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{aligned}{\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 )&={\sqrt {2}}\,q^{\frac {1}{24}}\prod _{n>0}(1+q^{n})={\frac {{\sqrt {2}}\,\eta (2\tau )}{\eta (\tau )}}\end{aligned}}

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

{\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

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, MR 1415803, doi:10.1090/S0025-5718-97-00854-5

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Weber modular function

Definition

Relation to theta functions

Relation to j-function

See also

References