HOME

TheInfoList

Click Here for Items Related To - Weber Modular Function

OR:

Weber Modular Function on: [Wikipedia] [Google] [Amazon]

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 mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...

in ''f'', ''f''₁ and ''f''₂. Some authors use a non-equivalent definition of "modular functions". studied by

Heinrich Martin Weber Heinrich Martin Weber (5 March 1842, Heidelberg, Germany – 17 May 1913, Straßburg, Alsace-Lorraine, German Empire, now Strasbourg, France) was a German mathematician. Weber's main work was in algebra, number theory, and analysis. He is ...

.

Definition

Let

q = e^

where ''τ'' is an element of the

upper half-plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...

. 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 In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string ...

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)\big 8 = \big sqrt2\big 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

\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^,

:

\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)

:

\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 In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of t ...

:

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

* * *

Notes

{{reflist, group=note Modular forms