j-invariant

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 $j(e^{2\pi i/3})=0$ and $j(i)=1728$. Rational functions of $j$ are modular, and in fact give all modular functions. Classically, the j-variant 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

Further information: Elliptic_curve#Elliptic_curves_over_the_complex_numbers and Modular_forms

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 lattice of rank 2. Multiplying this lattice by a complex number 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

$|\tau| \ge 1,$

$-\frac{1}{2} < \mathfrak{R}(\tau) \le \frac{1}{2},$

$-\frac{1}{2} < \mathfrak{R}(\tau) < 0 \Rightarrow |\tau| > 1.$

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 element of an imaginary quadratic field with positive imaginary part (so that j is defined) then $j(\tau)$ is an algebraic integer{citation needed}. 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.^{[citation needed]}.

A remarkable property of the q-series for j is the product formula; if p and q are small enough we have

$j(p)-j(q) = \left({1 \over p} - {1 \over q}\right) \prod_{n,m=1}^{\infty}(1-p^n q^m)^{c_{nm}}.$^{[citation needed]}

Alternate Expressions

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

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

[x = m*(1 - m)]

J is unchanged when m is replaced by

m, m⁻¹, 1 − m, 1 − m⁻¹, (1 − m)⁻¹, m / (m − 1)

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. For example, the value of j at $4 i$ simplifies(!) to:

$\begin{align} j(4 i) & = \frac{1}{108} \left( \frac{33}{8} \left( \left( \sqrt{- 3} + 1 \ \right) \sqrt[3]{21 \sqrt{- 3 / 8} - 1} - \frac{11 \left( \sqrt{- 3} - 1 \right)}{\sqrt[3]{42 \sqrt{- 6} - 8}} \right) + 1 \right) \times \\ & {} \quad \times \left( \frac{256}{16 - 33 \left( 11 \left( \sqrt{- 3} - 1 \ \right) / \sqrt[3]{21 \sqrt{- 3 / 8} - 1} - \left( \sqrt{- 3} + 1 \ \right) \sqrt[3]{42 \sqrt{- 6} - 8} \right)} - 1 \right)^3. \end{align}$

The j-invariant vanishes at the "corner" of the fundamental domain at $(1 + i \sqrt{3}) / 2$. Here are a few more special values (only the first four of which are well known):

$\begin{align} j(i) &= j(\tfrac{1+i}{2}) = 1 \\ j(\sqrt{2}i) &= \left( \tfrac{5}{3} \right)^3\\ j(2i) &= \left( \tfrac{11}{2} \right)^3\\ j(2\sqrt{2}i) &= \left(\tfrac{5(19+13\sqrt{2})}{6} \right)^3\\ j(4i) &= \left(\tfrac{724+513\sqrt{2}}{4}\right)^3\\ j\left(\tfrac{1+2i}{2}\right) &= \left(\tfrac{724-513\sqrt{2}}{4}\right)^3\\ j\left(\tfrac{1+2\sqrt{2}i}{3}\right) &= \left(\tfrac{5(19-13\sqrt{2})}{6}\right)^3\\ j(3i) &= (2+\sqrt{3})^2\left(\tfrac{21+20\sqrt{3}}{3}\right)^3\\ j(2\sqrt{3}i) &= \tfrac{125}{16}(30+17\sqrt{3})^3\\ j\left(\tfrac{1+7\sqrt{3}i}{2}\right) & = \tfrac{-1}{7}\left(40(651+142\sqrt{21})\right)^3\\ j\left(\tfrac{1+3\sqrt{11}i}{10}\right) &= \tfrac{64}{27}(23-4\sqrt{33})^2(-77+15\sqrt{33})^3\\ j(\sqrt{21}i) &= \left(\tfrac{(5+3\sqrt{3})(3+\sqrt{7})}{2}\right)^2\left(\tfrac{65+34\sqrt{3}+26\sqrt{7}+15\sqrt{21}}{2}\right)^3\\ j\left(\tfrac{\sqrt{30}i}{1}\right) &= (8+5\sqrt{2}+\sqrt{5}+\sqrt{10})^2\left(\tfrac{565+360\sqrt{2}+249\sqrt{5}+160\sqrt{10}}{2}\right)^3\\ j\left(\tfrac{\sqrt{30}i}{2}\right) &= (8+5\sqrt{2}-\sqrt{5}-\sqrt{10})^2\left(\tfrac{565+360\sqrt{2}-249\sqrt{5}-160\sqrt{10}}{2}\right)^3\\ j\left(\tfrac{\sqrt{30}i}{5}\right) &= (8-5\sqrt{2}+\sqrt{5}-\sqrt{10})^2\left(\tfrac{565-360\sqrt{2}+249\sqrt{5}-160\sqrt{10}}{2}\right)^3\\ j\left(\tfrac{\sqrt{30}i}{10}\right) &= (8-5\sqrt{2}-\sqrt{5}+\sqrt{10})^2\left(\tfrac{565-360\sqrt{2}-249\sqrt{5}+160\sqrt{10}}{2}\right)^3\\ j(\sqrt{70}i) &= \left(1+9\left(\tfrac{303+220\sqrt{2}+139\sqrt{5}+96\sqrt{10}}{2}\right)^2\right)^3\\ j\left(\tfrac{1+\sqrt{1435}i}{2}\right) &= \left(1-9\left(9892538+4424079\sqrt{5}+1544955\sqrt{41}+690925\sqrt{205}\right)^2\right)^3 \end{align}$

References

• Apostol, Tom M. (1976), Modular functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics 41, New York: Springer-Verlag, MR 0422157 . Provides a very readable introduction and various interesting identities.
  • Apostol, Tom M. (1990), Modular functions and Dirichlet Series in Number Theory (2nd ed.), ISBN 0-387-97127-0, MR 1027834 
• Berndt, Bruce C.; Chan, Heng Huat (1999), "Ramanujan and the modular j-invariant", Canadian Mathematical Bulletin 42 (4): 427–440, doi:10.4153/CMB-1999-050-1, MR 1727340 . Provides a variety of interesting algebraic identities, including the inverse as a hypergeometric series.
• Cox, David A. (1989), Primes of the Form x^2 + ny^2: Fermat, Class Field Theory, and Complex Multiplication, New York: Wiley-Interscience Publication, John Wiley & Sons Inc., MR 1028322  Introduces the j-invariant and discusses the related class field theory.
• Conway, John Horton; Norton, Simon (1979), "Monstrous moonshine", Bulletin of the London Mathematical Society 11 (3): 308–339, doi:10.1112/blms/11.3.308, MR 0554399 . Includes a list of the 175 genus-zero modular functions.
• Petersson, Hans (1932), "Über die Entwicklungskoeffizienten der automorphen Formen", Acta Mathematica 58 (1): 169–215, doi:10.1007/BF02547776, MR 1555346 .
• Rademacher, Hans (1938), "The Fourier coefficients of the modular invariant j(τ)", American Journal of Mathematics (The Johns Hopkins University Press) 60 (2): 501–512, doi:10.2307/2371313, JSTOR 2371313, MR 1507331 .
• Rankin, Robert A. (1977), Modular forms and functions, Cambridge: Cambridge University Press, ISBN 0-521-21212-X, MR 0498390 . Provides a short review in the context of modular forms.
• Schneider, Theodor (1937), "Arithmetische Untersuchungen elliptischer Integrale", Math. Annalen 113: 1–13, doi:10.1007/BF01571618, MR 1513075 .