The Grothendieck Festschrift pp 435-483 | Cite as

# Euler Systems

## 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 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.

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