## Operator (811)

### BibTeX

@MISC{Nikolaev811operator,

author = {Igor Nikolaev},

title = {Operator},

year = {811}

}

### OpenURL

### Abstract

algebras and torsion points of elliptic curves with complex multiplication

### Citations

99 |
Quadratic differentials and foliations
- Hubbard, Masur
- 1979
(Show Context)
Citation Context ... a Teichmüller space of the torus. The moduli space of the torus will be denoted by M1 := H/SL2(Z). Let Ω be a space of the closed differentials on torus. By a fundamental result of Hubbard and Masur =-=[4]-=-, there exists a homeomorphism hω0 : H → Ω, which depends on an initial differential ω0 (a measured foliation) in the space Ω. One can normalize hω0 to be unique by choosing ω0, such that lim µ→∞ h−1 ... |

69 |
The Arithmetic of Elliptic Curves, GTM 106
- Silverman
- 1986
(Show Context)
Citation Context ...spectively. The elliptic curves E1 2 , E2 and E−1 are pairwise isomorphic (over Q) and we shall fix ECM = E−1 : y2 = x3 −x. It is well known that in this case Etors ∼ ( ) 5 2 = Z2 ⊕ Z2, see p. 311 of =-=[11]-=-. Let A = and F(E−1) = GA. Let 2 1 us calculate Abx−1(GA): A − I = ( ) 4 2 ∼ 2 0 ( ) 2 4 ∼ 0 2 ( ) 2 0 , (22) 0 2 where ∼ are the elementary transformations, see section 2, item (iv). Thus, Abx−1(GA) ... |

29 | of Elliptic Curves - Rubin, Silverberg |

25 |
An Introduction to K-Theory for C ∗ -Algebras
- Rørdam, Larsen, et al.
- 2000
(Show Context)
Citation Context ...gument. (i) For a convenience of the reader with background in the elliptic curves, let us recall some useful definitions from the theory of operator algebras. An introductory account can be found in =-=[8]-=-. By the C ∗ -algebra (an operator algebra) one understands a noncommutative Banach algebra with an involution. Namely, a C ∗ -algebra A is an algebra over the field of complex numbers C with a norm a... |

24 |
multiplication and noncommutative geometry
- Manin, Real
(Show Context)
Citation Context ...stably isomorphic if and only if θ ′ ≡ θ mod GL(2, Z), i.e. θ ′ = (aθ+b) / (cθ+d), where a, b, c, d ∈ Z and ad−bc = ±1. The Aθ is said to have a real multiplication, if θ is a quadratic irrationality =-=[6]-=-. We shall denote by AθRM the Effros-Shen algebra with a real multiplication. B. The GA and AθRM are stably isomorphic. Let θ be a quadratic irrationality. By the Lagrange theorem, the continued fract... |

18 |
Approximately finite C ∗ -algebras and continued fractions
- Effros, Shen
- 1980
(Show Context)
Citation Context ...li diagram: a0 a1 ❜ ❜ ❜ ❜ � ❅❜ ❜ ❜ ❅ �❅ �❅ �❅ � . . . . . . Figure 1: The Effros-Shen algebra Aθ. where ai indicate the multiplicity of the edges of the graph. We shall call Aθ an Effros-Shen algebra =-=[2]-=-. It is known that the Effros-Snen algebras Aθ, Aθ ′ are stably isomorphic if and only if θ ′ ≡ θ mod GL(2, Z), i.e. θ ′ = (aθ+b) / (cθ+d), where a, b, c, d ∈ Z and ad−bc = ±1. The Aθ is said to have ... |

13 |
Strong shift equivalence theory and the shift equivalence problem
- Wagoner
- 1999
(Show Context)
Citation Context ...rphic if and only if the matrices A and A ′ are similar. Proof. By Theorem 6.4 of [1], (G, G + A ) and (G, G+ A ′) are order-isomorphic if and only if the matrices A and A ′ are shift equivalent, see =-=[13]-=- for a definition of the shift equivalence. On the other hand, since the matrices A and A ′ are unimodular, the shift equivalence between A and A ′ coincides with a similarity of the matrices in the g... |

9 |
Progress in the theory of complex algebraic curves
- Eisenbud, Harris
- 1989
(Show Context)
Citation Context ...c Effros-Shen algebras. Proof. See [7]. □ D. The universal family of curves over M1. Recall that by a family of smooth projective curves one understands a map f : X → B whose fibers are smooth curves =-=[3]-=-. The family f : X → B is universal if every other family f ′ : X ′ → B ′ is a pull back of f along some map φ : B ′ → B. We wish to assign to each point p ∈ M1 a smooth elliptic curve, so that the re... |

9 |
Basic Notions of Algebra
- Shafarevich
- 1990
(Show Context)
Citation Context ...p GLn(Z), i.e. A ′ = BAB −1 for a B ∈ GLn(Z). The rest of the proof follows from the structure theorem for the finitely generated modules given by the matrix A over a principal ideal domain, see e.g. =-=[10]-=-, p. 43. The result says the normal form of the module (in our case – over Z[x]) is independent of the particular choice of a matrix in the similarity class of A. The normal form, evaluated at x = 0, ... |

5 | Remark on the rank of elliptic curves
- Nikolaev
(Show Context)
Citation Context ... elliptic curve defined over an algebraic number field K. By the Mordell-Weil theorem, E(K) is a finitely generated abelian group, whose torsion subgroup we shall denote by Etors(K). It is known from =-=[7]-=- that there exists a covariant functor, F, defined on the space of elliptic curves, which maps isomorphic elliptic curves to the stably isomorphic AF-algebras. In particular, F(ECM) = GA for a two by ... |

3 |
Symbolic Dynamic Systems, available at http://www.math.ku.dk/symbdyn
- Jensen
(Show Context)
Citation Context ...he following table: R = Ok ECM j(ECM) Etors ( A ) Abx−1(GA) 5 2 Z2 ⊕ Z2 2 1 k = Q( √ −1) y 2 = x 3 − x 1728 Z2 ⊕ Z2 Acknowledgments. An extensive use of the online software created by Ole Lund Jensen =-=[5]-=- is kindly acknowledged. 2 The data does not include the verification of the rank conjecture, which is out of scope of the present note. However, for the sake of completeness, let us make the pertinen... |