## A (811)

### Abstract

conjecture on the torsion points of elliptic curves with the complex multiplication

### Citations

143 |
Quadratic differentials
- Strebel
- 1984
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Citation Context ...orphic differential on the complex torus S. (Note that ω is just a constant times dz, and hence its vertical trajectory structure is just a family of the parallel lines of a slope θ, see e.g. Strebel =-=[14]-=-, pp. 54–55.) Therefore, Φ T 2 consists of the equivalence classes of the non-singular measured foliations on the two-dimensional torus. It is well known (the Denjoy theory), that every such foliation... |

102 |
On the classification of inductive limits of sequences of semi-simple finite-dimensional algebras
- Elliott
- 1976
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Citation Context ...abelian monoid VC(A) defines the AF-algebra up to an isomorphism and is known as a dimension group of A. We shall use a standard dictionary existing between the AF-algebras and their dimension groups =-=[4]-=-. Instead of dealing with the AF-algebra GA, we shall work with its dimension group (G, G + A ), where G ∼ = Zn is the lattice and G + A is a positive cone inside the lattice given by a sequence of th... |

99 |
Quadratic differentials and foliations
- Hubbard, Masur
- 1979
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Citation Context ...measured foliation F. Note that, if F is a measured foliation with the simple zeroes (a generic case), then EF ∼ = R n − 0, while T(g) ∼ = R n , where n = 6g − 6 if g ≥ 2 and n = 2 if g = 1. Theorem (=-=[5]-=-) The restriction of p to EF defines a homeomorphism (an embedding) hF : EF → T(g). C. The Teichmüller space and measured foliations. The Hubbard-Masur result implies that the measured foliations para... |

69 |
The Arithmetic of Elliptic Curves, GTM 106
- Silverman
- 1986
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Citation Context ...ormulated. 1 Preliminaries For a convenience of the reader, we briefly review the AF-algebras, the elliptic curves and a Teichmüller functor between the two. An original account can be found in [10], =-=[12]-=- and [8], respectively. 1.1 AF-algebras A. The C ∗ -algebras. By the C ∗ -algebra one understands a noncommutative Banach algebra with an involution. Namely, a C ∗ -algebra A is an algebra over the co... |

25 |
An Introduction to K-Theory for C ∗ -Algebras
- Rørdam, Larsen, et al.
- 2000
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Citation Context ... are formulated. 1 Preliminaries For a convenience of the reader, we briefly review the AF-algebras, the elliptic curves and a Teichmüller functor between the two. An original account can be found in =-=[10]-=-, [12] and [8], respectively. 1.1 AF-algebras A. The C ∗ -algebras. By the C ∗ -algebra one understands a noncommutative Banach algebra with an involution. Namely, a C ∗ -algebra A is an algebra over ... |

24 |
multiplication and noncommutative geometry
- Manin, Real
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Citation Context ...y if θ′ ≡ θ mod GL(2, Z), Snen algebras Aθ, Aθ ′ 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 =-=[7]-=-. We shall denote by AθRM the Effros-Shen algebra with a real multiplication. 4a0 a1 ❜ ❜ ❜ ❜ � ❅❜ ❜ ❜ ❅ �❅ �❅ �❅ � . . . . . . Figure 1: The Effros-Shen algebra Aθ. B. The GA and AθRM are stably isom... |

13 |
Strong shift equivalence theory and the shift equivalence problem
- Wagoner
- 1999
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Citation Context ...nd (GA′, σA ′) are isomorphic if and only if the matrices A and A ′ are similar. Proof. By Theorem 6.4 of [1], (GA, σA) ∼ = (GA′, σA ′) if and only if the matrices A and A ′ are shift equivalent, see =-=[15]-=- 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... |

10 |
Basic Notions of Algebra
- Shafarevich
- 1999
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Citation Context ...up 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. =-=[11]-=-, p. 43. The result says the normal form of the module (in our case – over Q[x]) is independent of the particular choice of a matrix in the similarity class of A. The normal form, evaluated at x = 0, ... |

9 |
Progress in the theory of complex algebraic curves
- Eisenbud, Harris
- 1989
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Citation Context ... will be denoted by M1 := H/SL2(Z). (5) 5B. 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... |

5 | Remark on the rank of elliptic curves
- Nikolaev
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Citation Context ...enote by Etors(K). There exists a covariant functor, F, which maps isomorphic elliptic curves to the stably isomorphic AF-algebras [8]. In particular, F(ECM) = GA for a two by two hyperbolic matrix A =-=[9]-=-. We use the functor to formulate a conjecture on the group Etors(K). C. The structure of the paper. The note is organized as follows. The preliminary facts are brought together in section 1. The theo... |

3 | Symbolic Dynamic Systems, available at http://www.math.ku.dk/symbdyn - Jensen |

2 | On a Teichmüller functor between the categories of complex tori and the Effros-Shen algebras
- Nikolaev
- 2009
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Citation Context ...a finitely generated abelian group, whose torsion subgroup we shall denote by Etors(K). There exists a covariant functor, F, which maps isomorphic elliptic curves to the stably isomorphic AF-algebras =-=[8]-=-. In particular, F(ECM) = GA for a two by two hyperbolic matrix A [9]. We use the functor to formulate a conjecture on the group Etors(K). C. The structure of the paper. The note is organized as follo... |

1 |
Dimensions and C ∗ -Algebras, in: Conf. Board of the Math. Sciences, Regional conference series
- Effros
- 1981
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Citation Context ...one inside the lattice given by a sequence of the simplicial dimension 3groups: Z n A −→ Z n A −→ Z n A −→ . . . (2) There exists a natural automorphism, σA, of the dimension group (GA, G + A ), see =-=[1]-=-, p.37. It can be defined as follows. Let λA > 1 be the Perron-Frobenius eigenvalue and vA ∈ Rn + the corresponding eigenvector of the matrix A. It is known that G + A is defined by the inequality Zv(... |