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Adelic Eisenstein classes and divisibility properties of Stickelberger elements

Alexandros Galanakis, Michael Spieß

Abstract

Nori's Eisenstein cohomology classes and their integral refinements due to Beilinson, Kings and Levin can be used to obtain simple proofs of the rationality and integrality properties of special values of abelian $L$-functions of totally real fields. Here we introduce an adelic refinement of these constructions. This will be used to establish new divisibility properties of Stickelberger elements associated to abelian extensions of totally real fields.

Adelic Eisenstein classes and divisibility properties of Stickelberger elements

Abstract

Nori's Eisenstein cohomology classes and their integral refinements due to Beilinson, Kings and Levin can be used to obtain simple proofs of the rationality and integrality properties of special values of abelian -functions of totally real fields. Here we introduce an adelic refinement of these constructions. This will be used to establish new divisibility properties of Stickelberger elements associated to abelian extensions of totally real fields.
Paper Structure (22 sections, 49 theorems, 363 equations)

This paper contains 22 sections, 49 theorems, 363 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Locally polynomial functions and distributions
  4. Polynomial functions and polynomial distributions on lattices
  5. Locally polynomial functions and distributions
  6. Change of ${\mathscr L}$
  7. Lattice topology, sheaves and cohomology
  8. $({\mathscr L}, \widetilde{\Gamma})$--spaces and lattice topology
  9. Sheaves on $({\mathscr L}, \widetilde{\Gamma})$-spaces
  10. Stalks
  11. $\widetilde{\Gamma}$-stable closed subspaces
  12. Sheaf Cohomology
  13. Adelic Eisenstein Classes
  14. The sheaf of locally polynomial distributions
  15. Adelic Eisenstein Classes
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $k\in {\mathbb Z}_{\ge 0}$ and let ${\mathfrak p}\in S$ be a fixed place. Let $T$ be a finite set of nonarchimedean places of $F$ disjoint from $S$ such that the map galoisres is injective. Then we have

Theorems & Definitions (126)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • proof
  • ...and 116 more