j-invariant

In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for special linear group SL(2, Z) defined on the upper half-plane of complex numbers.

Classically, the j-invariant was studied as a parameterization of elliptic curves over

The function cannot be continued analytically beyond the upper half-plane due to the natural boundary at the real line.

The j-invariant can then be directly expressed in terms of the Eisenstein series as, with no numerical factor other than 1728.

This implies a third way to define the modular discriminant,[1] For example, using the definitions above and

has the exact value, implying the transcendental numbers, but yielding the algebraic number (in fact, an integer), In general, this can be motivated by viewing each τ as representing an isomorphism class of elliptic curves.

This is due to the corresponding cubic polynomial having distinct roots.

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 numbers C exactly once.

Thus, j has the property of mapping the fundamental region to the entire complex plane.

This means j provides a bijection from the set of elliptic curves over C to the complex plane.

The j-invariant has many remarkable properties: These classical results are the starting point for the theory of complex multiplication.

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.

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.

Stronger results are now known, for example if e2πiτ is algebraic then the following three numbers are algebraically independent, and thus at least two of them transcendental: Several remarkable properties of j have to do with its q-expansion (Fourier series expansion), written as a Laurent series in terms of q = e2πiτ, which begins: 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: The asymptotic formula for the coefficient of qn is given by as can be proved by the Hardy–Littlewood circle method.

[4][5] 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, first made by John McKay, was the starting point for moonshine theory.

The study of the Moonshine conjecture led John Horton Conway and Simon P. Norton to look at the genus-zero modular functions.

If they are normalized to have the form then John G. Thompson showed that there are only a finite number of such functions (of some finite level), and Chris J. Cummins later showed that there are exactly 6486 of them, 616 of which have integral coefficients.

The j(τ) can then be rapidly computed, 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.

We also have the discriminant 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 is equal to The inverse function of the j-invariant can be expressed in terms of the hypergeometric function 2F1 (see also the article Picard–Fuchs equation).

[note 1] Method 2: Solving the quartic in γ, then for any of the four roots, Method 3: Solving the cubic in β, then for any of the three roots, Method 4: Solving the quadratic in α, then, One root gives τ, and the other gives −⁠1/τ⁠, but since j(τ) = j(−⁠1/τ⁠), it makes no 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 applied in high-precision calculations of elliptic function periods even as their ratios become unbounded.

[citation needed] 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 for j of order 2 is cubic.

[11] The Chudnovsky brothers found in 1987,[12] a proof of which uses the fact that For similar formulas, see the Ramanujan–Sato series.

-invariant is only sensitive to isomorphism classes of elliptic curves over the complex numbers, or more generally, an algebraically closed field.

This can be shown using Cardano's formula to show that in that case the solutions to