Modus ponendo tollens

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at the identity element of A is a direct sum of one-dimensional modules). Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.

It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in algebraic number theory, allowing some features of the theory of cyclotomic fields to be carried over to wider areas of application.

David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science.

Example

An example of an elliptic curve with complex multiplication is

C/Z[i]θ

where Z[i] is the Gaussian integer ring, and θ is any non-zero complex number. Any such complex torus has the Gaussian integers as endomorphism ring. It is known that the corresponding curves can all be written as

Y² = 4X³ − aX,

having an order 4 automorphism sending

Y → −iY, X → −X

in line with the action of i on the Weierstrass elliptic functions.

This is a typical example of an elliptic curve with complex multiplication. Over the complex numbers, all elliptic curves with complex multiplication can be similarly constructed. That is, as quotients of the complex plane by some order in the ring of integers in an imaginary quadratic field.

Abstract theory of endomorphisms

The ring of endomorphisms of an elliptic curve can be of one of three forms:the integers Z; an order in an imaginary quadratic number field; or an order in a definite quaternion algebra over Q.

When the field of definition is a finite field, there are always non-trivial endomorphisms of an elliptic curve, coming from the Frobenius map, so the complex multiplication case is in a sense typical (and the terminology isn't often applied). But when the base field is a number field, complex multiplication is the exception. It is known that, in a general sense, the case of complex multiplication is the hardest to resolve for the Hodge conjecture.

Kronecker and abelian extensions

Kronecker first postulated that the values of elliptic functions at torsion points should be enough to generate all abelian extensions for imaginary quadratic fields, an idea that went back to Eisenstein in some cases, and even to Gauss. This became known as the Kronecker Jugendtraum; and was certainly what had prompted Hilbert's remark above, since it makes explicit class field theory in the way the roots of unity do for abelian extensions of the rational number field, via Shimura's reciprocity law.

Indeed, let K be an imaginary quadratic field with class field H. Let E be an elliptic curve with complex multiplication by the integers of K, defined over H. Then the maximal abelian extension of K is generated by the x-coordinates of the points of finite order on some Weierstrass model for E over H.

Many generalisations have been sought of Kronecker's ideas; they do however lie somewhat obliquely to the main thrust of the Langlands philosophy, and there is no definitive statement currently known.

Sample consequence

It is no accident that

$e^{\pi \sqrt{163}}=262537412640768743.99999999999925007\dots$

or equivalently,

$e^{\pi \sqrt{163}}=640320^3+743.99999999999925007\dots$

is so close to an integer. This remarkable fact is explained by the theory of complex multiplication, together with some knowledge of modular forms, and the fact that

$\mathbf{Z}\left[\frac{1+\sqrt{-163}}{2}\right]$

is a unique factorization domain.

Here $(1+\sqrt{-163})/2$ satisfies α² = α − 41. In general, S[α] denotes the set of all polynomial expressions in α with coefficients in S, which is the smallest ring containing α and S. Because α satisfies this quadratic equation, the required polynomials can be limited to degree one.

Alternatively,

$e^{\pi \sqrt{163}}=12^3(231^2-1{)}^3+743.99999999999925007\dots$

an internal structure due to certain Eisenstein series, and with similar simple expressions for the other Heegner numbers.

Singular moduli

The points of the upper half-plane τ which correspond to the period ratios of elliptic curves over the complex numbers with complex multiplication are precisely the imaginary quadratic numbers. The corresponding modular invariants j(τ) are the singular moduli, coming from an older terminology in which "singular" referred to the property of having non-trivial endomorphisms rather than referring to a singular curve.

The modular function j(τ) is algebraic on imaginary quadratic numbers τ: these are the only algebraic numbers in the upper half-plane for which j is algebraic.

If Λ is a lattice with period ratio τ then we write j(Λ) for j(τ). If further Λ is an ideal a in a ring of integers O of a quadratic imaginary field K then we write j(a) for the corresponding singular modulus. The values j(a) are then real algebraic integers, and generate the Hilbert class field H of K: the field extension degree [H:K] = h is the class number of K and the H/K is a Galois extension with Galois group isomorphic to the ideal class group of K. The class group acts on the values j(a) by [b] : j(a) → j(ab).

In particular, if K has class number one, then j(a) = j(O) is a rational integer: for example, j(Z[i]) = j(i) = 1728.

See also

• Abelian variety of CM-type, higher dimensions
• Algebraic Hecke character
• Heegner point
• Hilbert's twelfth problem
• Lubin–Tate formal group, local fields
• Drinfeld shtuka, global function field case

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Borel, A.; Chowla, S.; Herz, C. S.; Iwasawa, K.; Serre, J.-P. Seminar on complex multiplication. Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957-58. Lecture Notes in Mathematics, No. 21 Springer-Verlag, Berlin-New York, 1966
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• Complex multiplication from PlanetMath.org
• Examples of elliptic curves with complex multiplication from PlanetMath.org
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