{1}(q)}. For example, using τ = − 5 , − 13 , − 37 {\displaystyle \tau ={\sqrt {-5}},\,{\sqrt {-13}},\,{\sqrt {-37}}}, which are some of the square-free discriminants with class number 2,

G 5 = ( 1 + 5 2 ) 1 / 4 , G 13 = ( 3 + 13 2 ) 1 / 4 , G 37 = ( 6 + 37 ) 1 / 4 , {\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}}}

and one can easily get f ( τ ) = 2 1 / 4 G n {\displaystyle {\mathfrak {f}}(\tau )=2^{1/4}G_{n}} from these, as well as the more complicated examples found in Ramanujan's Notebooks.

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 / 2 ) 2 = 2 η 2 ( 2 τ ) η ( τ ) , ϑ 00 ( 0 ; τ ) = θ 3 ( q ) = ∑ n = − ∞ ∞ q n 2 = η 5 ( τ ) η 2 ( τ 2 ) η 2 ( 2 τ ) = η 2 ( τ + 1 2 ) η ( τ + 1 ) , ϑ 01 ( 0 ; τ ) = θ 4 ( q ) = ∑ n = − ∞ ∞ ( − 1 ) n q n 2 = η 2 ( τ 2 ) η ( τ ) . {\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}}}

Dividing them by η ( τ ) {\displaystyle \eta (\tau )}, and also noting that η ( τ ) = e − π i 12 η ( τ + 1 ) {\displaystyle \eta (\tau )=e^{\frac {-\pi i}{\,12}}\eta (\tau +1)}, then they are just squares of the Weber functions f i ( q ) {\displaystyle {\mathfrak {f}}_{i}(q)}

θ 2 ( q ) η ( τ ) = f 2 ( q ) 2 , θ 4 ( q ) η ( τ ) = f 1 ( q ) 2 , θ 3 ( q ) η ( τ ) = f ( q ) 2 , {\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,

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};}

therefore,

f 2 ( q ) 8 + f 1 ( q ) 8 = f ( q ) 8 . {\displaystyle {\mathfrak {f}}_{2}(q)^{8}+{\mathfrak {f}}_{1}(q)^{8}={\mathfrak {f}}(q)^{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}}. Also, since,

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, plus the fact that f 2 ( q ) 2 f 1 ( q ) 2 f ( q ) 2 = θ 2 ( q ) η ( τ ) θ 4 ( q ) η ( τ ) θ 3 ( q ) η ( τ ) = 2 {\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}, then

j ( τ ) = ( f ( τ ) 16 + f 1 ( τ ) 16 + f 2 ( τ ) 16 2 ) 3 = ( f ( q ) 16 + f 1 ( q ) 16 + f 2 ( q ) 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}=\left({\frac {{\mathfrak {f}}(q)^{16}+{\mathfrak {f}}_{1}(q)^{16}+{\mathfrak {f}}_{2}(q)^{16}}{2}}\right)^{3}}

since f i ( τ ) = f i ( q ) {\displaystyle {\mathfrak {f}}_{i}(\tau )={\mathfrak {f}}_{i}(q)} and have the same formulas in terms of the Dedekind eta function η ( τ ) {\displaystyle \eta (\tau )}.