# Weber modular function

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In [mathematics](/source/mathematics), the '''Weber modular functions''' are a family of three functions ''f'', ''f''<sub>1</sub>, and ''f''<sub>2</sub>,<ref group="note>''f'', ''f''<sub>1</sub> and ''f''<sub>2</sub> are not [modular functions](/source/Modular_form) (per the Wikipedia definition), but every modular function is a [rational function](/source/rational_function) in ''f'', ''f''<sub>1</sub> and ''f''<sub>2</sub>. Some authors use a non-equivalent definition of "modular functions".</ref> studied by [Heinrich Martin Weber](/source/Heinrich_Martin_Weber).

==Definition==

Let <math>q = e^{2\pi i \tau}</math> where ''τ'' is an element of the [upper half-plane](/source/upper_half-plane). Then the Weber functions are

:<math>\begin{align}
\mathfrak{f}(\tau) &= q^{-\frac{1}{48}}\prod_{n>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>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>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"''.<ref group="note>https://www.math.ucla.edu/~wdduke/preprints/bams4.pdf ''Continued Fractions and Modular Functions'', W. Duke, pp 22-23</ref> The function <math>\eta(\tau)</math> is the [Dedekind eta function](/source/Dedekind_eta_function) and <math>(e^{2\pi i\tau})^{\alpha}</math> should be interpreted as <math>e^{2\pi i\tau\alpha}</math>. The descriptions as <math>\eta</math> quotients immediately imply

:<math>\mathfrak{f}(\tau)\mathfrak{f}_1(\tau)\mathfrak{f}_2(\tau) =\sqrt{2}.</math>

The transformation ''τ''&nbsp;→&nbsp;–1/''τ'' fixes ''f'' and exchanges ''f''<sub>1</sub> and ''f''<sub>2</sub>. So the 3-dimensional complex vector space with basis ''f'', ''f''<sub>1</sub> and ''f''<sub>2</sub> is acted on by the group SL<sub>2</sub>('''Z''').

==Alternative infinite product==

Alternatively, let <math>q = e^{\pi i \tau}</math> be the [nome](/source/Nome_(mathematics)), 

:<math>\begin{align}
\mathfrak{f}(q) &= q^{-\frac{1}{24}}\prod_{n>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>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>0}(1+q^{2n})= \frac{\sqrt2\,\eta(2\tau)}{\eta(\tau)}.
\end{align}</math>

The form of the [infinite product](/source/infinite_product) has slightly changed. But since the eta quotients remain the same, then <math>\mathfrak{f}_i(\tau) = \mathfrak{f}_i(q)</math> as long as the second uses the nome <math>q = e^{\pi i \tau}</math>. The utility of the second form is to show connections and consistent notation with the [https://mathworld.wolfram.com/Ramanujang-andG-Functions.html Ramanujan G- and g-functions] and the [Jacobi theta functions](/source/Theta_function), both of which conventionally uses the nome.

==Relation to the Ramanujan G and g functions==

Still employing the nome <math>q = e^{\pi i \tau}</math>, define the [https://mathworld.wolfram.com/Ramanujang-andG-Functions.html Ramanujan G- and g-functions] as

:<math>\begin{align}
2^{1/4}G_n &= q^{-\frac{1}{24}}\prod_{n>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>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>\tau=\sqrt{-n}.</math> Then,

:<math>\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>G_n</math> and <math>g_n</math> which implies similar relations between <math>\mathfrak{f}(q)</math> and <math>\mathfrak{f}_1(q)</math>. For example, his identity,

:<math>(G_n^8-g_n^8)(G_n\,g_n)^8 = \tfrac14,</math>

leads to

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

:<math>\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>\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](/source/Theta_function) is traditionally the nome <math>q = e^{\pi i \tau},</math> 

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

:<math>\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](/source/Jacobi_identity) with even subscripts on the LHS,

:<math>\theta_2(q)^4+\theta_4(q)^4 = \theta_3(q)^4;</math>

therefore,

:<math>\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](/source/cubic_equation)

:<math>j(\tau)=\frac{(x-16)^3}{x}</math>

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

:<math>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>\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>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>\mathfrak{f}_i(\tau) = \mathfrak{f}_i(q)</math> and have the same formulas in terms of the Dedekind eta function <math>\eta(\tau)</math>.

==See also== 

*[Ramanujan–Sato series, level 4](/source/Ramanujan%E2%80%93Sato_series)

==References==

*{{Citation | last1=Duke | first1=William | author1-link=William Duke (mathematician) | title=Continued Fractions and Modular Functions | year=2005 | url=https://www.math.ucla.edu/~wdduke/preprints/bams4.pdf | publisher=Bull. Amer. Math. Soc. 42}}
*{{Citation | last1=Weber | first1=Heinrich Martin | author1-link=Heinrich Martin Weber | title=Lehrbuch der Algebra | orig-date=1898 | url=https://archive.org/details/lehrbuchderalgeb03webeuoft | publisher=AMS Chelsea Publishing | location=New York | language=German | edition=3rd | isbn=978-0-8218-2971-4 | year=1981 | volume=3}}
*{{citation|mr=1415803
|last1=Yui|first1= Noriko |last2= Zagier|first2= Don
|title=On the singular values of Weber modular functions
|journal = Mathematics of Computation
|volume= 66
|issue=220|year=1997
|pages=1645–1662
|doi=10.1090/S0025-5718-97-00854-5
|doi-access=free
}}

===Notes===
{{reflist|group=note}}

Category:Modular forms

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Adapted from the Wikipedia article [Weber modular function](https://en.wikipedia.org/wiki/Weber_modular_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Weber_modular_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.