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

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

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

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

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

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

The form of the infinite product has slightly changed. But since the eta quotients remain the same, then ${\mathfrak {f}}_{i}(\tau )={\mathfrak {f}}_{i}(q)$ as long as the second uses the nome $q=e^{\pi i\tau }$. 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.

Relation to the Ramanujan G and g functions

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

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

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

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

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

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

leads to

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

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

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

and one can easily get ${\mathfrak {f}}(\tau )=2^{1/4}G_{n}$ 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 $q=e^{\pi i\tau },$

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

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

${\begin{aligned}{\frac {\theta _{2}(q)}{\eta (\tau )}}&={\mathfrak {f}}_{2}(q)^{2},\\[4pt]{\frac {\theta _{4}(q)}{\eta (\tau )}}&={\mathfrak {f}}_{1}(q)^{2},\\[4pt]{\frac {\theta _{3}(q)}{\eta (\tau )}}&={\mathfrak {f}}(q)^{2},\end{aligned}}$

with even-subscript theta functions purposely listed first. Using the well-known Jacobi identity with even subscripts on the LHS,

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

therefore,

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

Relation to j-function

The three roots of the cubic equation

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

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

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

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

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

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

See also

• Ramanujan–Sato series, level 4

References

• Duke, William (2005), Continued Fractions and Modular Functions (PDF), Bull. Amer. Math. Soc. 42
• Weber, Heinrich Martin (1981) [1898], Lehrbuch der Algebra (in German), vol. 3 (3rd ed.), New York: AMS Chelsea Publishing, 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