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Rational functions of j are modular, and in fact give all modular functions. Classically, the j-invariant was studied as a parameterization of elliptic curves over C, but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).
Definition
Real part of the j-invariant as a function of the nomeq on the unit diskPhase of the j-invariant as a function of the nome q on the unit disk
While the j-invariant can be defined purely in terms of certain infinite sums (see g2, g3 below), these can be motivated by considering isomorphism classes of elliptic curves. Every elliptic curve E over C is a complex torus, and thus can be identified with a rank 2 lattice; i.e., two-dimensional lattice of C. This is done by identifying opposite edges of each parallelogram in the lattice. It turns out that multiplying the lattice by complex numbers, which corresponds to rotating and scaling the lattice, preserves the isomorphism class of the elliptic curve, and thus we can consider the lattice generated by 1 and some τ in H (where H is the Upper half-plane). Conversely, if we define
then this lattice corresponds to the elliptic curve over C defined by y2 = 4x3 − g2x - g3 via the Weierstrass elliptic functions. Then the j-invariant is defined as
where the modular discriminantΔ is
It can be shown that Δ is a modular form of weight twelve, and g2 one of weight four, so that its third power is also of weight twelve. Thus their quotient, and therefore j, is a modular function of weight zero, in particular a meromorphic function H → C invariant under the action of SL(2, Z). As explained below, j is surjective, which means that it gives a bijection between isomorphism classes of elliptic curves over C and the complex numbers.
The fundamental region
The fundamental domain of the modular group acting on the upper half plane.
The two transformations τ → τ + 1 and τ → -τ−1 together generate a group called the modular group, which we may identify with the projective special linear groupPSL(2, Z). By a suitable choice of transformation belonging to this group,
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
The function j(τ) when restricted to this region still takes on every value in the complex numbersC exactly once. In other words, for every c in C, there is a unique 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.
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 C(j).
The field extension Q[j(τ), τ]/Q(τ) is abelian, that is, it has an abelian Galois group.
Let Λ be the lattice in C generated by {1, τ}. It is easy to see that all of the elements of Q(τ) which fix Λ under multiplication form a ring with units, called an order. The other lattices with generators {1, τ′}, associated in like manner to the same order define the algebraic conjugatesj(τ′) of j(τ) over Q(τ). Ordered by inclusion, the unique maximal order in Q(τ) is the ring of algebraic integers of Q(τ), and values of τ having it as its associated order lead to unramified extensions of Q(τ).
These classical results are the starting point for the theory of complex multiplication.
Transcendence properties
In 1937 Theodor Schneider proved the aforementioned result that if τ is a quadratic irrational number in the upper half plane then j(τ) is an algebraic integer. In addition he proved that if τ is an algebraic number but not imaginary quadratic then j(τ) is transcendental.
The j function has numerous other transcendental properties. Kurt Mahler conjectured a particular transcendence result that is often referred to as Mahler's conjecture, though it was proved as a corollary of results by Yu. V. Nesterenko and Patrice Phillipon in the 1990s. Mahler's conjecture was that if τ was in the upper half plane then exp(2πiτ) and j(τ) were never both simultaneously algebraic. Stronger results are now known, for example if exp(2πiτ) is algebraic then the following three numbers are algebraically independent, and thus at least two of them transcendental:
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:
Note that j has a simple pole at the cusp, so its q-expansion has no terms below q−1.
More remarkably, the Fourier coefficients for the positive exponents of q are the dimensions of the graded part of an infinite-dimensional graded algebra representation of the monster group called the moonshine module – specifically, the coefficient of qn is the dimension of grade-n part of the moonshine module, the first example being the Griess algebra, which has dimension 196,884, corresponding to the term 196884q. 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
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.[4]
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
be a plane elliptic curve over any field. Then we may define
and
the latter expression is the discriminant of the curve.
The j-invariant for the elliptic curve may now be defined as
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
Inverse function
The inverse function of the j-invariant can be expressed in terms of the hypergeometric function2F1 (see also the article Picard–Fuchs equation). Explicitly, given a number N, to solve the equation j(τ) = N for τ can be done in at least four ways.
One root gives τ, and the other gives 1/τ, but since j(τ) = j(1/τ), then it doesn't make a difference which α is chosen. The latter three methods can be found in Ramanujan's theory of elliptic functions to alternative bases.
The inversion is highly relevant to applications via enabling high-precision calculations of elliptic functions periods even as their ratios become unbounded. A related result is the expressibility via quadratic radicals of the values of j at the points of the imaginary axis whose magnitudes are powers of 2 (thus permitting compass and straightedge constructions). The latter result is hardly evident since the modular equation of level 2 is cubic.
Here are a few more special values given in terms of the alternative notation (only the first four of which are well known):
Several special values were calculated in 2014:[10]
and let,
All preceding values are real. A complex conjugate pair might be inferred exploiting the symmetry described in the reference, along with the values for and , given above:
Four more special values are given as two complex conjugate pairs:[11]
^Cummins, C.J. (2004). "Congruence subgroups of groups commensurable with PSL(2,Z)$ of genus 0 and 1". Exp. Math. 13 (3): 361–382. ISSN1058-6458. Zbl1099.11022.
^Girondo, Ernesto; González-Diez, Gabino (2012), Introduction to compact Riemann surfaces and dessins d'enfants, London Mathematical Society Student Texts, vol. 79, Cambridge: Cambridge University Press, p. 267, ISBN978-0-521-74022-7, Zbl1253.30001
^Lang, Serge (1987). Elliptic functions. Graduate Texts in Mathematics. Vol. 112. New-York ect: Springer-Verlag. pp. 299–300. ISBN978-1-4612-9142-8. Zbl0615.14018.
Apostol, Tom M. (1976), Modular functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics, vol. 41, New York: Springer-Verlag, MR0422157. Provides a very readable introduction and various interesting identities.
Cox, David A. (1989), Primes of the Form x^2 + ny^2: Fermat, Class Field Theory, and Complex Multiplication, New York: Wiley-Interscience Publication, John Wiley & Sons Inc., MR1028322 Introduces the j-invariant and discusses the related class field theory.
Rankin, Robert A. (1977), Modular forms and functions, Cambridge: Cambridge University Press, ISBN0-521-21212-X, MR0498390. Provides a short review in the context of modular forms.