J-invariant

In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that
Rational functions of are modular, and in fact give all modular functions. Classically, the -invariant was studied as a parameterization of elliptic curves over , but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).
Definition.
The -invariant can be defined as a function on the upper half-plane 
where:
This can be motivated by viewing each as representing an isomorphism class of elliptic curves. Every elliptic curve over is a complex torus, and thus can be identified with a rank 2 lattice; that is, a two-dimensional lattice of . This lattice can be rotated and scaled (operations that preserve the isomorphism class), so that it is generated by and . This lattice corresponds to the elliptic curve formula_6 (see Weierstrass elliptic functions).
Note that is defined everywhere in as the modular discriminant is non-zero. This is due to the corresponding cubic polynomial having distinct roots.
The fundamental region.
It can be shown that is a modular form of weight twelve, and one of weight four, so that its third power is also of weight twelve. Thus their quotient, and therefore , is a modular function of weight zero, in particular a holomorphic function invariant under the action of . Quotienting out by its centre yields the modular group, which we may identify with the projective special linear group .
By a suitable choice of transformation belonging to this group,
we may reduce to a value giving the same value for , and lying in the fundamental region for , which consists of values for satisfying the conditions
The function when restricted to this region still takes on every value in the complex numbers exactly once. In other words, for every in , there is a unique τ in the fundamental region such that . Thus, has the property of mapping the fundamental region to the entire complex plane.
Additionally two values produce the same elliptic curve iff for some . This means provides a bijection from the set of elliptic curves over to the complex plane.
As a Riemann surface, the fundamental region has genus , and every (level one) modular function is a rational function in ; and, conversely, every rational function in is a modular function. In other words, the field of modular functions is .
Class field theory and.
The -invariant has many remarkable properties:
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 is a quadratic irrational number in the upper half plane then is an algebraic integer. In addition he proved that if is an algebraic number but not imaginary quadratic then is transcendental.
The 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 was in the upper half plane then and were never both simultaneously algebraic. Stronger results are now known, for example if is algebraic then the following three numbers are algebraically independent, and thus at least two of them transcendental:
The -expansion and moonshine.
Several remarkable properties of have to do with its -expansion (Fourier series expansion), written as a Laurent series in terms of (the square of the nome), which begins:
Note that has a simple pole at the cusp, so its -expansion has no terms below .
All the Fourier coefficients are integers, which results in several almost integers, notably Ramanujan's constant:
The asymptotic formula for the coefficient of is given by
as can be proved by the Hardy–Littlewood circle method.
Moonshine.
More remarkably, the Fourier coefficients for the positive exponents of 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 is the dimension of grade- part of the moonshine module, the first example being the Griess algebra, which has dimension 196,884, corresponding to the term . 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.
Alternate expressions.
We have
where and is the modular lambda function
a ratio of Jacobi theta functions , and is the square of the elliptic modulus . The value of is unchanged when is replaced by any of the six values of the cross-ratio:
The branch points of are at , so that is a Belyi function.
Expressions in terms of theta functions.
Define the nome and the Jacobi theta function,
from which one can derive the auxiliary theta functions. Let,
where and are alternative notations, and . Then,
for Weierstrass invariants , , and Dedekind eta function . We can then express in a form which can rapidly be computed.
Algebraic definition.
So far we have been considering 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 perform successive transformations to get the above equation into the standard form (note that this transformation can only be made when the characteristic of the field is not equal to 2 or 3). The resulting coefficients are:
where and . We also have the discriminant
The -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
Inverse function.
The inverse function of the -invariant can be expressed in terms of the hypergeometric function (see also the article Picard–Fuchs equation). Explicitly, given a number , to solve the equation for can be done in at least four ways.
Method 1: Solving the sextic in ,
where , and is the modular lambda function so the sextic can be solved as a cubic in . Then,
Failure to classify elliptic curves over other fields.
The formula_28-invariant is only sensitive to isomorphism classes of elliptic curves over the complex numbers, or more generally, an algebraically closed field. Over other fields there exist examples of elliptic curves whose formula_28-invariant is the same, but are non-isomorphic. For example, let formula_30 be the elliptic curves associated to the polynomialsformula_31both have formula_28-invariant formula_33. Then, the rational points of formula_34 can be computed asformula_35sinceformula_36. There are no rational solutions with formula_37. This can be shown using Cardano's formula to show that in that case the solutions to formula_38 are all irrational.
On the other hand, on the set of pointsformula_39 the equation for formula_34 gives the equation of formula_41Dividing the equation by formula_42 to eliminate the formula_43 solution givesformula_44which can be rewritten as the quadratic equationformula_45Using the quadratic formula, this givesformula_46hence it is a rational number.
If these curves are considered over formula_47, there is an isomorphism formula_48 sendingformula_49