# j-invariant

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Klein's j-invariant in the complex plane

In mathematics, Klein's j-invariant, regarded as a function of a complex variable τ, is a modular function of weight zero for $SL(2,\mathbb{Z})$ defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that $j\left(e^{\frac{2}{3}\pi i}\right) = 0$ and $j(i) = 1728$. Rational functions of $j$ are modular, and in fact give all modular functions. Classically, the j-invariant was studied as a parameterization of elliptic curves over $\mathbb{C}$, but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).

## Definition

Real part of the j-invariant as a function of the nome q on the unit disk
Phase of the j-invariant as a function of the nome q on the unit disk

While the j-invariant can be defined purely in terms of certain infinite sums (see $g_2, g_3$ below), these can be motivated by considering isomorphism classes of elliptic curves. Every elliptic curve $E$ over $\mathbb{C}$ is a complex torus, and thus can be identified with a rank 2 lattice; i.e., two-dimensional lattice of $\mathbb{C}$. This is done by identifying opposite edges of each parallelogram in the lattice. It turns out that multiplying the lattice by complex numbers, which corresponds to rotating and scaling the lattice, preserves the isomorphism class of the elliptic curve, and thus we can consider the lattice generated by 1 and some $\tau \in \mathbb{H}$ (where $\mathbb{H}$ is the Upper half-plane). Conversely, if we define

$g_2 = 60\sum_{(m,n) \neq (0,0)} (m + n\tau)^{-4},\qquad g_3 = 140\sum_{(m,n) \neq (0,0)} (m + n\tau)^{-6}$

then this lattice corresponds to the elliptic curve over $\mathbb{C}$ defined by $y^2 = 4x^3 - g_2X - g_3$ via the Weierstrass elliptic functions. Then the j-invariant is defined as

$j(\tau) = 1728{g_2^3 \over \Delta}$

where the modular discriminant $\Delta$ is

$\Delta = g_2^3 - 27g_3^2$

It can be shown that $\Delta$ is a modular form of weight twelve, and $g_2$ one of weight four, so that its third power is also of weight twelve. Thus their quotient, and therefore $j$, is a modular function of weight zero, in particular a meromorphic function $\mathbb{H} \rightarrow \mathbb{C}$ invariant under the action of $\operatorname{SL}(2,\mathbb{Z})$. As explained below, $j$ is surjective, which means that it gives a bijection between isomorphism classes of elliptic curves over $\mathbb{C}$ and the complex numbers.

## The fundamental region

The fundamental domain of the modular group acting on the upper half plane.

The two transformations τ → τ + 1 and τ → τ−1 together generate a group called the modular group, which we may identify with the projective special linear group $PSL_2(\mathbb{Z})$. By a suitable choice of transformation belonging to this group, τ → (aτ + b)/(cτ + d), with ad − bc = 1, we may reduce τ to a value giving the same value for j, and lying in the fundamental region for j, which consists of values for τ satisfying the conditions

\begin{align} |\tau| &\ge 1 \\ -\frac{1}{2} &< \mathfrak{R}(\tau) \le \frac{1}{2} \\ -\frac{1}{2} &< \mathfrak{R}(\tau) < 0 \Rightarrow |\tau| > 1 \end{align}

The function j(τ) when restricted to this region still takes on every value in the complex numbers $\mathbb{C}$ exactly once. In other words, for every $c\in\mathbb{C}$, there is a unique τ in the fundamental region such that c=j(τ). Thus, j has the property of mapping the fundamental region to the entire complex plane.

As a Riemann surface, the fundamental region has genus 0, and every (level one) modular function is a rational function in j; and, conversely, every rational function in j is a modular function. In other words the field of modular functions is $\mathbb{C}(j)$.

## Class field theory and j

The j-invariant has many remarkable properties. One of these is that if τ is any of the singular moduli, that is, any element of an imaginary quadratic field with positive imaginary part (so that j is defined) then $j(\tau)$ is an algebraic integer.[1] The field extension

$\mathbb{Q}[j(\tau), \tau]/\mathbb{Q}(\tau)$

is abelian, meaning with abelian Galois group. We have a lattice in the complex plane defined by 1 and τ, and it is easy to see that all of the elements of the field $\mathbb{Q}(\tau)$ which send lattice points to other lattice points under multiplication form a ring with units, called an order. The other lattices with generators 1 and τ' associated in like manner to the same order define the algebraic conjugates $j(\tau')$ of $j(\tau)$ over $\mathbb{Q}(\tau)$. The unique maximal order under inclusion of $\mathbb{Q}(\tau)$ is the ring of algebraic integers of $\mathbb{Q}(\tau)$, and values of τ having it as its associated order lead to unramified extensions of $\mathbb{Q}(\tau)$. These classical results are the starting point for the theory of complex multiplication.

## Transcendence properties

In 1937 Theodor Schneider proved the aforementioned result that if $\tau$ is a quadratic irrational number in the upper half plane then j($\tau$) is an algebraic integer. In addition he proved that if $\tau$ is an algebraic number but not imaginary quadratic then j($\tau$) is transcendental.

The j function has numerous other transcendental properties. Kurt Mahler conjectured a particular transcendence result that is often referred to as Mahler's conjecture, though it was proved as a corollary of results by Yu. V. Nesternko and Patrice Phillipon in the 1990s. Mahler's conjecture was that the if $\tau$ was in the upper half plane then exp(2πi$\tau$) and j($\tau$) were never both simultaneously algebraic. Stronger results are now known, for example if exp(2πi$\tau$) is algebraic then the following three numbers are algebraically independent, and thus transcendental:

$j(\tau), \frac{j^\prime(\tau)}{\pi}, \frac{j^{\prime\prime}(\tau)}{\pi^2}$

## The q-expansion and moonshine

Several remarkable properties of j have to do with its q-expansion (Fourier series expansion, written as a Laurent series in terms of q= exp(2πiτ)), which begins:

$j(\tau) = {1 \over q} + 744 + 196884 q + 21493760 q^2 + 864299970 q^3 + 20245856256 q^4 + \cdots$

Note that j has a simple pole at the cusp, so its q-expansion has no terms below q−1.

All the Fourier coefficients are integers, which results in several almost integers, notably Ramanujan's constant: $e^{\pi \sqrt{163}} \approx 640320^3 + 744$.

### Moonshine

More remarkably, the Fourier coefficients for the positive exponents of q are the dimensions of the graded part of an infinite-dimensional graded algebra representation of the monster group called the moonshine module – specifically, the coefficient of $q^n$ is the dimension of grade-n part of the moonshine module, the first example being the Griess algebra, which has dimension 196,884, corresponding to the term $196884 q^1.$ This startling observation was the starting point for moonshine theory.

The study of the Moonshine conjecture led J.H. Conway and Simon P. Norton to look at the genus-zero modular functions. If they are normalized to have the form

$q^{-1} + {O}(q)$

then Thompson showed that there are only a finite number of such functions (of some finite level), and Cummins later showed that there are exactly 6486 of them, 616 of which have integral coefficients.[2]

## Alternate Expressions

We have

$J(m) = \frac{4}{27} \frac{(1 - \lambda + \lambda^2)^3}{\lambda^2 (1 - \lambda)^2}$

where λ is the modular lambda function and so

$J(x) = \frac{4}{27} \frac{(1 - x)^3}{x^2}$

where x = λ*(1 - λ).

The value of J is unchanged when λ is replaced by any of the six values of the cross-ratio:[3]

$\left\lbrace { \lambda, \frac{1}{1-\lambda}, \frac{\lambda-1}{\lambda}, \frac{1}{\lambda}, \frac{\lambda}{\lambda-1}, 1-\lambda } \right\rbrace$

The branch points of J are at {0,1,∞}, so that J is a Belyi function.[4]

## Expressions in terms of theta functions

$g_2(\tau) = {2\pi^4 \over 3} \left[\vartheta(0; \tau)^8+\vartheta_{01}(0;\tau)^8 + \vartheta_{10}(0; \tau)^8\right]$
$\Delta(\tau) = 16\pi^{12} \left[\vartheta(0; \tau) \vartheta_{01}(0; \tau) \vartheta_{10}(0; \tau)\right]^8$

We can express it in terms of Jacobi's theta functions, in which form it can very rapidly be computed.

$j(\tau) = 32 {[\vartheta(0; \tau)^8 + \vartheta_{01}(0; \tau)^8 + \vartheta_{10}(0; \tau)^8]^3 \over [\vartheta(0;\tau) \vartheta_{01}(0; \tau) \vartheta_{10}(0; \tau)]^8}$

## Algebraic definition

So far we have been considering j as a function of a complex variable. However, as an invariant for isomorphism classes of elliptic curves, it can be defined purely algebraically. Let

$y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$

be a plane elliptic curve over any field. Then we may define

$b_2 = a_1^2 + 4a_2,\quad b_4 = a_1a_3 + 2a_4$
$b_6 = a_3^2 + 4a_6,\quad b_8 = a_1^2a_6 - a_1a_3a_4 + a_2a_3^2 + 4a_2a_6 - a_4^2$
$c_4 = b_2^2 - 24b_4,\quad c_6 = -b_2^3 + 36b_2b_4 - 216b_6$

and

$\Delta = -b_2^2b_8 + 9b_2b_4b_6 - 8b_4^3 - 27b_6^2$

the latter expression is the discriminant of the curve.

The j-invariant for the elliptic curve may now be defined as

$j = {c_4^3 \over \Delta}$

In the case that the field over which the curve is defined has characteristic different from 2 or 3, this definition can also be written as

$j= 1728{c_4^3 \over c_4^3-c_6^2}$

## Inverse and special values

The inverse of the j-invariant can be expressed in terms of the hypergeometric function ${}_2F_1$ (see main article Picard–Fuchs equation). The inversion of the j-invariant was reported by Semjon Adlaj (CCRAS, Moscow, Russia) on May 30, 2011 prior to a talk given at the 14-TH WORKSHOP ON COMPUTER ALGEBRA (Dubna, Russia). The inversion is highly relevant to applications via enabling high precision calculations of elliptic functions periods even as their ratios become unbounded. A related result is the expressability via quadratic radicals of the values of j at the points of the imaginary axis whose magnitudes are powers of 2 (thus permitting compass and straightedge constructions). The latter result is hardly evident since the modular equation of level 2 is cubic.

The j-invariant vanishes at the "corner" of the fundamental domain at $\frac{1}{2}\left(1 + i \sqrt{3}\right)$. Here are a few more special values (only the first four of which are well known; in what follows, j means j/1728 throughout):

\begin{align} j\left(i\right) &= j\left( \tfrac{1 + i}{2} \right) = 1 \\ j\left(\sqrt{2}i\right) &= \left( \tfrac{5}{3} \right)^3\\ j\left(2i\right) &= \left( \tfrac{11}{2} \right)^3\\ j\left(2\sqrt{2}i\right) &= \left( \tfrac{5}{6}\left[19 + 13\sqrt{2}\right] \right)^3\\ j\left(4i\right) &= \left( \tfrac{1}{4}\left[724 + 513\sqrt{2}\right] \right)^3\\ j\left( \tfrac{1 + 2i}{2} \right) &= \left(\tfrac{1}{4}\left[724 - 513\sqrt{2}\right]\right)^3\\ j\left( \tfrac{1 + 2\sqrt{2}i}{3} \right) &= \left(\tfrac{5}{6}\left[19 - 13\sqrt{2}\right] \right)^3\\ j\left(3i\right) &= \left(2 + \sqrt{3}\right)^2\left(\tfrac{1}{3}\left[21 + 20\sqrt{3}\right] \right)^3\\ j\left(2\sqrt{3}i\right) &= \tfrac{125}{16}\left(30 + 17\sqrt{3}\right)^3\\ j\left( \tfrac{1 + 7\sqrt{3}i}{2} \right) &= -\tfrac{1}{7} \left(40\left[651 + 142\sqrt{21}\right]\right)^3\\ j\left( \tfrac{1 + 3\sqrt{11}i}{10} \right) &= \tfrac{64}{27} \left(23 - 4\sqrt{33}\right)^2 \left(-77 + 15\sqrt{33}\right)^3\\ j\left(\sqrt{21}i\right) &= \left( \tfrac{1}{2}\left[5 + 3\sqrt{3}\right]\left[3 + \sqrt{7}\right] \right)^2 \left(\tfrac{1}{2}\left[65 + 34\sqrt{3} + 26\sqrt{7} + 15\sqrt{21}\right] \right)^3\\ j\left( \tfrac{\sqrt{30}i}{1} \right) &= \left( \tfrac{1}{2} \left[10 + 7\sqrt{2} + 4\sqrt{5} + 3\sqrt{10} \right] \right)^4 \left( 55 + 30\sqrt{2} + 12\sqrt{5} + 10\sqrt{10} \right)^3\\ j\left( \tfrac{\sqrt{30}i}{2} \right) &= \left( \tfrac{1}{2} \left[10 + 7\sqrt{2} - 4\sqrt{5} - 3\sqrt{10} \right] \right)^4 \left( 55 + 30\sqrt{2} - 12\sqrt{5} - 10\sqrt{10} \right)^3\\ j\left( \tfrac{\sqrt{30}i}{5} \right) &= \left( \tfrac{1}{2} \left[10 - 7\sqrt{2} + 4\sqrt{5} - 3\sqrt{10} \right] \right)^4 \left( 55 - 30\sqrt{2} + 12\sqrt{5} - 10\sqrt{10} \right)^3\\ j\left( \tfrac{\sqrt{30}i}{10} \right) &= \left( \tfrac{1}{2} \left[10 - 7\sqrt{2} - 4\sqrt{5} + 3\sqrt{10} \right] \right)^4 \left( 55 - 30\sqrt{2} - 12\sqrt{5} + 10\sqrt{10} \right)^3\\ j(\sqrt{70}i) &= \left( 1 + 9\left[ \tfrac{1}{2}\left(303 + 220\sqrt{2} + 139\sqrt{5} + 96\sqrt{10}\right)\right]^2 \right)^3\\ j(\sqrt{105}i) &= \tfrac{1}{16384} \left( 1445 + 591\sqrt{3} + 461\sqrt{5} + 549\sqrt{7} + 375\sqrt{15} + 225\sqrt{21} + 173\sqrt{35} + 141\sqrt{105} \right)^4 \left( 3105 + 1860\sqrt{3} + 853\sqrt{5} + 980\sqrt{7} + 620\sqrt{15} + 765\sqrt{21} + 300\sqrt{35} + 225\sqrt{105} \right)^3\\ j\left( \tfrac{1 + \sqrt{1435}i}{2} \right) &= \left( 1 - 9\left[ 9892538 + 4424079\sqrt{5} + 1544955\sqrt{41} + 690925\sqrt{205} \right]^2 \right)^3 \end{align}

## References

1. ^ Silverman, Joseph H. (1986). The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 106. Springer-Verlag. p. 339. ISBN 0-387-96203-4. Zbl 0585.14026.
2. ^ Cummins, C.J. (2004). "Congruence subgroups of groups commensurable with PSL(2,Z)\$ of genus 0 and 1". Exp. Math. 13 (3): 361–382. ISSN 1058-6458. Zbl 1099.11022.
3. ^ Chandrasekharan, K. (1985), Elliptic Functions, Grundlehren der mathematischen Wissenschaften 281, Springer-Verlag, p. 110, ISBN 3-540-15295-4, Zbl 0575.33001
4. ^ Girondo, Ernesto; González-Diez, Gabino (2012), Introduction to compact Riemann surfaces and dessins d'enfants, London Mathematical Society Student Texts 79, Cambridge: Cambridge University Press, p. 267, ISBN 978-0-521-74022-7, Zbl 1253.30001