j-Function

The $j$-function is defined as

$$j(q)\equiv 1728J(\sqrt{q}\,),$$ (1)

where

$$J(q)\equiv {4\over 27} {[1-\lambda(q)+\lambda^2(q)]^3\over \lambda^2(q)[1-\lambda(q)]^2}$$ (2)

is Klein's Absolute Invariant, $\lambda(q)$ the Elliptic Lambda Function

$$\lambda(q)\equiv k^2(q) =\left[{\vartheta_2(q)\over \vartheta_3(q)}\right]^4,$$ (3)

and $\vartheta_i$ a Theta Function. This function can also be specified in terms of the Weber Functions $f$, $f_1$, $f_2$, $\gamma_2$, and $\gamma_3$ as

$$j(z) = {[f^{24}(z)-16]^3\over f^{24}(z)}$$ (4)

$$= {[{f_1}^{24}(z)+16]^3\over {f_1}^{24}(z)}$$ (5)

$$= {[{f_2}^{24}(z)+16]^3\over {f_2}^{24}(z)}$$ (6)

$$= {\gamma_2}^3(z)$$ (7)

$$= {\gamma_3}^2(z)+1728$$ (8)

(Weber 1902, p. 179; Atkin and Morain 1993).

The $j$-function is a Meromorphic function on the upper half of the Complex Plane which is invariant with respect to the Special Linear Group ${\it SL}(2,Z)$. It has a Fourier Series

$$j(q) = \sum_{n=-\infty}^\infty a_n q^n,$$ (9)

for the Nome

$$q \equiv e^{2\pi it}$$ (10)

with $\Im[t]>0$. The coefficients in the expansion of the $j$-function satisfy:

1. $a_n = 0$ for $n<-1$ and $a_{-1}=1$,

2. all $a_n$s are Integers with fairly limited growth with respect to $n$, and

3. $j(q)$ is an Algebraic Number, sometimes a Rational Number, and sometimes even an Integer at certain very special values of $q$ (or $t$).

The latter result is the end result of the massive and beautiful theory of Complex multiplication and the first step of Kronecker's so-called ``Jugendtraum.''

Then all of the Coefficients in the Laurent Series

$j(q)={1\over q}+744+196884q+21493760q^2$
$+864299970q^3+20245856256q^4+333202640600q^5+\ldots\quad$
	(11)

(Sloane's A000521) are Positive Integers (Rankin 1977). Let $d$ be a Positive Squarefree Integer, and define

$$t\equiv\cases{ i\sqrt{d} & for $d\equiv 1{\rm\ or\ }2\ \lef... ...t{d}\,) & for $d\equiv 3\ \left({{\rm mod\ } {4}}\right)$.\cr}$$ (12)

Then the Nome is

$$q\equiv e^{i\pi\tau} = \left\{\begin{array}{ll} e^{2\pi i(i\sqrt{d})} & \mbox{for $d\equ... ...)/2} & \mbox{for $d\equiv 3\ \left({{\rm mod\ } {4}}\right)$}\end{array}\right.$$

$$= \left\{\begin{array}{ll} e^{-2\pi\sqrt{d}} & \mbox{for $d\equiv 1... ...d}} & \mbox{for $d\equiv 3\ \left({{\rm mod\ } {4}}\right)$.}\end{array}\right.$$ (13)

It then turns out that $j(q)$ is an Algebraic Integer of degree $h(-d)$, where $h(-d)$ is the Class Number of the Discriminant $-d$ of the Quadratic Field $\Bbb{Q}(\sqrt{n}\,)$ (Silverman 1986). The first term in the Laurent Series is then $q^{-1}=e^{-2\pi\sqrt{n}}$ or $-e^{-\pi\sqrt{n}}$, and all the later terms are Powers of $q^{-1}$, which are small numbers. The larger $n$, the faster the series converges. If $h(-d)=1$, then $j(q)$ is a Algebraic Integer of degree 1, i.e., just a plain Integer. Furthermore, the Integer is a perfect Cube.

The numbers whose Laurent Series give Integers are those with Class Number 1. But these are precisely the Heegner Numbers $-1$, $-2$, $-3$, $-7$, $-11$, $-19$, $-43$, $-67$, $-163$. The greater (in Absolute Value) the Heegner Number $d$, the closer to an Integer is the expression $e^{\pi\sqrt{-n}}$, since the initial term in $j(q)$ is the largest and subsequent terms are the smallest. The best approximations with $h(-d)=1$ are therefore

$$e^{\pi\sqrt{43}} \approx 960^3+744-2.2\times 10^{-4}$$ (14)

$$e^{\pi\sqrt{67}} \approx 5280^3+744-1.3\times 10^{-6}$$ (15)

$$e^{\pi\sqrt{163}} \approx 640320^3+744-7.5\times 10^{-13}.$$ (16)

The exact values of $j(q)$ corresponding to the Heegner Numbers are

$$j(-e^{-\pi}) = 12^3$$ (17)

$$j(e^{-2\pi\sqrt{2}}\,) = 20^3$$ (18)

$$j(-e^{-\pi\sqrt{3}}\,) = 0^3$$ (19)

$$j(-e^{-\pi\sqrt{7}}\,) = -15^3$$ (20)

$$j(-e^{-\pi\sqrt{11}}\,) = -32^3$$ (21)

$$j(-e^{-\pi\sqrt{19}}\,) = -96^3$$ (22)

$$j(-e^{-\pi\sqrt{43}}\,) = -960^3$$ (23)

$$j(-e^{-\pi\sqrt{67}}\,) = -5280^3$$ (24)

$$j(-e^{-\pi\sqrt{163}}\,) = -640320^3.$$ (25)

(The number 5280 is particularly interesting since it is also the number of feet in a mile. ) The Almost Integer generated by the last of these, $e^{\pi\sqrt{163}}$ (corresponding to the field $\Bbb{Q}(\sqrt{-163}\,)$ and the Imaginary quadratic field of maximal discriminant), is known as the Ramanujan Constant.

$e^{\pi\sqrt{22}}$, $e^{\pi\sqrt{37}}$, and $e^{\pi\sqrt{58}}$ are also Almost Integers. These correspond to binary quadratic forms with discriminants $-88$, $-148$, and $-232$, all of which have Class Number two and were noted by Ramanujan (Berndt 1994).

It turns out that the $j$-function also is important in the Classification Theorem for finite simple groups, and that the factors of the orders of the Sporadic Groups, including the celebrated Monster Group, are also related.

See also Almost Integer, Klein's Absolute Invariant, Weber Functions

References

Atkin, A. O. L. and Morain, F. ``Elliptic Curves and Primality Proving.'' Math. Comput. 61, 29-68, 1993.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 90-91, 1994.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 117-118, 1987.

Cohn, H. Introduction to the Construction of Class Fields. New York: Dover, p. 73, 1994.

Conway, J. H. and Guy, R. K. ``The Nine Magic Discriminants.'' In The Book of Numbers. New York: Springer-Verlag, pp. 224-226, 1996.

Morain, F. ``Implementation of the Atkin-Goldwasser-Kilian Primality Testing Algorithm.'' Rapport de Récherche 911, INRIA, Oct. 1988.

Rankin, R. A. Modular Forms. New York: Wiley, 1985.

Rankin, R. A. Modular Forms and Functions. Cambridge, England: Cambridge University Press, p. 199, 1977.

Serre, J. P. Cours d'arithmétique. Paris: Presses Universitaires de France, 1970.

Silverman, J. H. The Arithmetic of Elliptic Curves. New York: Springer-Verlag, p. 339, 1986.

Sloane, N. J. A. Sequence A000521/M5477 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Weber, H. Lehrbuch der Algebra, Vols. I-II. New York: Chelsea, 1979.

Weisstein, E. W. ``$j$-Function.'' Mathematica notebook jFunction.m.