Weber modular function

Let ${\displaystyle q=e^{2\pi i\tau }}$ where τ is an element of the upper half-plane. Then the Weber functions are

${\displaystyle {\begin{aligned}{\mathfrak {f}}(\tau )&=q^{-{\frac {1}{48}}}\prod _{n>0}(1+q^{n-1/2})={\frac {\eta ^{2}(\tau )}{\eta {\big (}{\tfrac {\tau }{2}}{\big )}\eta (2\tau )}}=e^{-{\frac {\pi i}{24}}}{\frac {\eta {\big (}{\frac {\tau +1}{2}}{\big )}}{\eta (\tau )}},\\{\mathfrak {f}}_{1}(\tau )&=q^{-{\frac {1}{48}}}\prod _{n>0}(1-q^{n-1/2})={\frac {\eta {\big (}{\tfrac {\tau }{2}}{\big )}}{\eta (\tau )}},\\{\mathfrak {f}}_{2}(\tau )&={\sqrt {2}}\,q^{\frac {1}{24}}\prod _{n>0}(1+q^{n})={\frac {{\sqrt {2}}\,\eta (2\tau )}{\eta (\tau )}}.\end{aligned}}}$

These are also the definitions in Duke's paper "Continued Fractions and Modular Functions".^{[note 2]} The function ${\displaystyle \eta (\tau )}$ is the Dedekind eta function and ${\displaystyle (e^{2\pi i\tau })^{\alpha }}$ should be interpreted as ${\displaystyle e^{2\pi i\tau \alpha }}$. The descriptions as ${\displaystyle \eta }$ quotients immediately imply

${\displaystyle {\mathfrak {f}}(\tau ){\mathfrak {f}}_{1}(\tau ){\mathfrak {f}}_{2}(\tau )={\sqrt {2}}.}$

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).

Alternatively, let ${\displaystyle q=e^{\pi i\tau }}$ be the nome,

${\displaystyle {\begin{aligned}{\mathfrak {f}}(q)&=q^{-{\frac {1}{24}}}\prod _{n>0}(1+q^{2n-1})={\frac {\eta ^{2}(\tau )}{\eta {\big (}{\tfrac {\tau }{2}}{\big )}\eta (2\tau )}},\\{\mathfrak {f}}_{1}(q)&=q^{-{\frac {1}{24}}}\prod _{n>0}(1-q^{2n-1})={\frac {\eta {\big (}{\tfrac {\tau }{2}}{\big )}}{\eta (\tau )}},\\{\mathfrak {f}}_{2}(q)&={\sqrt {2}}\,q^{\frac {1}{12}}\prod _{n>0}(1+q^{2n})={\frac {{\sqrt {2}}\,\eta (2\tau )}{\eta (\tau )}}.\end{aligned}}}$

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.

Still employing the nome ${\displaystyle q=e^{\pi i\tau }}$, define the Ramanujan G- and g-functions as

${\displaystyle {\begin{aligned}2^{1/4}G_{n}&=q^{-{\frac {1}{24}}}\prod _{n>0}(1+q^{2n-1})={\frac {\eta ^{2}(\tau )}{\eta {\big (}{\tfrac {\tau }{2}}{\big )}\eta (2\tau )}},\\2^{1/4}g_{n}&=q^{-{\frac {1}{24}}}\prod _{n>0}(1-q^{2n-1})={\frac {\eta {\big (}{\tfrac {\tau }{2}}{\big )}}{\eta (\tau )}}.\end{aligned}}}$

The eta quotients make their connection to the first two Weber functions immediately apparent. In the nome, assume ${\displaystyle \tau ={\sqrt {-n}}.}$ Then,

${\displaystyle {\begin{aligned}2^{1/4}G_{n}&={\mathfrak {f}}(q)={\mathfrak {f}}(\tau ),\\2^{1/4}g_{n}&={\mathfrak {f}}_{1}(q)={\mathfrak {f}}_{1}(\tau ).\end{aligned}}}$

Ramanujan found many relations between ${\displaystyle G_{n}}$ and ${\displaystyle g_{n}}$ which implies similar relations between ${\displaystyle {\mathfrak {f}}(q)}$ and ${\displaystyle {\mathfrak {f}}_{1}(q)}$. For example, his identity,

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

leads to

${\displaystyle {\big [}{\mathfrak {f}}^{8}(q)-{\mathfrak {f}}_{1}^{8}(q){\big ]}{\big [}{\mathfrak {f}}(q)\,{\mathfrak {f}}_{1}(q){\big ]}^{8}={\big [}{\sqrt {2}}{\big ]}^{8}.}$

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

${\displaystyle {\begin{aligned}G_{5}&=\left({\frac {1+{\sqrt {5}}}{2}}\right)^{1/4},\\G_{13}&=\left({\frac {3+{\sqrt {13}}}{2}}\right)^{1/4},\\G_{37}&=\left(6+{\sqrt {37}}\right)^{1/4},\end{aligned}}}$

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

The argument of the classical Jacobi theta functions is traditionally the nome ${\displaystyle q=e^{\pi i\tau },}$

${\displaystyle {\begin{aligned}\vartheta _{10}(0;\tau )&=\theta _{2}(q)=\sum _{n=-\infty }^{\infty }q^{(n+1/2)^{2}}={\frac {2\eta ^{2}(2\tau )}{\eta (\tau )}},\\[2pt]\vartheta _{00}(0;\tau )&=\theta _{3}(q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\;=\;{\frac {\eta ^{5}(\tau )}{\eta ^{2}\left({\frac {\tau }{2}}\right)\eta ^{2}(2\tau )}}={\frac {\eta ^{2}\left({\frac {\tau +1}{2}}\right)}{\eta (\tau +1)}},\\[3pt]\vartheta _{01}(0;\tau )&=\theta _{4}(q)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}={\frac {\eta ^{2}\left({\frac {\tau }{2}}\right)}{\eta (\tau )}}.\end{aligned}}}$

Dividing them by ${\displaystyle \eta (\tau )}$, and also noting that ${\displaystyle \eta (\tau )=e^{\frac {-\pi i}{\,12}}\eta (\tau +1)}$, 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,

${\displaystyle \theta _{2}(q)^{4}+\theta _{4}(q)^{4}=\theta _{3}(q)^{4};}$

therefore,

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

The three roots of the cubic equation

${\displaystyle j(\tau )={\frac {(x-16)^{3}}{x}}}$

where j(τ) is the j-function are given by ${\displaystyle x_{i}={\mathfrak {f}}(\tau )^{24},-{\mathfrak {f}}_{1}(\tau )^{24},-{\mathfrak {f}}_{2}(\tau )^{24}}$. Also, since,

${\displaystyle j(\tau )=32{\frac {{\Big (}\theta _{2}(q)^{8}+\theta _{3}(q)^{8}+\theta _{4}(q)^{8}{\Big )}^{3}}{{\Big (}\theta _{2}(q)\,\theta _{3}(q)\,\theta _{4}(q){\Big )}^{8}}}}$

and using the definitions of the Weber functions in terms of the Jacobi theta functions, plus the fact that ${\displaystyle {\mathfrak {f}}_{2}(q)^{2}\,{\mathfrak {f}}_{1}(q)^{2}\,{\mathfrak {f}}(q)^{2}={\frac {\theta _{2}(q)}{\eta (\tau )}}{\frac {\theta _{4}(q)}{\eta (\tau )}}{\frac {\theta _{3}(q)}{\eta (\tau )}}=2}$, then

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

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

• Ramanujan–Sato series, level 4