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The Grothendieck Festschrift: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck
Pierre Cartier ; Nicholas M. Katz ; Yuri I. Manin ; Luc Illusie ; Gérard Laumon ; Kenneth A. Ribet (eds.)
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Disponibilidad
| Institución detectada | Año de publicación | Navegá | Descargá | Solicitá |
|---|---|---|---|---|
| No detectada | 2007 | SpringerLink |
Información
Tipo de recurso:
libros
ISBN impreso
978-0-8176-64567-0
ISBN electrónico
978-0-8176-4575-5
Editor responsable
Springer Nature
País de edición
Reino Unido
Fecha de publicación
2007
Información sobre derechos de publicación
© Birkhäuser Boston 2007
Cobertura temática
Tabla de contenidos
Euler Systems
V. A. Kolyvagin
In this paper we study Euler systems defined by the characterizing condition AX1, perhaps with the addition of other conditions (AX2 and AX3 systems, see §1). Our main purpose is to apply them to determine the structure of the class groups of certain algebraic number fields R , and the Mordell-Weil groups and Shafarevich-Tate groups of Weil curves. In the case of the class group Cl of a field R , Theorem 7 of §2 says that, if the Galois group G of R is annihilated by l − 1, where l is a rational prime, and if ψ is a homomorphism from G to the group of ( l — l)-th roots of unity in Z _l, then (under certain conditions on R and ψ ) any Euler system associated to R which is non-degenerate (in its (l, ψ )-component) determines the structure of the ψ -component of Cl ⊗ Z _l, i.e., it determines the set of integers n _i, n _i ≥ n _ i +1, such that $$ (Cl \otimes Z_l )_\psi \simeq \sum\nolimits_{i = 1}^{i_0 } {Z/l^{n,} } $$ as an abelian group. Theorem 7 also shows how the Euler system determines bases of ( Cl ⊗ Z _l)_ψ consisting of prime divisor classes, the expansions of certain prime divisor classes in these bases, and also certain representations of primary numbers. For example, this holds for the cyclotomic field K _l = Q (ζ_l) (see below) with odd characters ψ and the system of Gauss sums, or with even characters ψ and the system of cyclotomic units. As a corollary we find that the order of X = ( Cl ⊗ Z _l)_ψ is bounded from above by the predicted explicit order [ X ]?; and this, along with formulas for the class number, enables us in several cases (cyclotomic fields, fields which are abelian extensions of an imaginary quadratic field) to prove that [ X ] and [ X ]? are equal.
Palabras clave: Elliptic Curve; Prime Divisor; Multiplicative Group; Infinite Order; Euler System.
Pp. 435-483
Descent for Transfer Factors
R. Langlands; D. Shelstad
In [ I ] we introduced the notion of transfer from a group over a local field to an associated endoscopic group, but did not prove its existence, nor do we do so in the present paper. Nonetheless we carry out what is probably an unavoidable step in any proof of existence: reduction to a local statement at the identity in the centralizer of a semisimple element, a favorite procedure of Harish Chandra that he referred to as descent.
Palabras clave: Conjugacy Class; Galois Group; Transfer Factor; Dynkin Diagram; Borel Subgroup.
Pp. 485-563