- J-invariant
nome q on the unit disk
In
mathematics , Klein's "j"-invariant, regarded as a function of a complex variable τ, is amodular function defined on theupper half-plane of complex numbers. We can express it in terms of Jacobi'stheta function s, in which form it can very rapidly be computed.We have
::j( au) = 32 { [vartheta(0; au)^8+vartheta_{01}(0; au)^8+vartheta_{10}(0; au)^8] ^3 over [vartheta(0; au) vartheta_{01}(0; au) vartheta_{10}(0; au)] ^8}={g_2^3 over Delta}.
The numerator and denominator above are in terms of the invariant g_2 of the
Weierstrass elliptic functions ::g_2( au) = frac{vartheta(0; au)^8+vartheta_{01}(0; au)^8+vartheta_{10}(0; au)^8}{2}
and the
modular discriminant ::Delta( au) = frac{ [vartheta(0; au) vartheta_{01}(0; au) vartheta_{10}(0; au)] ^8}{2}.
These have the properties that
::g_2( au+1)=g_2( au),; g_2left(-frac{1}{ au} ight)= au^4g_2( au)
::Delta( au+1) = Delta( au),; Deltaleft(-frac{1}{ au} ight) = au^{12} Delta( au)
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( au+1)=j( au),; jleft(-frac{1}{ au} ight) = j( au).
The fundamental region
The two transformations au ightarrow au+1 and au ightarrow -frac{1}{ au} together generate a group called the
modular group , which we may identify with theprojective linear group PSL_2(mathbb{Z}). By a suitable choice of transformation belonging to this group, au ightarrow frac{a au+b}{c au+d}, with "ad" − bc = 1, we may reduce τ to a value giving the same value for "j", and lying in thefundamental region for "j", which consists of values for τ satisfying the conditions::au| ge 1,
::frac{1}{2} < mathfrak{R}( au) le frac{1}{2},
::frac{1}{2} < mathfrak{R}( au) < 0 Rightarrow | au| > 1.
The function "j"(τ) takes on every value in the
complex number s mathbb{C} exactly once in this region. In other words, for every cinmathbb{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 ofelliptic curve s.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( au) is analgebraic integer . The field extension:mathbb{Q} [j( au), au] /mathbb{Q}( au)
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}( au) 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( au') of j( au) over mathbb{Q}( au). The unique maximal order under inclusion of mathbb{Q}( au) is the ring of algebraic integers of mathbb{Q}( au), and values of τ having it as its associated order lead tounramified extension s of mathbb{Q}( au). These classical results are the starting point for the theory ofcomplex multiplication .The "q"-expansion and moonshine
Several remarkable properties of "j" have to do with its "q"-expansion (
Fourier series expansion, written as aLaurent series in terms of q=exp(2 pi i au)), which begins::j( au) = {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.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 formoonshine theory .The study of the Moonshine conjecture led
J.H. Conway andSimon 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 Cumminslater showed that there are exactly 6486 of them, 616 of which have integral coefficients(see [http://www.expmath.org/expmath/volumes/13/13.3/cummins/cumminstables.pdf here] for the complete list).
Growth
By a theorem of Petersson and Rademacher, the rate of growth of ln(cn) is asymptotically
:ln(c_n) sim 4pi 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} ight) 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 + 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_4^3over c_4^3-c_6^2}.
Inverse
The inverse of the j-invariant can be expressed in terms of the
hypergeometric series 2F_1. See main articlePicard-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", [http://www.journals.cms.math.ca/cgi-bin/vault/public/view/berndt7376/body/PDF/berndt7376.pdf 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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