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

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.
210, 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 Unpublished manuscript.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.