Weber modular function

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

==Definition==

Let <math>q = e^{2\pi i \tau}</math> where ''τ'' is an element of the 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 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'' → –1/''τ'' fixes ''f'' and exchanges ''f''1 and ''f''2. So the 3-dimensional complex vector space with basis ''f'', ''f''1 and ''f''2 is acted on by the group SL2('''Z''').

==Alternative infinite product==

Alternatively, let <math>q = e^{\pi i \tau}</math> be the nome,

:<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 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, 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 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 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 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

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

where ''j''(''τ'') is the j-function 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

==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