J-invariant
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Image:J-inv-real.jpeg 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 can express it in terms of Jacobi's theta functions, in which form it can very rapidly be computed.
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) = \frac{\vartheta(0;\tau)^8+\vartheta_{01}(0;\tau)^8+\vartheta_{10}(0;\tau)^8}{2}</math>
and the modular discriminant
- <math>\Delta(\tau) = \frac{[\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\left(-\frac{1}{\tau}\right)=\tau^4g_2(\tau)</math>
- <math>\Delta(\tau+1) = \Delta(\tau),\; \Delta\left(-\frac{1}{\tau}\right) = \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\left(-\frac{1}{\tau}\right) = j(\tau)</math>
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The fundamental region
The two transformations <math>\tau \rightarrow \tau+1</math> and <math>\tau \rightarrow -\frac{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 \frac{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>-\frac{1}{2} < \mathfrak{R}(\tau) \le \frac{1}{2} </math>
- <math>-\frac{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
Image:J-inv-phase.jpeg 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>\frac{\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 led 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 over any field. Then we may define
- <math>b_2 = a_1^2+4a_2,\quad b_4=a_1a_3+2a_4,</math>
- <math>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,</math>
- <math>c_4 = b_2^2-24b_4,\quad c_6 = -b_2^3+36b_2b_4-216b_6</math>
and
- <math>\Delta = -b_2^2b_8+9b_2b_4b_6-8b_4^3-27b_6^2;</math>
the latter expression is the discriminant of the curve.
The j-invariant for the elliptic curve may now be defined as
- <math>j = {c_4^3 \over \Delta}.</math>
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
- <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. (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.)de:j-Funktion