Klein,s Absolute Invariant
المؤلف:
Cohn, H
المصدر:
Introduction to the Construction of Class Fields. New York: Dover
الجزء والصفحة:
...
23-4-2019
2839
Klein's Absolute Invariant
Let 
and 
be periods of a doubly periodic function, with 
the half-period ratio a number with ![I[tau]!=0](http://mathworld.wolfram.com/images/equations/KleinsAbsoluteInvariant/Inline4.gif)
. Then Klein's absolute invariant (also called Klein's modular function) is defined as
 |
(1)
|
where 
and 
are the invariants of the Weierstrass elliptic function with modular discriminant
 |
(2)
|
(Klein 1877). If 
, where 
is the upper half-plane, then
 |
(3)
|
is a function of the ratio 
only, as are 
, 
, and 
. Furthermore, 
, 
, 
, and 
are analytic in 
(Apostol 1997, p. 15).
Klein's absolute invariant is implemented in the Wolfram Language as KleinInvariantJ[tau].
The function 
is the same as the j-function, modulo a constant multiplicative factor.
Every rational function of 
is a modular function, and every modular function can be expressed as a rational functionof 
(Apostol 1997, p. 40).
Klein's invariant can be given explicitly by
(Klein 1878-1879, Cohn 1994), where 
is the elliptic lambda function
![lambda(tau)=[(theta_2(0,q))/(theta_3(0,q))]^4,](http://mathworld.wolfram.com/images/equations/KleinsAbsoluteInvariant/NumberedEquation4.gif) |
(6)
|
is a Jacobi theta function, the 
are Eisenstein series, and 
is the nome. Klein's invariant can also be simply expressed in terms of the five Weber functions 
, 
, 
, 
, and 
.
is invariant under a unimodular transformation, so
 |
(7)
|
and 
is a modular function. 
takes on the special values
satisfies the functional equations
It satisfies a number of beautiful multiple-argument identities, including the duplication formula
with
and 
the Dedekind eta function, the triplication formula
with
and the quintuplication formula
with
Plotting the real or imaginary part of 
in the complex plane produces a beautiful fractal-like structure, illustrated above.
REFERENCES:
Apostol, T. M. "Klein's Modular Function 
," "Invariance of 
Under Unimodular Transformation," "The Fourier Expansions of 
and 
," "Special Values of 
," and "Modular Functions as Rational Functions of 
." §1.12-1.13, 1.15, and 2.5-2.6 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 15-18, 20-22, and 39-40, 1997.
Brezhnev, Y. V. "Uniformisation: On the Burnside Curve 
." 9 Dec 2001. http://arxiv.org/abs/math.CA/0111150.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 115 and 179, 1987.
Cohn, H. Introduction to the Construction of Class Fields. New York: Dover, p. 73, 1994.
Klein, F. "Sull' equazioni dell' Icosaedro nella risoluzione delle equazioni del quinto grado [per funzioni ellittiche]." Reale Istituto Lombardo, Rendiconto, Ser. 2 10, 1877.
Klein, F. "Über die Transformation der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades." Math. Ann.14, 111-172, 1878-1879.
Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. Unpublished manuscript. 1999.
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