J-invariant

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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 (0;tau)^8+vartheta_(0;tau)^8]^3 \over [vartheta(0;tau) vartheta_(0;tau) vartheta_(0;tau)]^8}=]

The numerator and denominator above are in terms of the invariant [g_2] of the Weierstrass elliptic functions

:[g_2(\tau) = \frac(0;\tau)^8+\vartheta_(0;\tau)^8}]

and the modular discriminant

:[\Delta(\tau) = \frac(0;tau) vartheta_(0;tau)]^8}]

These have the properties that

:[g_2(\tau+1)=g_2(\tau),\; g_2\left(-\frac\right)=\tau^4g_2(\tau)]

:[\Delta(\tau+1) = \Delta(\tau),\; \Delta\left(-\frac\right) = \tau^ \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+1)=j(\tau),\; j\left(-\frac\right) = j(\tau)]

The fundamental region

The two transformations [\tau \rightarrow \tau+1] and [\tau \rightarrow -\frac] together generate a group called the modular group, which we may identify with the projective linear group [PSL_2(\mathbb)]. By a suitable choice of transformation belonging to this group, [\tau \rightarrow \frac], 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 < \mathfrak(\tau) \le \frac ]

:[-\frac < \mathfrak(\tau) < 0 \Rightarrow |\tau| > 1 ]

The function j(τ) takes on every value in the complex numbers [\mathbb] exactly once in this region. In other words, for every [c\in\mathbb], 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(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.

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[j(tau),tau]/\mathbb(\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(\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(\tau)]. The unique maximal order under inclusion of [\mathbb(\tau)] is the ring of algebraic integers of [\mathbb(\tau)], and values of τ having it as its associated order lead to unramified extensions of [\mathbb(\tau)]. 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 [q=\exp(2 \pi i \tau)] the Fourier series becomes a Laurent series in q, [\sum c_n q^n], where the values for [c_n] for n < -1 are all zero, and where the [c_n] are integers. The first few terms of it are

:[j(q) = + 744 + 196884 q + 21493760 q^2 + 864299970 q^3 + 20245856256 q^4 + \cdots]

as we may easily find by substituting q for [\exp(2 \pi i \tau)] in the definition for j with which we started. The coefficients [c_n] 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.

By a theorem of Petersson and Rademacher, the rate of growth of ln(c_n) is asymtotically

[\ln(c_n) \sim 4\pi \sqrt - \frac \ln(n) - \frac \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( - \right) \prod_^(1-p^n q^m)^}]

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(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 (see [here] for the complete list).

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 = .]

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.]

Inverse

The inverse of the j-invariant can be expressed in terms of the hypergeometric series [_2F_1]. 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)
• 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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