J-invariant

From Academic Kids

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

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J-inv-real.jpeg
Real part of the j-invariant as a function of the nome q on the unit disk
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J-inv-phase.jpeg
Phase of the j-invariant as a function of the nome q on the unit disk
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J-inv-modulus.jpeg
Modulus of the j-invariant as a function of the nome q on the unit disk

We have

<math>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}

={g_2^3 \over \Delta}<math>

The numerator and denominator above are in terms of the invariant <math>g_2<math> of the Weierstrass elliptic functions

<math>g_2(\tau) = (\vartheta(0;\tau)^8+\vartheta_{01}(0;\tau)^8+\vartheta_{10}(0;\tau)^8)/2<math>

and the modular discriminant

<math>\Delta(\tau) = (\vartheta(0;\tau) \vartheta_{01}(0;\tau) \vartheta_{10}(0;\tau))^8/2<math>

These have the properties that

<math>g_2(\tau+1)=g_2(\tau),\; g_2(-1/\tau)=\tau^4g_2(\tau)<math>
<math>\Delta(\tau+1) = \Delta(\tau),\; \Delta(-1/\tau) = \tau^{12} \Delta(\tau)<math>

and possess the analytic properties making them modular forms. Δ is a modular form of weight twelve by the above, and <math>g_2<math> 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

<math>j(\tau+1)=j(\tau),\; j(-1/\tau) = j(\tau)<math>
Contents

The fundamental region

The two transformations <math>\tau \rightarrow \tau+1<math> and <math>\tau \rightarrow - 1/\tau<math> together generate a group called the modular group, which we may identify with the projective linear group <math>PSL_2(\mathbb{Z})<math>. By a suitable choice of transformation belonging to this group, <math>\tau \rightarrow (a\tau+b)/(c\tau+d)<math>, 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

<math>|\tau| \ge 1 <math>
<math>-1/2 < \mathfrak{R}(\tau) \le 1/2 <math>
<math>-1/2 < \mathfrak{R}(\tau) < 0 \Rightarrow |\tau| > 1 <math>

The function j(τ) takes on every value in the complex numbers <math>\mathbb{C}<math> exactly once in this region. In other words, for every <math>c\in\mathbb{C}<math>, 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 <math>\mathbb{C}(j)<math>.

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.

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 <math>j(\tau)<math> is an algebraic integer. The field extension <math>\mathbb{Q}(j(\tau),\tau)/\mathbb{Q}(\tau)<math> 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 <math>\mathbb{Q}(\tau)<math> 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 <math>j(\tau')<math> of <math>j(\tau)<math> over <math>\mathbb{Q}(\tau)<math>. The unique maximal order under inclusion of <math>\mathbb{Q}(\tau)<math> is the ring of algebraic integers of <math>\mathbb{Q}(\tau)<math>, and values of τ having it as its associated order lead to unramified extensions of <math>\mathbb{Q}(\tau)<math>. These classical results are the starting point for the theory of complex multiplication.

The q-series and moonshine

Another remarkable property of j has to do with what is called its q-series. If we fix the imaginary part of τ and vary the real part, we obtain a periodic complex function of a real variable with period 1. The Fourier coefficients for these functions are extremely interesting. If we perform the substitution <math>q=\exp(2 \pi i \tau)<math> the Fourier series becomes a Laurent series in q, <math>\sum c_n q^n<math>, where the values for <math>c_n<math> for n < -1 are all zero, and where the <math>c_n<math> are integers. The first few terms of it are

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

as we may easily find by substituting q for <math>\exp(2 \pi i \tau)<math> in the definition for j with which we started. The coefficients <math>c_n<math> for the positive exponents of q are the dimensions of the grade-n part of an infinite-dimensional graded algebra representation of the monster group called the moonshine module, a fact which may be taken as the starting point for moonshine theory.

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

<math>j(p)-j(q) = \left({1 \over p} - {1 \over q}\right) \prod_{n,m=1}^{\infty}(1-p^n q^m)^{c_{nm}}<math>

The study of the Moonshine conjecture lead J.H. Conway and S.P. Norton to look at the genus-zero modular functions. There are 175 such functions, of which j(q) is but one. All have the form

<math>q+c+\mathcal{O}(q^{-1})<math>.

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

<math>y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6<math>

be a plane elliptic curve in any field of characteristic neither 2 nor 3 in which the coefficients lie. Then we may define

<math>c_4 = (a_1^2+4 a_2)^2-24(2a_4+a_1 a_3)<math>
<math>c_6 = -(a_1^2+4a_2)^3+36(a_1^2+4 a_2)(2a_4+a_1 a_3)-216(a_3^2 +4a_6)<math>

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

<math>j = 1728 {c_4^3 \over c_4^3-c_6^2}<math>

Inverse

The inverse of the j-invariant can be expressed in terms of the hypergeometric series <math>{}_2F_1<math>. See main article Picard-Fuchs equation.

References

  • Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 (Provides a very readable introduction and various interesting identities)
  • Robert A. Rankin, Modular forms and functions, (1977) Cambridge University Press, Cambridge. ISBN 0-521-21212-X (Provides short review in the context of modular forms.)
  • Bruce C. Berndt and Heng Huat Chan, Ramanujan and the Modular j-Invariant, Canadian Mathematical Bulletin, Vol. 42(4), 1999 pp 427-440. (http://www.journals.cms.math.ca/cgi-bin/vault/public/view/berndt7376/body/PDF/berndt7376.pdf) (Provides a variety of interesting algebraic identities, including the inverse as a hypergeometric series).
  • John Horton Conway and S.P.Norton, Monstrous Moonshine, Bulletin of the London Mathematical Society, Vol. 11, (1979) pp.308-339. (A list of the 175 genus-zero modular functions.)
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