j-invariant

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Klein's j-invariant in the complex plane
Klein's j-invariant in the complex plane
Real part of the j-invariant as a function of the nome q on the unit disk
Real part of the j-invariant as a function of the nome q on the unit disk
Modulus of the j-invariant as a function of the nome q on the unit disk
Modulus 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
Phase of the j-invariant as a function of the nome q on the unit disk

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

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}.

The numerator and denominator above are in terms of the invariant g2 of the Weierstrass elliptic functions

g_2(\tau) = \frac{\vartheta(0;\tau)^8+\vartheta_{01}(0;\tau)^8+\vartheta_{10}(0;\tau)^8}{2}

and the modular discriminant

\Delta(\tau) = \frac{[\vartheta(0;\tau) \vartheta_{01}(0;\tau) \vartheta_{10}(0;\tau)]^8}{2}.

These have the properties that

g_2(\tau+1)=g_2(\tau),\; g_2\left(-\frac{1}{\tau}\right)=\tau^4g_2(\tau)
\Delta(\tau+1) = \Delta(\tau),\; \Delta\left(-\frac{1}{\tau}\right) = \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 g2 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+1)=j(\tau),\; j\left(-\frac{1}{\tau}\right) = j(\tau).

Contents

[edit] The fundamental region

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

[edit] 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(τ) 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(τ') of j(τ) 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.

[edit] 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(q) = {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 640,320^3+744.

[edit] Moonshine

More remarkably, the Fourier coefficients 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. 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+\mathcal{O}(q^{-1}).

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 (see here for the complete list).

[edit] Growth

By a theorem of Petersson and Rademacher, the rate of growth of ln(cn) is asymptotically

\ln(c_n) \sim 4\pi \sqrt{n} - \frac{3}{4} \ln(n) - \frac{1}{2} \ln(2),

which entails by the root test that the q-series converges absolutely if 0<|q|<1. In the case of a p-adic field, since the coefficients are integers we have that the series converges when 0<|q|p<1.

Still another 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}}.

[edit] 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}.

[edit] Inverse

The inverse of the j-invariant can be expressed in terms of the hypergeometric series 2F1. See main article Picard-Fuchs equation.

[edit] 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)
  • 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 Simon Norton, Monstrous Moonshine, Bulletin of the London Mathematical Society, Vol. 11, (1979) pp.308-339. (A list of the 175 genus-zero modular functions)
  • Hans Petersson, Über die Entwicklungskoeffizienten der automorphen formen, Acta Math. 58 (1932), 169-215
  • Hans Rademacher, The Fourier Coefficients of the Modular Invariant j(tau), Amer. J. Math. 60 (1938), 501-512
  • 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)
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