Weber modular function

In mathematics, the Weber modular functions are a family of three modular 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.

$\begin{align} \mathfrak{f}(\tau) &= q^{-\frac{1}{48}}\prod_{n>0}(1+q^{n-\frac{1}{2}}) = e^{-\frac{\pi\rm{i}}{24}}\frac{\eta\big(\frac{\tau+1}{2}\big)}{\eta(\tau)}=\frac{\eta^2(\tau)}{\eta\big(\tfrac{\tau}{2}\big)\eta(2\tau)}\\ \mathfrak{f}_1(\tau) &= q^{-\frac{1}{48}}\prod_{n>0}(1-q^{n-\frac{1}{2}}) = \frac{\eta\big(\tfrac{\tau}{2}\big)}{\eta(\tau)}\\ \mathfrak{f}_2(\tau) &= \sqrt2\, q^{-\frac{1}{24}}\prod_{n>0}(1+q^{n})= \frac{\sqrt2\,\eta(2\tau)}{\eta(\tau)} \end{align}$

where $\eta (\tau )$ is the Dedekind eta function. Note the $\eta$ quotients immediately imply that

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

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).

Relation to theta functions

Let the argument of the Jacobi theta function be the nome $q = e^{\pi i \tau}$. Then,

$\begin{align} \mathfrak{f}(\tau) &= \sqrt{\frac{\theta_3(0,q)}{\eta(\tau)}} \\ \mathfrak{f}_1(\tau) &= \sqrt{\frac{\theta_4(0,q)}{\eta(\tau)}} \\ \mathfrak{f}_2(\tau) &= \sqrt{\frac{\theta_2(0,q)}{\eta(\tau)}} \\ \end{align}$

Thus,

$\mathfrak{f}_1(\tau)^8+\mathfrak{f}_2(\tau)^8 = \mathfrak{f}(\tau)^8$

which is simply a consequence of the well known identity,

$\theta_2(0,q)^4+\theta_4(0,q)^4 = \theta_3(0,q)^4$

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,

$j(\tau)=32\frac{\Big(\theta_2(0,q)^8+\theta_3(0,q)^8+\theta_4(0,q)^8\Big)^3}{\Big(\theta_2(0,q)\theta_3(0,q)\theta_4(0,q)\Big)^8}$

then,

$j(\tau)=\left(\frac{\mathfrak{f}(\tau)^{16}+\mathfrak{f}_1(\tau)^{16}+\mathfrak{f}_2(\tau)^{16}}{2}\right)^3$

See also

• Ramanujan–Sato series, level 4

References

• Weber, Heinrich Martin (1981) [1898], Lehrbuch der Algebra (in German), 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: 1645–1662, doi:10.1090/S0025-5718-97-00854-5, MR 1415803