Weber modular function explained

In mathematics, the Weber modular functions are a family of three functions f, f₁, and f₂,^[1] studied by Heinrich Martin Weber.

Definition

Let

q = e ^{2\pi} where τ is an element of the upper half-plane. Then the Weber functions are\begin{align} ak{f}(\tau) & = q ^{- (1)/(48)} \prod _{n>0} ( 1+ q ^{n-1/2} ) = (η ^{2(\tau)})/(η ( \tfrac{\tau) { 2 } ) η ( 2 \tau)} = e ^{- (\pi i)/(24)} (η ( (\tau +1)/() 2))/(η (\tau)) ,\\ ak{f} _{1(\tau)} & = q ^{- (1)/(48)} \prod _{n>0} ( 1- q ^{n-1/2} ) = (η ( \tfrac{\tau)/(2) ) }{ η (\tau)},\\ ak{f} _{2(\tau)} & = \sqrt 2 q ^{(1)/(24)} \prod _{n>0} ( 1+ q ^{n} ) = (\sqrt 2η ( 2 \tau))/(η (\tau)) . \end{align}

These are also the definitions in Duke's paper "Continued Fractions and Modular Functions".^[2] The function

η (\tau) is the Dedekind eta function and (e ^{2\pi} ) ^{\alpha} should be interpreted as e ^{2\pi} . The descriptions as η quotients immediately implyak{f}(\tau) ak{f} _{1(\tau)ak{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).

Alternative infinite product

Alternatively, let

q = e ^{\pi} be the nome, \begin{align} ak{f}(q) & = q ^{- (1)/(24)} \prod _{n>0} ( 1+ q ^{2n-1} ) = (η ^{2(\tau)})/(η ( \tfrac{\tau) { 2 } ) η ( 2 \tau)},\\ ak{f} _{1(q)} & = q ^{- (1)/(24)} \prod _{n>0} ( 1- q ^{2n-1} ) = (η ( \tfrac{\tau)/(2) ) }{ η (\tau)},\\ ak{f} _{2(q)} & = \sqrt 2 q ^{(1)/(12)} \prod _{n>0} ( 1+ q ^{2n} ) = (\sqrt 2η ( 2 \tau))/(η (\tau)) . \end{align}

The form of the infinite product has slightly changed. But since the eta quotients remain the same, then

ak{f} _{i(\tau)} = ak{f} _{i(q)} as long as the second uses the nome q = e ^{\pi} . The utility of the second form is to show connections and consistent notation with the 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

q = e ^{\pi} , define the Ramanujan G- and g-functions as\begin{align} 2 ^{1/4} G _{n} & = q ^{- (1)/(24)} \prod _{n>0} ( 1+ q ^{2n-1} ) = (η ^{2(\tau)})/(η ( \tfrac{\tau) { 2 } ) η ( 2 \tau)},\\ 2 ^{1/4} g _{n} & = q ^{- (1)/(24)} \prod _{n>0} ( 1- q ^{2n-1} ) = (η ( \tfrac{\tau)/(2) ) }{ η (\tau)}. \end{align}

The eta quotients make their connection to the first two Weber functions immediately apparent. In the nome, assume

\tau = \sqrt{ - n}. Then,\begin{align} 2 ^{1/4} G _{n} & = ak{f}(q) = ak{f}(\tau),\\ 2 ^{1/4} g _{n} & = ak{f} _{1(q)} = ak{f} _{1(\tau). \end{align}}

Ramanujan found many relations between

G _{n} and g _{n} which implies similar relations between ak{f}(q) and ak{f} _{1(q)} . For example, his identity,(G ^{8 )(G} _{ng} ^{8} ^{n)} = \tfrac 14 ,

leads to

[ ak{f} ^{8 (q) ]} _{1} [ ak{f}(q) ak{f} ^{8} _{1 (q) ]} = [ \sqrt 2 ] ^{8.}

For many values of n, Ramanujan also tabulated

G _{n} for odd n, and g _{n} for even n. This automatically gives many explicit evaluations of ak{f}(q) and ak{f} _{1(q)} . For example, using \tau = \sqrt{ -5 },\sqrt{ -13 },\sqrt{ -37 } , which are some of the square-free discriminants with class number 2,\begin{align} G _{5} & = \left( (1+ \sqrt{ 5)/() }\right)^,\\G_ &= \left(\frac\right)^,\\G_ &= \left(6+\sqrt\right)^,\end

and one can easily get

ak{f}(\tau) = 2 ^{1/4} G _{n} 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

q = e ^{\pi} , \begin{align} \vartheta _{10} ( 0 ;\tau)& = \theta _{2(q)=\sum} ^{inf ty} ^{n =-inf ty} q ^{(n +1 / 2 ) ^{2}} = (2η ^{2(2\tau)})/(η (\tau)) ,\\[ 2 pt] \vartheta _{00} ( 0 ;\tau)& = \theta _{3(q)=\sum} ^{inf ty} ^{n =-inf ty} q ^{n ^{2}} = (η ^{5(\tau)})/(η ^{2 \left( (\tau)/(2)} \right) η ^{2(2\tau)}) = (η ^{2 \left( (\tau +1)/(2)} \right))/(η (\tau +1 )) ,\\[ 3 pt] \vartheta _{01} ( 0 ;\tau)& = \theta _{4(q)=\sum} ^{inf ty} ^{n =-inf ty} ( -1 ) ^{n} q ^{n ^{2}} = (η ^{2 \left( (\tau)/(2)} \right))/(η (\tau)) . \end{align}

Dividing them by

η (\tau) , and also noting that η (\tau) = e ^{(- \pi i)/(12)} η (\tau +1 ) , then they are just squares of the Weber functions ak{f} _{i(q)} \begin{align} (\theta _{2(q)})/(η (\tau)) & = ak{f} ^{2 ,\\[ 4 pt] (\theta _{4(q)})/(η (\tau))} _{2 (q)} & = ak{f} ^{2 ,\\[ 4 pt] (\theta _{3(q)})/(η (\tau))} _{1 (q)} & = ak{f}(q) ^{2, \end{align}}

with even-subscript theta functions purposely listed first. Using the well-known Jacobi identity with even subscripts on the LHS,

\theta ^{4} _{4 (q)} = \theta ^{4 ;} _{3 (q)}

therefore,

ak{f} ^{8} _{1 (q)} = ak{f}(q) ^{8.}

Relation to j-function

The three roots of the cubic equation

j(\tau) = ((x -16 ) ^{3})/(x)

where j(τ) is the j-function are given by

x _{i} = ak{f}(\tau) ^{24} , - ak{f} ^{24} _{1 (\tau)} , - ak{f} ^{24} _{2 (\tau)} . Also, since,j(\tau) =32 (( \theta ^{8 )} _{4 (q)} ^{3})/(( \theta _{2 (q)\theta} _{3(q)\theta} ^{8} ^{4 (q) )})

and using the definitions of the Weber functions in terms of the Jacobi theta functions, plus the fact that

ak{f} ^{2} _{2 (q)} ak{f} ^{2 ak{f}(q)} _{1 (q)} ^{2} = (\theta _{2(q)})/(η (\tau)) (\theta _{4(q)})/(η (\tau)) (\theta _{3(q)})/(η (\tau)) = 2 , thenj(\tau) = \left( (ak{f)/((\tau)) ^{16} + ak{f} ^{16} _{1 (\tau)} + ak{f} ^{16} _{2 (\tau)} }\right)^3 = \left(\frac\right)^3

since

ak{f} _{i(\tau)} = ak{f} _{i(q)} and have the same formulas in terms of the Dedekind eta function η (\tau) .

See also

• Ramanujan–Sato series, level 4

References

Notes

Notes and References

1. f, f₁ and f₂ are not modular functions (per the Wikipedia definition), but every modular function is a rational function in f, f₁ and f₂. Some authors use a non-equivalent definition of "modular functions".
2. https://www.math.ucla.edu/~wdduke/preprints/bams4.pdf Continued Fractions and Modular Functions, W. Duke, pp 22-23