![KleinInvariantJ](https://upload.wikimedia.org/wikipedia/commons/3/37/KleinInvariantJ.jpg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
's -invariant or function, regarded as a function of a
complex variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
, is a
modular function
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
of weight zero for defined on the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to t ...
of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. It is the unique such function which is
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
away from a simple pole at the
cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
such that
:
Rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s of are modular, and in fact give all modular functions. Classically, the -invariant was studied as a parameterization of
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s over , but it also has surprising connections to the symmetries of the
Monster group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
246320597611213317192329314147 ...
(this connection is referred to as
monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...
).
Definition
![J-inv-phase](https://upload.wikimedia.org/wikipedia/commons/8/86/J-inv-phase.jpeg)
The -invariant can be defined as a function on the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to t ...
:
with the third definition implying
can be expressed as a
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
, also since
1728 =
.
The given functions are the
modular discriminant
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the ...
,
Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string t ...
, and modular invariants,
:
:
where
,
are
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
,
:
and
,
are
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generaliz ...
,
:
and
(the square of the
nome). The -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,
:
For example, using the definitions above and
, then the Dedekind eta function
has the
exact value,
:
implying the
transcendental numbers
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes o ...
,
:
but yielding the
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
(in fact, an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
),
:
In general, 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
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the ...
).
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
![ModularGroup-FundamentalDomain](https://upload.wikimedia.org/wikipedia/commons/a/a8/ModularGroup-FundamentalDomain.svg)
It can be shown that is a
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
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
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
, which we may identify with the
projective special linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...
.
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
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
for , which consists of values for satisfying the conditions
:
The function when restricted to this region still takes on every value in the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s 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 mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
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
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
with positive imaginary part (so that is defined), then is an
algebraic integer
In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
. These special values are called
singular moduli.
* The field extension is abelian, that is, it has an abelian
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
.
* 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 conjugate
In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element , over a field extension , are the roots of the minimal polynomial of over . Conjugate elements are commonly called conju ...
s 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 extension
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
s of .
These classical results are the starting point for the theory of
complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
.
Transcendence properties
In 1937
Theodor Schneider __NOTOC__
Theodor Schneider (7 May 1911, Frankfurt am Main – 31 October 1988, Freiburg im Breisgau) was a German mathematician, best known for providing proof of what is now known as the Gelfond–Schneider theorem.
Schneider studied from 19 ...
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
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
but not imaginary quadratic then is transcendental.
The function has numerous other transcendental properties.
Kurt Mahler
Kurt Mahler FRS (26 July 1903, Krefeld, Germany – 25 February 1988, Canberra, Australia) was a German mathematician who worked in the fields of transcendental number theory, diophantine approximation, ''p''-adic analysis, and the geometry of ...
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
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
expansion), written as a
Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
in terms of , which begins:
:
Note that has a
simple pole
In complex analysis (a branch of mathematics), a pole is a certain type of singularity (mathematics), singularity of a complex-valued function of a complex number, complex variable. In some sense, it is the simplest type of singularity. Technical ...
at the cusp, so its -expansion has no terms below .
All the Fourier coefficients are integers, which results in several
almost integer
In recreational mathematics, an almost integer (or near-integer) is any number that is not an integer but is very close to one. Almost integers are considered interesting when they arise in some context in which they are unexpected.
Almost inte ...
s, notably
Ramanujan's constant In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factoriza ...
:
:
.
The
asymptotic formula
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as bec ...
for the coefficient of is given by
:
,
as can be proved by the
Hardy–Littlewood circle method
In mathematics, the Hardy–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy and J. E. Littlewood, who developed it in a series of papers on Waring's problem.
History
The initial idea is usually at ...
.
Moonshine
More remarkably, the Fourier coefficients for the positive exponents of are the dimensions of the graded part of an infinite-dimensional
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
representation of the
monster group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
246320597611213317192329314147 ...
called the ''
moonshine module
The monster vertex algebra (or moonshine module) is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R. Borcherds used it to prove the monstrous moonshine conjectures, by apply ...
'' – specifically, the coefficient of is the dimension of grade- part of the moonshine module, the first example being the
Griess algebra In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fisc ...
, which has dimension 196,884, corresponding to the term . This startling observation, first made by
John McKay, was the starting point for
moonshine theory
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 197 ...
.
The study of the Moonshine conjecture led
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
and
Simon P. Norton to look at the genus-zero modular functions. If they are normalized to have the form
:
then
John G. Thompson
John Griggs Thompson (born October 13, 1932) is an American mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992, and the Abel Prize in 2008.
...
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
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
, and is the square of the elliptic modulus .
[Chandrasekharan (1985) p.108] The value of is unchanged when is replaced by any of the six values of the
cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, th ...
:
:
The branch points of are at , so that is a
Belyi function
In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve ''C'', defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at t ...
.
Expressions in terms of theta functions
Define the
nome and the
Jacobi theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
,
:
from which one can derive the
auxiliary theta functions. Let,
:
where and are alternative notations, and . Then we have the for
modular invariants , ,
:
and modular discriminant,
:
with
Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string t ...
. The can then be rapidly 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
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
:
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
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X\t ...
of the -invariant can be expressed in terms of the
hypergeometric function (see also the article
Picard–Fuchs equation In mathematics, the Picard–Fuchs equation, named after Émile Picard and Lazarus Fuchs, is a linear ordinary differential equation whose solutions describe the periods of elliptic curves.
Definition
Let
:j=\frac
be the j-invariant with g_2 and ...
). Explicitly, given a number , to solve the equation for can be done in at least four ways.
Method 1: Solving the
sextic
In algebra, a sextic (or hexic) polynomial is a polynomial of degree six.
A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More precis ...
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 , where is the
arithmetic–geometric mean.
[The equality holds if the arithmetic–geometric mean of ]complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
(such that ) is defined as follows: Let , , , where the signs are chosen such that for all . If , the sign is chosen such that . Then . When are positive real (with ), this definition coincides with the usual definition of the arithmetic–geometric mean for positive real numbers. Se
The Arithmetic-Geometric Mean of Gauss
by David A. Cox
David Archibald Cox (born September 23, 1948 in Washington, D.C.) is a retired American mathematician, working in algebraic geometry.
Cox graduated from Rice University with a bachelor's degree in 1970 and his Ph.D. in 1975 at Princeton Universit ...
.
Method 2: Solving the
quartic in ,
:
then for any of the four
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
,
:
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
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
to alternative bases.
The inversion is 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
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
). The latter result is hardly evident since the
modular equation for of order 2 is cubic.
Pi formulas
The
Chudnovsky brothers
David Volfovich Chudnovsky (born January 22, 1947 in Kyiv) and Gregory Volfovich Chudnovsky (born April 17, 1952 in Kyiv) are Ukrainian-born American mathematicians and engineers known for their world-record mathematical calculations and developing ...
found in 1987,
[.]
:
a proof of which uses the fact that
:
For similar formulas, see the
Ramanujan–Sato series In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as,
:\frac = \frac \sum_^\infty \frac \frac
to the form
:\frac = \sum_^\infty s(k) \frac
by using other well-defined sequences of integers s(k) obeying a cer ...
.
Special values
The -invariant vanishes at the "corner" of the
fundamental domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
:
:
Here are a few more special values given in terms of the alternative notation , the first four well known:
:
Failure to classify elliptic curves over other fields
The
-invariant is only sensitive to isomorphism classes of elliptic curves over the complex numbers, or more generally, an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
. Over other fields there exist examples of elliptic curves whose
-invariant is the same, but are non-isomorphic. For example, let
be the elliptic curves associated to the polynomials
both having
-invariant
. Then, the rational points of
can be computed as:
since
There are no rational solutions with
. This can be shown using
Cardano's formula
In algebra, a cubic equation in one variable is an equation of the form
:ax^3+bx^2+cx+d=0
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the ...
to show that in that case the solutions to
are all irrational.
On the other hand, on the set of points
the equation for
becomes
. Dividing by
to eliminate the
solution, the quadratic formula gives the rational solutions:
If these curves are considered over
, there is an isomorphism
sending
References
Notes
Other
*. 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.
*{{citation, first=Theodor, last=Schneider, author-link=Theodor Schneider, title=Arithmetische Untersuchungen elliptischer Integrale, journal=Math. Annalen, volume=113, year=1937, pages=1–13, mr=1513075, doi=10.1007/BF01571618, s2cid=121073687.
Modular forms
Elliptic functions
Moonshine theory