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

[math]\displaystyle{ \begin{align} \mathfrak{f}(\tau) &= q^{-\frac{1}{48}}\prod_{n\gt 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\gt 0}(1-q^{n-1/2}) = \frac{\eta\big(\tfrac{\tau}{2}\big)}{\eta(\tau)},\\ \mathfrak{f}_2(\tau) &= \sqrt2\, q^{\frac{1}{24}}\prod_{n\gt 0}(1+q^{n})= \frac{\sqrt2\,\eta(2\tau)}{\eta(\tau)}. \end{align} }[/math]

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

[math]\displaystyle{ \mathfrak{f}(\tau)\mathfrak{f}_1(\tau)\mathfrak{f}_2(\tau) =\sqrt{2}. }[/math]

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 [math]\displaystyle{ q = e^{\pi i \tau} }[/math] be the nome,

[math]\displaystyle{ \begin{align} \mathfrak{f}(q) &= q^{-\frac{1}{24}}\prod_{n\gt 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\gt 0}(1-q^{2n-1}) = \frac{\eta\big(\tfrac{\tau}{2}\big)}{\eta(\tau)},\\ \mathfrak{f}_2(q) &= \sqrt2\, q^{\frac{1}{12}}\prod_{n\gt 0}(1+q^{2n})= \frac{\sqrt2\,\eta(2\tau)}{\eta(\tau)}. \end{align} }[/math]

The form of the infinite product has slightly changed. But since the eta quotients remain the same, then [math]\displaystyle{ \mathfrak{f}_i(\tau) = \mathfrak{f}_i(q) }[/math] as long as the second uses the nome [math]\displaystyle{ q = e^{\pi i \tau} }[/math]. 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 [math]\displaystyle{ q = e^{\pi i \tau} }[/math], define the Ramanujan G- and g-functions as

[math]\displaystyle{ \begin{align} 2^{1/4}G_n &= q^{-\frac{1}{24}}\prod_{n\gt 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\gt 0}(1-q^{2n-1}) = \frac{\eta\big(\tfrac{\tau}{2}\big)}{\eta(\tau)}. \end{align} }[/math]

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

[math]\displaystyle{ \begin{align} 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{align} }[/math]

Ramanujan found many relations between [math]\displaystyle{ G_n }[/math] and [math]\displaystyle{ g_n }[/math] which implies similar relations between [math]\displaystyle{ \mathfrak{f}(q) }[/math] and [math]\displaystyle{ \mathfrak{f}_1(q) }[/math]. For example, his identity,

[math]\displaystyle{ (G_n^8-g_n^8)(G_n\,g_n)^8 = \tfrac14, }[/math]

leads to

[math]\displaystyle{ \big[\mathfrak{f}^8(q)-\mathfrak{f}_1^8(q)\big] \big[\mathfrak{f}(q)\,\mathfrak{f}_1(q)\big]^8 = \big[\sqrt2\big]^8. }[/math]

For many values of n, Ramanujan also tabulated [math]\displaystyle{ G_n }[/math] for odd n, and [math]\displaystyle{ g_n }[/math] for even n. This automatically gives many explicit evaluations of [math]\displaystyle{ \mathfrak{f}(q) }[/math] and [math]\displaystyle{ \mathfrak{f}_1(q) }[/math]. For example, using [math]\displaystyle{ \tau = \sqrt{-5},\,\sqrt{-13},\,\sqrt{-37} }[/math], which are some of the square-free discriminants with class number 2,

[math]\displaystyle{ \begin{align} 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{align} }[/math]

and one can easily get [math]\displaystyle{ \mathfrak{f}(\tau) = 2^{1/4}G_n }[/math] 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 [math]\displaystyle{ q = e^{\pi i \tau}, }[/math]

[math]\displaystyle{ \begin{align} \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{align} }[/math]

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

[math]\displaystyle{ \begin{align} \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{align} }[/math]

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

[math]\displaystyle{ \theta_2(q)^4+\theta_4(q)^4 = \theta_3(q)^4; }[/math]

therefore,

[math]\displaystyle{ \mathfrak{f}_2(q)^8+\mathfrak{f}_1(q)^8 = \mathfrak{f}(q)^8. }[/math]

Relation to j-function

The three roots of the cubic equation

[math]\displaystyle{ j(\tau)=\frac{(x-16)^3}{x} }[/math]

where j(τ) is the j-function are given by [math]\displaystyle{ x_i = \mathfrak{f}(\tau)^{24}, -\mathfrak{f}_1(\tau)^{24}, -\mathfrak{f}_2(\tau)^{24} }[/math]. Also, since,

[math]\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} }[/math]

and using the definitions of the Weber functions in terms of the Jacobi theta functions, plus the fact that [math]\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 }[/math], then

[math]\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 }[/math]

since [math]\displaystyle{ \mathfrak{f}_i(\tau) = \mathfrak{f}_i(q) }[/math] and have the same formulas in terms of the Dedekind eta function [math]\displaystyle{ \eta(\tau) }[/math].

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