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J-invariant

Index 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 special linear group defined on the upper half-plane of complex numbers. [1]

Table of Contents

  1. 75 relations: Algebraic integer, Algebraic number, Algebraically closed field, Almost integer, Arithmetic–geometric mean, Asymptotic analysis, Belyi's theorem, Cambridge University Press, Canadian Mathematical Bulletin, Chudnovsky algorithm, Complex analysis, Complex multiplication, Complex number, Conjugate element (field theory), Cross-ratio, Cube (algebra), Cubic equation, Cubic function, Cusp (singularity), David A. Cox, Dedekind eta function, Elliptic curve, Elliptic function, Felix Klein, Fourier series, Fundamental domain, Galois group, Graded ring, Graduate Texts in Mathematics, Griess algebra, Hardy–Ramanujan–Littlewood circle method, Heegner number, Holomorphic function, Hypergeometric function, Integer, Inverse function, John G. Thompson, John Horton Conway, John McKay (mathematician), Kurt Mahler, Laurent series, Mathematics, Modular equation, Modular form, Modular group, Modular lambda function, Monster group, Monster vertex algebra, Monstrous moonshine, Nome (mathematics), ... Expand index (25 more) »

  2. Elliptic functions
  3. Moonshine theory

Algebraic integer

In algebraic number theory, an algebraic integer is a complex number that is integral over the integers.

See J-invariant and Algebraic integer

Algebraic number

An algebraic number is a number that is a root of a non-zero polynomial (of finite degree) in one variable with integer (or, equivalently, rational) coefficients.

See J-invariant and Algebraic number

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.

See J-invariant and Algebraically closed field

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.

See J-invariant and Almost integer

Arithmetic–geometric mean

In mathematics, the arithmetic–geometric mean (AGM or agM) of two positive real numbers and is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. J-invariant and arithmetic–geometric mean are elliptic functions.

See J-invariant and Arithmetic–geometric mean

Asymptotic analysis

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.

See J-invariant and Asymptotic analysis

Belyi's theorem

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 three points only.

See J-invariant and Belyi's theorem

Cambridge University Press

Cambridge University Press is the university press of the University of Cambridge.

See J-invariant and Cambridge University Press

Canadian Mathematical Bulletin

The Canadian Mathematical Bulletin (Bulletin Canadien de Mathématiques) is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society.

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Chudnovsky algorithm

The Chudnovsky algorithm is a fast method for calculating the digits of pi, based on Ramanujan's pi formulae.

See J-invariant and Chudnovsky algorithm

Complex analysis

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.

See J-invariant and Complex analysis

Complex multiplication

In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. J-invariant and complex multiplication are elliptic functions.

See J-invariant and Complex multiplication

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^.

See J-invariant and Complex number

Conjugate element (field theory)

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.

See J-invariant and Conjugate element (field theory)

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.

See J-invariant and Cross-ratio

Cube (algebra)

In arithmetic and algebra, the cube of a number is its third power, that is, the result of multiplying three instances of together.

See J-invariant and Cube (algebra)

Cubic equation

In algebra, a cubic equation in one variable is an equation of the form ax^3+bx^2+cx+d.

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Cubic function

In mathematics, a cubic function is a function of the form f(x).

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Cusp (singularity)

In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction.

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David A. Cox

David Archibald Cox (born September 23, 1948, in Washington, D.C.) is a retired American mathematician, working in algebraic geometry.

See J-invariant and David A. Cox

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. J-invariant and Dedekind eta function are elliptic functions and modular forms.

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Elliptic curve

In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point.

See J-invariant and Elliptic curve

Elliptic function

In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. J-invariant and elliptic function are elliptic functions.

See J-invariant and Elliptic function

Felix Klein

Felix Christian Klein (25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations between geometry and group theory.

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Fourier series

A Fourier series is an expansion of a periodic function into a sum of trigonometric functions.

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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.

See J-invariant and Fundamental domain

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.

See J-invariant and Galois group

Graded ring

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.

See J-invariant and Graded ring

Graduate Texts in Mathematics

Graduate Texts in Mathematics (GTM) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.

See J-invariant and Graduate Texts in Mathematics

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 M as its automorphism group.

See J-invariant and Griess algebra

Hardy–Ramanujan–Littlewood circle method

In mathematics, the Hardy–Ramanujan–Littlewood circle method is a technique of analytic number theory.

See J-invariant and Hardy–Ramanujan–Littlewood circle method

Heegner number

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 has class number 1.

See J-invariant and Heegner number

Holomorphic function

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.

See J-invariant and Holomorphic function

Hypergeometric function

In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases.

See J-invariant and Hypergeometric function

Integer

An integer is the number zero (0), a positive natural number (1, 2, 3,...), or the negation of a positive natural number (−1, −2, −3,...). The negations or additive inverses of the positive natural numbers are referred to as negative integers.

See J-invariant and Integer

Inverse function

In mathematics, the inverse function of a function (also called the inverse of) is a function that undoes the operation of.

See J-invariant and Inverse function

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.

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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.

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John McKay (mathematician)

John K. S. McKay (18 November 1939 – 19 April 2022) was a British-Canadian mathematician and academic who worked at Concordia University, known for his discovery of monstrous moonshine, his joint construction of some sporadic simple groups, for the McKay conjecture in representation theory, and for the McKay correspondence relating certain finite groups to Lie groups.

See J-invariant and John McKay (mathematician)

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 numbers.

See J-invariant and Kurt Mahler

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.

See J-invariant and Laurent series

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See J-invariant and Mathematics

Modular equation

In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problems. J-invariant and modular equation are modular forms.

See J-invariant and Modular equation

Modular form

In mathematics, a modular form is a (complex) analytic function on the upper half-plane, \,\mathcal\,, that satisfies. J-invariant and modular form are modular forms.

See J-invariant and Modular form

Modular group

In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of matrices with integer coefficients and determinant 1. J-invariant and modular group are modular forms.

See J-invariant and Modular group

Modular lambda function

In mathematics, the modular lambda function λ(τ)\lambda(\tau) is not a modular function (per the Wikipedia definition), but every modular function is a rational function in \lambda(\tau). J-invariant and modular lambda function are elliptic functions and modular forms.

See J-invariant and Modular lambda function

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 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. J-invariant and monster group are moonshine theory.

See J-invariant and Monster group

Monster vertex algebra

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.

See J-invariant and Monster vertex algebra

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. J-invariant and monstrous moonshine are moonshine theory.

See J-invariant and Monstrous moonshine

Nome (mathematics)

In mathematics, specifically the theory of elliptic functions, the nome is a special function that belongs to the non-elementary functions. J-invariant and nome (mathematics) are elliptic functions.

See J-invariant and Nome (mathematics)

Order (ring theory)

In mathematics, an order in the sense of ring theory is a subring \mathcal of a ring A, such that.

See J-invariant and Order (ring theory)

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. J-invariant and Picard–Fuchs equation are elliptic functions and modular forms.

See J-invariant and Picard–Fuchs equation

Proceedings of the National Academy of Sciences of the United States of America

Proceedings of the National Academy of Sciences of the United States of America (often abbreviated PNAS or PNAS USA) is a peer-reviewed multidisciplinary scientific journal.

See J-invariant and Proceedings of the National Academy of Sciences of the United States of America

Projective 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 associated projective space P(V).

See J-invariant and Projective linear group

Quadratic equation

In mathematics, a quadratic equation is an equation that can be rearranged in standard form as ax^2 + bx + c.

See J-invariant and Quadratic equation

Quadratic field

In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.

See J-invariant and Quadratic field

Quartic function

In algebra, a quartic function is a function of the form where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial.

See J-invariant and Quartic function

Ramanujan–Sato series

In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as, to the form by using other well-defined sequences of integers s(k) obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients \tbinom, and A,B,C employing modular forms of higher levels.

See J-invariant and Ramanujan–Sato series

Ramification (mathematics)

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.

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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.

See J-invariant and Rational function

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold.

See J-invariant and Riemann surface

Sextic equation

In algebra, a sextic (or hexic) polynomial is a polynomial of degree six.

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Simon P. Norton

Simon Phillips Norton (28 February 1952 – 13 February 2019) Obituary: Daily Telegraph was a mathematician in Cambridge, England, who worked on finite simple groups.

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Special linear group

In mathematics, the special linear group of degree n over a commutative ring R is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.

See J-invariant and Special linear group

Springer Science+Business Media

Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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Srinivasa Ramanujan

Srinivasa Ramanujan (22 December 188726 April 1920) was an Indian mathematician.

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Straightedge and compass construction

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 idealized ruler and a pair of compasses.

See J-invariant and Straightedge and compass construction

Theodor Schneider

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.

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Theta function

In mathematics, theta functions are special functions of several complex variables. J-invariant and theta function are elliptic functions.

See J-invariant and Theta function

Transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients.

See J-invariant and Transcendental number

Upper half-plane

In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with. J-invariant and upper half-plane are modular forms.

See J-invariant and Upper half-plane

Weierstrass elliptic function

In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. J-invariant and Weierstrass elliptic function are elliptic functions and modular forms.

See J-invariant and Weierstrass elliptic function

Zero of a function

In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) vanishes at x; that is, the function f attains the value of 0 at x, or equivalently, x is a solution to the equation f(x).

See J-invariant and Zero of a function

Zeros and poles

In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable.

See J-invariant and Zeros and poles

1728 (number)

1728 is the natural number following 1727 and preceding 1729.

See J-invariant and 1728 (number)

See also

Elliptic functions

Moonshine theory

References

[1] https://en.wikipedia.org/wiki/J-invariant

Also known as Elliptic modular function, J invariant, Klein J-invariant, Klein absolute invariant, Klein modular function, Klein's absolute invariant, Klein's modular function.

, Order (ring theory), Picard–Fuchs equation, Proceedings of the National Academy of Sciences of the United States of America, Projective linear group, Quadratic equation, Quadratic field, Quartic function, Ramanujan–Sato series, Ramification (mathematics), Rational function, Riemann surface, Sextic equation, Simon P. Norton, Special linear group, Springer Science+Business Media, Srinivasa Ramanujan, Straightedge and compass construction, Theodor Schneider, Theta function, Transcendental number, Upper half-plane, Weierstrass elliptic function, Zero of a function, Zeros and poles, 1728 (number).