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

$\begin{array}{rl}\mathfrak{𝔣}(\tau )& =q^{-\frac{1}{48}}\prod _{n>0}(1+q^{n-1/2})=\frac{{\eta }^2(\tau )}{\eta \left(\frac{\tau }{2}\right)\eta (2\tau )}=e^{-\frac{\pi i}{24}}\frac{\eta \left(\frac{\tau +1}{2}\right)}{\eta (\tau )},\\ {\mathfrak{𝔣}}_1(\tau )& =q^{-\frac{1}{48}}\prod _{n>0}(1-q^{n-1/2})=\frac{\eta \left(\frac{\tau }{2}\right)}{\eta (\tau )},\\ {\mathfrak{𝔣}}_2(\tau )& =\sqrt{2}{q}^{\frac{1}{24}}\prod _{n>0}(1+q^n)=\frac{\sqrt{2}\eta (2\tau )}{\eta (\tau )}.\end{array}$

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,

$\begin{array}{rl}\mathfrak{𝔣}(q)& =q^{-\frac{1}{24}}\prod _{n>0}(1+q^{2n-1})=\frac{{\eta }^2(\tau )}{\eta \left(\frac{\tau }{2}\right)\eta (2\tau )},\\ {\mathfrak{𝔣}}_1(q)& =q^{-\frac{1}{24}}\prod _{n>0}(1-q^{2n-1})=\frac{\eta \left(\frac{\tau }{2}\right)}{\eta (\tau )},\\ {\mathfrak{𝔣}}_2(q)& =\sqrt{2}{q}^{\frac{1}{12}}\prod _{n>0}(1+q^{2n})=\frac{\sqrt{2}\eta (2\tau )}{\eta (\tau )}.\end{array}$

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

$\begin{array}{rl}{2}^{1/4}{G}_n& =q^{-\frac{1}{24}}\prod _{n>0}(1+q^{2n-1})=\frac{{\eta }^2(\tau )}{\eta \left(\frac{\tau }{2}\right)\eta (2\tau )},\\ 2^{1/4}{g}_n& =q^{-\frac{1}{24}}\prod _{n>0}(1-q^{2n-1})=\frac{\eta \left(\frac{\tau }{2}\right)}{\eta (\tau )}.\end{array}$

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

$\begin{array}{rl}{2}^{1/4}{G}_n& =\mathfrak{𝔣}(q)=\mathfrak{𝔣}(\tau ),\\ 2^{1/4}{g}_n& ={\mathfrak{𝔣}}_1(q)={\mathfrak{𝔣}}_1(\tau ).\end{array}$

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,

$(G_n^8-g_n^8)(G_n{g}_n{)}^8=\frac{1}{4},$

leads to

$[{\mathfrak{𝔣}}^8(q)-{\mathfrak{𝔣}}_1^8(q)][\mathfrak{𝔣}(q){\mathfrak{𝔣}}_1(q){]}^8=[\sqrt{2}{]}^8.$

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,

$\begin{array}{rl}{G}_5& ={\left(\frac{1+\sqrt{5}}{2}\right)}^{1/4},\\ G_{13}& ={\left(\frac{3+\sqrt{13}}{2}\right)}^{1/4},\\ G_{37}& ={(6+\sqrt{37})}^{1/4},\end{array}$

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

$\begin{array}{rl}{\vartheta }_{10}(0;\tau )& ={\theta }_2(q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{(n+1/2{)}^2}=\frac{2{\eta }^2(2\tau )}{\eta (\tau )},\\ {\vartheta }_{00}(0;\tau )& ={\theta }_3(q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{n^2}=\frac{{\eta }^5(\tau )}{{\eta }^2\left(\frac{\tau }{2}\right){\eta }^2(2\tau )}=\frac{{\eta }^2\left(\frac{\tau +1}{2}\right)}{\eta (\tau +1)},\\ {\vartheta }_{01}(0;\tau )& ={\theta }_4(q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^n{q}^{n^2}=\frac{{\eta }^2\left(\frac{\tau }{2}\right)}{\eta (\tau )}.\end{array}$

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 ${\mathfrak{𝔣}}_i(q)$

$\begin{array}{rl}\frac{{\theta }_2(q)}{\eta (\tau )}& ={\mathfrak{𝔣}}_2(q{)}^2,\\ \frac{{\theta }_4(q)}{\eta (\tau )}& ={\mathfrak{𝔣}}_1(q{)}^2,\\ \frac{{\theta }_3(q)}{\eta (\tau )}& =\mathfrak{𝔣}(q{)}^2,\end{array}$

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,

$j(\tau )=32\frac{({\theta }_2(q{)}^8+{\theta }_3(q{)}^8+{\theta }_4(q{)}^8{)}^3}{({\theta }_2(q){\theta }_3(q){\theta }_4(q){)}^8}$

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, Bull. Amer. Math. Soc. 42, https://www.math.ucla.edu/~wdduke/preprints/bams4.pdf 
• Weber, Heinrich Martin (1981) [1898] (in German), Lehrbuch der Algebra, 3 (3rd ed.), New York: AMS Chelsea Publishing, ISBN 978-0-8218-2971-4, https://archive.org/details/lehrbuchderalgeb03webeuoft 
• 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