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Date: 12-10-2018
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Date: 18-8-2018
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Date: 22-5-2019
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Let and
be periods of a doubly periodic function, with
the half-period ratio a number with
. Then Klein's absolute invariant (also called Klein's modular function) is defined as
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(1) |
where and
are the invariants of the Weierstrass elliptic function with modular discriminant
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(2) |
(Klein 1877). If , where
is the upper half-plane, then
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(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
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(4) |
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(5) |
(Klein 1878-1879, Cohn 1994), where is the elliptic lambda function
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(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
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(7) |
and is a modular function.
takes on the special values
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(8) |
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(9) |
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(10) |
satisfies the functional equations
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(11) |
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(12) |
It satisfies a number of beautiful multiple-argument identities, including the duplication formula
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(13) |
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(14) |
with
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(15) |
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(16) |
and the Dedekind eta function, the triplication formula
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(17) |
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(18) |
with
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(19) |
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(20) |
and the quintuplication formula
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(21) |
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(22) |
with
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(23) |
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(24) |
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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