1. Definition

2. The fundamental region

3. Class field theory and {{mvar|j}}

4. Transcendence properties

5. The {{mvar|q}}-expansion and moonshine
    Moonshine 

6. Inverse function

7. Pi formulas

8. Special values

9. References

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 {{math|SL(2, Z)}} 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 {{mvar|j}} are modular, and in fact give all modular functions. Classically, the {{mvar|j}}-invariant was studied as a parameterization of elliptic curves over {{math|C}}, but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).

Definition

While the {{mvar|j}}-invariant can be defined purely in terms of certain infinite sums (see {{math|g₂, g₃}} below), these can be motivated by considering isomorphism classes of elliptic curves. Every elliptic curve {{mvar|E}} over {{math|C}} is a complex torus, and thus can be identified with a rank 2 lattice; i.e., two-dimensional lattice of {{math|C}}. This is done by identifying opposite edges of each parallelogram in the lattice. However, multiplying the lattice by a complex number, which corresponds to rotating and scaling the lattice, preserves the isomorphism class of the elliptic curve, so we can always arrange for the lattice to be generated by {{math|1}} and some {{mvar|τ}} in {{math|H}} (where {{math|H}} is the upper half-plane). Conversely, if we define

then this lattice corresponds to the elliptic curve over {{math|C}} defined by {{math|y² {{=}} 4x³ − g₂x − g₃}} via the Weierstrass elliptic functions. Then the {{mvar|j}}-invariant is defined as

where the modular discriminant {{math|Δ}} is

It can be shown that {{math|Δ}} is a modular form of weight twelve, and {{math|g₂}} one of weight four, so that its third power is also of weight twelve. Thus their quotient, and therefore {{mvar|j}}, is a modular function of weight zero, in particular a holomorphic function {{math|H → C}} invariant under the action of {{math|SL(2, Z)}}. As explained below, {{mvar|j}} is surjective, which means that it gives a bijection between isomorphism classes of elliptic curves over {{math|C}} and the complex numbers.

The fundamental region

The two transformations {{math|τ → τ + 1}} and {{math|τ → -τ⁻¹}} together generate the special linear group {{math|SL(2, Z)}}. Quotienting out by its centre {{mvar| { ± I }}} yields the modular group, which we may identify with the projective special linear group {{math|PSL(2, Z)}}. By a suitable choice of transformation belonging to this group,

we may reduce {{mvar|τ}} to a value giving the same value for {{mvar|j}}, and lying in the fundamental region for {{mvar|j}}, which consists of values for {{mvar|τ}} satisfying the conditions

The function {{math|j(τ)}} when restricted to this region still takes on every value in the complex numbers {{math|C}} exactly once. In other words, for every {{mvar|c}} in {{math|C}}, there is a unique τ in the fundamental region such that {{math|c {{=}} j(τ)}}. Thus, {{mvar|j}} has the property of mapping the fundamental region to the entire complex plane.

As a Riemann surface, the fundamental region has genus {{math|0}}, and every (level one) modular function is a rational function in {{mvar|j}}; and, conversely, every rational function in {{mvar|j}} is a modular function. In other words, the field of modular functions is {{math|C(j)}}.

Class field theory and {{mvar|j}}

The {{mvar|j}}-invariant has many remarkable properties:

• If {{mvar|τ}} is any CM point, that is, any element of an imaginary quadratic field with positive imaginary part (so that {{mvar|j}} is defined), then {{math|j(τ)}} is an algebraic integer.^[1] These special values are called singular moduli.
• The field extension {{math|Q[j(τ), τ]/Q(τ)}} is abelian, that is, it has an abelian Galois group.
• Let {{math|Λ}} be the lattice in {{math|C}} generated by {{math|{1, τ}.}} It is easy to see that all of the elements of {{math|Q(τ)}} which fix {{math|Λ}} under multiplication form a ring with units, called an order. The other lattices with generators {{math|{1, τ′},}} associated in like manner to the same order define the algebraic conjugates {{math|j(τ′)}} of {{math|j(τ)}} over {{math|Q(τ)}}. Ordered by inclusion, the unique maximal order in {{math|Q(τ)}} is the ring of algebraic integers of {{math|Q(τ)}}, and values of {{mvar|τ}} having it as its associated order lead to unramified extensions of {{math|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 {{mvar|τ}} is a quadratic irrational number in the upper half plane then {{math|j(τ)}} is an algebraic integer. In addition he proved that if {{mvar|τ}} is an algebraic number but not imaginary quadratic then {{math|j(τ)}} is transcendental.

The {{mvar|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 {{mvar|τ}} was in the upper half plane then {{math|exp(2πiτ)}} and {{math|j(τ)}} were never both simultaneously algebraic. Stronger results are now known, for example if {{math|exp(2πiτ)}} is algebraic then the following three numbers are algebraically independent, and thus at least two of them transcendental:

The {{mvar|q}}-expansion and moonshine

Several remarkable properties of {{mvar|j}} have to do with its {{mvar|q}}-expansion (Fourier series expansion), written as a Laurent series in terms of {{math|q {{=}} exp(2πiτ)}} (the square of the nome), which begins:

Note that {{mvar|j}} has a simple pole at the cusp, so its {{mvar|q}}-expansion has no terms below {{math|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 {{math|qⁿ}} is given by

,

as can be proved by Hardy–Littlewood circle method.^[2][3]

Moonshine

More remarkably, the Fourier coefficients for the positive exponents of {{mvar|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 {{math|qⁿ}} is the dimension of grade//Griess algebra">Griess algebra, which has dimension {{math|196,884}}, corresponding to the term {{math|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.^{[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 {{mvar|j}}-invariant can be expressed in terms of the hypergeometric function {{math|₂F₁}} (see also the article Picard–Fuchs equation). Explicitly, given a number {{mvar|N}}, to solve the equation {{math|j(τ) {{=}} N}} for {{mvar|τ}} can be done in at least four ways.

Method 1: Solving the sextic in {{mvar|λ}},

where {{math|x {{=}} λ(1−λ)}}, and {{mvar|λ}} is the modular lambda function so the sextic can be solved as a cubic in {{mvar|x}}. Then,

for any of the six values of {{mvar|λ}}.

Method 2: Solving the quartic in {{mvar|γ}},

then for any of the four roots,

Method 3: Solving the cubic in {{mvar|β}},

then for any of the three roots,

Method 4: Solving the quadratic in {{mvar|α}},

then,

One root gives {{mvar|τ}}, and the other gives {{math|-1/τ}}, but since {{math|j(τ) {{=}} j(-1/τ)}}, it makes no difference which {{mvar|α}} 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 {{mvar|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.

Pi formulas

The Chudnovsky brothers found in 1987,^[9]

which uses the fact that . For similar formulas, see the Ramanujan–Sato series.

Special values

The {{mvar|j}}-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):

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]

References

• {{citation|first=Tom M.|last=Apostol|authorlink=Tom M. Apostol|title=Modular functions and Dirichlet Series in Number Theory|mr=0422157|year=1976|publisher=Springer-Verlag|series=Graduate Texts in Mathematics|volume=41|location=New York}}. Provides a very readable introduction and various interesting identities.
  • {{citation|first=Tom M.|last=Apostol|authorlink=Tom M. Apostol|title=Modular functions and Dirichlet Series in Number Theory|volume=41|edition=2nd |year=1990 |isbn=978-0-387-97127-8|mr=1027834|doi=10.1007/978-1-4612-0999-7|series=Graduate Texts in Mathematics}}
• {{citation|doi=10.4153/CMB-1999-050-1|first1=Bruce C.|last1=Berndt|author1-link=Bruce C. Berndt|first2=Heng Huat|last2=Chan|title=Ramanujan and the modular j-invariant|url=http://www.journals.cms.math.ca/cgi-bin/vault/public/view/berndt7376/body/PDF/berndt7376.pdf|journal=Canadian Mathematical Bulletin|volume=42|issue=4|year=1999|pages=427–440|mr=1727340|deadurl=yes|archiveurl=https://web.archive.org/web/20070929105913/http://www.journals.cms.math.ca/cgi-bin/vault/public/view/berndt7376/body/PDF/berndt7376.pdf|archivedate=2007-09-29|df=}}. Provides a variety of interesting algebraic identities, including the inverse as a hypergeometric series.
• {{citation|first=David A.|last=Cox|authorlink=David A. Cox|title=Primes of the Form x^2 + ny^2: Fermat, Class Field Theory, and Complex Multiplication|mr=1028322|year=1989|publisher= Wiley-Interscience Publication, John Wiley & Sons Inc.|location=New York}} Introduces the j-invariant and discusses the related class field theory.
• {{citation|first1=John Horton|last1=Conway|author1-link=John Horton Conway|first2=Simon|last2=Norton|author2-link=Simon P. Norton|title=Monstrous moonshine|journal=Bulletin of the London Mathematical Society|volume=11|issue=3|year=1979|pages=308–339|mr=0554399|doi=10.1112/blms/11.3.308}}. Includes a list of the 175 genus-zero modular functions.
• {{citation|first=Robert A.|last=Rankin|authorlink=Robert Alexander Rankin|title=Modular forms and functions|year=1977|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-21212-0|mr=0498390}}. Provides a short review in the context of modular forms.
• {{citation|first=Theodor|last=Schneider|authorlink=Theodor Schneider|title=Arithmetische Untersuchungen elliptischer Integrale|journal=Math. Annalen|volume=113|year=1937|pages=1–13|mr=1513075|doi=10.1007/BF01571618}}.

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