Part of the series Modern Birkhäuser Classics pp 435483
Euler Systems
 V. A. KolyvaginAffiliated withSteklov Institute
Abstract
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 MordellWeil groups and ShafarevichTate 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 nondegenerate (in its (l, ψ)component) determines the structure of the ψcomponent 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.
 Title
 Euler Systems
 Book Title
 The Grothendieck Festschrift
 Book Subtitle
 A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck
 Pages
 pp 435483
 Copyright
 1990
 DOI
 10.1007/9780817645755_11
 Print ISBN
 9780817645670
 Online ISBN
 9780817645755
 Series Title
 Modern Birkhäuser Classics
 Publisher
 Birkhäuser Boston
 Copyright Holder
 Birkhäuser Boston
 Additional Links
 Topics
 eBook Packages
 Editors

 Pierre Cartier ^{(1)}
 Nicholas M. Katz ^{(2)}
 Yuri I. Manin ^{(3)}
 Luc Illusie ^{(4)}
 Gérard Laumon ^{(4)}
 Kenneth A. Ribet ^{(5)}
 Editor Affiliations

 1. Institut des Hautes Études Scientifiques
 2. Department of Mathematics, Princeton University
 3. MaxPlanck Institut für Mathematik
 4. Département de Mathématiques, Université de ParisSud
 5. Department of Mathematics, University of California
 Authors

 V. A. Kolyvagin ^{(6)}
 Author Affiliations

 6. Steklov Institute, Vavilova 42, Moscow, U.S.S.R., 117966 GSP1
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