j-invariant

In mathematics, Klein's j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half-plane of complex numbers.

We have

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

The modular discriminant \Delta is defined as

\Delta=g_2^3-27g_3^2. \,

The numerator and denominator above are in terms of the modular invariants g_2 and g_3 of the Weierstrass elliptic functions

g_2= 60\sum_{(m,n) \neq (0,0)} (m %2B n\tau)^{-4},\qquad g_3=140\sum_{(m,n) \neq (0,0)} (m %2B n\tau)^{-6}

and the modular discriminant.

These have the properties that

g_2(\tau%2B1)=g_2(\tau),\; g_2(-\tau^{-1})=\tau^4g_2(\tau) \,
\Delta(\tau%2B1) = \Delta(\tau),\; \Delta(-\tau^{-1}) = \tau^{12} \Delta(\tau) \,

and possess the analytic properties making them modular forms. Δ is a modular form of weight twelve by the above, and g_2 one of weight four, so that its third power is also of weight twelve. The quotient is therefore a modular function of weight zero; this means j has the absolutely invariant property that

j(\tau%2B1)=j(\tau),\; j(-\tau^{-1}) = j(\tau).

Contents

Expressions in terms of theta functions

g_2(\tau) = \tfrac12 \left[\vartheta(0;\tau)^8%2B\vartheta_{01}(0;\tau)^8%2B\vartheta_{10}(0;\tau)^8\right]
\Delta(\tau) = \tfrac12 \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%2B\vartheta_{01}(0;\tau)^8%2B\vartheta_{10}(0;\tau)^8]^3 \over [\vartheta(0;\tau) \vartheta_{01}(0;\tau) \vartheta_{10}(0;\tau)]^8}.

The fundamental region

The two transformations \tau \rightarrow \tau%2B1 and \tau \rightarrow -\tau^{-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, \tau \rightarrow \frac{a\tau%2Bb}{c\tau%2Bd}, 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) takes on every value in the complex numbers \mathbb{C} exactly once in this region. In other words, for every c\in\mathbb{C}, there is a 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, and vice-versa.

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).

The values of j are in a one-to-one relationship with values of τ lying in the fundamental region, and each value for j corresponds to the field of elliptic functions with periods 1 and τ, for the corresponding value of τ; this means that j is in a one-to-one relationship with isomorphism classes of elliptic curves over the complex numbers.

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. 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(2 \pi i \tau)), which begins:

j(\tau) = {1 \over q} %2B 744 %2B 196884 q %2B 21493760 q^2 %2B 864299970 q^3 %2B 20245856256 q^4 %2B \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%2B744.

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}%2B{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..

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}}.

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 %2B a_1 xy %2B a_3 y = x^3 %2B a_2 x^2 %2B a_4 x %2B a_6\,

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

b_2 = a_1^2%2B4a_2,\quad b_4=a_1a_3%2B2a_4,
b_6=a_3^2%2B4a_6,\quad b_8=a_1^2a_6-a_1a_3a_4%2Ba_2a_3^2%2B4a_2a_6-a_4^2,
c_4 = b_2^2-24b_4,\quad c_6 = -b_2^3%2B36b_2b_4-216b_6

and

\Delta = -b_2^2b_8%2B9b_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} %2B 1 \ \right) \sqrt[3]{21 \sqrt{- 3 / 8} - 1} - \frac{11 \left( \sqrt{- 3} - 1 \right)}{\sqrt[3]{42 \sqrt{- 6} - 8}} \right) %2B 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} %2B 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 %2B i \sqrt{3}) / 2. Here are seven more special values (only the first four of which are well known):

\begin{align} j(i) &= j(\tfrac{1%2Bi}{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%2B13\sqrt{2})}{6} \right)^3\\ j(4i) &= \left(\tfrac{724%2B513\sqrt{2}}{4}\right)^3\\ j\left(\tfrac{1%2B2i}{2}\right) &= \left(\tfrac{724-513\sqrt{2}}{4}\right)^3\\ j\left(\tfrac{1%2B2\sqrt{2}i}{3}\right) &= \left(\tfrac{5(19-13\sqrt{2})}{6}\right)^3 \end{align}

References