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

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

where ${\displaystyle \eta (\tau )}$ is the Dedekind eta function. Note the descriptions as ${\displaystyle \eta }$ quotients immediately imply

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

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

${\displaystyle {\begin{aligned}{\mathfrak {f}}(\tau )&={\sqrt {\frac {\theta _{3}(0,q)}{\eta (\tau )}}}\\{\mathfrak {f}}_{1}(\tau )&={\sqrt {\frac {\theta _{4}(0,q)}{\eta (\tau )}}}\\{\mathfrak {f}}_{2}(\tau )&={\sqrt {\frac {\theta _{2}(0,q)}{\eta (\tau )}}}\\\end{aligned}}}$

Using the well-known identity,

${\displaystyle \theta _{2}(0,q)^{4}+\theta _{4}(0,q)^{4}=\theta _{3}(0,q)^{4}}$

thus,

${\displaystyle {\mathfrak {f}}_{1}(\tau )^{8}+{\mathfrak {f}}_{2}(\tau )^{8}={\mathfrak {f}}(\tau )^{8}}$

Relation to j-function

The three roots of the cubic equation,

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

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

${\displaystyle j(\tau )=32{\frac {{\Big (}\theta _{2}(0,q)^{8}+\theta _{3}(0,q)^{8}+\theta _{4}(0,q)^{8}{\Big )}^{3}}{{\Big (}\theta _{2}(0,q)\theta _{3}(0,q)\theta _{4}(0,q){\Big )}^{8}}}}$

then,

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

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 (220): 1645–1662, doi:10.1090/S0025-5718-97-00854-5, MR 1415803