Elliptic unit

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In mathematics, elliptic units are certain units of abelian extensions of imaginary quadratic fields constructed using singular values of modular functions, or division values of elliptic functions. They were introduced by Gilles Robert in 1973, and were used by John Coates and Andrew Wiles in their work on the Birch and Swinnerton-Dyer conjecture. Elliptic units are an analogue for imaginary quadratic fields of cyclotomic units. They form an example of an Euler system.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of an abelian group.

In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice. Such a doubly periodic function cannot be holomorphic, as it would then be a bounded entire function, and by Liouville's theorem every such function must be constant. In fact, an elliptic function must have at least two poles in a fundamental parallelogram, as it is easy to show using the periodicity that a contour integral around its boundary must vanish, implying that the residues of all simple poles must cancel.

A system of elliptic units may be constructed for an elliptic curve E with complex multiplication by the ring of integers R of an imaginary quadratic field F. For simplicity we assume that F has class number one. Let a be an ideal of R with generator α. For a Weierstrass model of E, define

Elliptic curve an algebraic curve of genus 1 equipped with a basepoint

In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form

In mathematics, complex multiplication (CM) 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. 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.

In mathematics, the ring of integers of an algebraic number field K is the ring of all integral elements contained in K. An integral element is a root of a monic polynomial with integer coefficients, xn + cn−1xn−1 + … + c0. This ring is often denoted by OK or . Since any integer number belongs to K and is an integral element of K, the ring Z is always a subring of OK.

where Δ is the discriminant and x is the X-coordinate on the Weierstrass model. The function Θ is independent of the choice of model, and is defined over the field of definition of E.

Let b be an ideal of R coprime to a and Q an R-generator of the b-torsion. Then Θa(Q) is defined over the ray class field K(b), and if b is not a prime power then Θa(Q) is a global unit: if b is a power of a prime p then Θa(Q) is a unit away from p.

In mathematics, a ray class field is an abelian extension of a global field associated with a ray class group of ideal classes or idele classes. Every finite abelian extension of a number field is contained in one of its ray class fields.

The function Θa satisfies a distribution relation for b = (β) coprime to a:

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References

Karl Rubin American mathematician

Karl Cooper Rubin is an American mathematician at University of California, Irvine as Thorp Professor of Mathematics. His research interest is in elliptic curves. He was the first mathematician (1986) to show that some elliptic curves over the rationals have finite Tate-Shafarevich groups. It is widely believed that these groups are always finite.

International Standard Book Number Unique numeric book identifier

The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.