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:

     (the modular discriminant)

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  (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:

If  is any CM point, that is, any element of an imaginary quadratic field with positive imaginary part (so that  is defined), then  is an algebraic integer. These special values are called singular moduli.
 The field extension  is abelian, that is, it has an abelian Galois group.
 Let  be the lattice in  generated by  It is easy to see that all of the elements of  which fix  under multiplication form a ring with units, called an order. The other lattices with generators  associated in like manner to the same order define the algebraic conjugates  of  over . Ordered by inclusion, the unique maximal order in  is the ring of algebraic integers of , and values of  having it as its associated order lead to unramified extensions of .

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,

for any of the six values of .

Method 2: Solving the quartic in ,

then for any of the four roots,

Method 3: Solving the cubic in ,

then for any of the three roots,

Method 4: Solving the quadratic in ,

then,

One root gives , and the other gives , but since , it makes no difference which  is chosen. The latter three methods can be found in Ramanujan's theory of elliptic functions to alternative bases.

The inversion applied in high-precision calculations of elliptic function periods even as their ratios become unbounded. A related result is the expressibility via quadratic radicals of the values of  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.

Pi formulas

The Chudnovsky brothers found in 1987,

which uses the fact that

For similar formulas, see the Ramanujan–Sato series.

Special values

The -invariant vanishes at the "corner" of the fundamental domain at

Here are a few more special values given in terms of the alternative notation  (only the first four of which are well known):

References

. Provides a very readable introduction and various interesting identities.

. Provides a variety of interesting algebraic identities, including the inverse as a hypergeometric series.
 Introduces the j-invariant and discusses the related class field theory.
. Includes a list of the 175 genus-zero modular functions.
. Provides a short review in the context of modular forms.
.

Category:Modular forms
Category:Elliptic functions
Category:Moonshine theory