Weber's Modular Function

In mathematics, the Weber modular functions are a family of three functions ''f'', ''f''₁, and ''f''₂,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". studied by Heinrich Martin Weber.

Definition

Let $q = e^$ where ''τ'' is an element of the upper half-plane. Then the Weber functions are

:$\begin \mathfrak(\tau) &= q^\prod_(1+q^) = \frac = e^\frac,\\ \mathfrak_1(\tau) &= q^\prod_(1-q^) = \frac,\\ \mathfrak_2(\tau) &= \sqrt2\, q^\prod_(1+q^)= \frac. \end$

These are also the definitions in Duke's paper ''"Continued Fractions and Modular Functions"''.Dedekind eta function and $(e^)^$ should be interpreted as $e^$. The descriptions as $\eta$ quotients immediately imply

:$\mathfrak(\tau)\mathfrak_1(\tau)\mathfrak_2(\tau) =\sqrt.$

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 $q = e^$ be the nome,

:$\begin \mathfrak(q) &= q^\prod_(1+q^) =\frac,\\ \mathfrak_1(q) &= q^\prod_(1-q^) = \frac,\\ \mathfrak_2(q) &= \sqrt2\, q^\prod_(1+q^)= \frac. \end$

The form of the infinite product has slightly changed. But since the eta quotients remain the same, then $\mathfrak_i(\tau) = \mathfrak_i(q)$ as long as the second uses the nome $q = e^$. The utility of the second form is to show connections and consistent notation with th
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 $q = e^$, define th
Ramanujan G- and g-functions
as

:$\begin 2^G_n &= q^\prod_(1+q^) = \frac,\\ 2^g_n &= q^\prod_(1-q^) = \frac. \end$

The eta quotients make their connection to the first two Weber functions immediately apparent. In the nome, assume $\tau=\sqrt.$ Then,

:$\begin 2^G_n &= \mathfrak(q) = \mathfrak(\tau),\\ 2^g_n &= \mathfrak_1(q) = \mathfrak_1(\tau). \end$

Ramanujan found many relations between $G_n$ and $g_n$ which implies similar relations between $\mathfrak(q)$ and $\mathfrak_1(q)$. For example, his identity,

:$(G_n^8-g_n^8)(G_n\,g_n)^8 = \tfrac14,$

leads to

:\big mathfrak^8(q)-\mathfrak_1^8(q)\big\big mathfrak(q)\,\mathfrak_1(q)\big8 = \big sqrt2\big8.

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 $\mathfrak(q)$ and $\mathfrak_1(q)$. For example, using $\tau = \sqrt,\,\sqrt,\,\sqrt$, which are some of the square-free discriminants with class number 2,

:$\begin G_5 &= \left(\frac\right)^,\\ G_ &= \left(\frac\right)^,\\ G_ &= \left(6+\sqrt\right)^, \end$

and one can easily get $\mathfrak(\tau) = 2^G_n$ 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 $q = e^,$

:\begin
\vartheta_(0;\tau)&=\theta_2(q)=\sum_^\infty q^ = \frac,\\ pt\vartheta_(0;\tau)&=\theta_3(q)=\sum_^\infty q^ \;=\; \frac = \frac,\\ pt\vartheta_(0;\tau)&=\theta_4(q)=\sum_^\infty (-1)^n q^ = \frac.
\end

Dividing them by $\eta(\tau)$, and also noting that $\eta(\tau) = e^\frac\eta(\tau+1)$, then they are just squares of the Weber functions $\mathfrak_i(q)$

:\begin
\frac &= \mathfrak_2(q)^2,\\ pt\frac &= \mathfrak_1(q)^2,\\ pt\frac &= \mathfrak(q)^2,
\end

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

:$\theta_2(q)^4+\theta_4(q)^4 = \theta_3(q)^4;$

therefore,

:$\mathfrak_2(q)^8+\mathfrak_1(q)^8 = \mathfrak(q)^8.$

Relation to j-function

The three roots of the cubic equation

:$j(\tau)=\frac$

where ''j''(''τ'') is the j-function are given by $x_i = \mathfrak(\tau)^, -\mathfrak_1(\tau)^, -\mathfrak_2(\tau)^$. Also, since,

:$j(\tau)=32\frac$

and using the definitions of the Weber functions in terms of the Jacobi theta functions, plus the fact that $\mathfrak_2(q)^2\, \mathfrak_1(q)^2\,\mathfrak(q)^2 = \frac \frac \frac = 2$, then

:$j(\tau)=\left(\frac\right)^3 = \left(\frac\right)^3$

since $\mathfrak_i(\tau) = \mathfrak_i(q)$ and have the same formulas in terms of the Dedekind eta function $\eta(\tau)$.

See also

* Ramanujan–Sato series, level 4

References

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Notes

{{reflist, group=note

Modular forms