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Shafarevich-Tate groups of abelian varieties

Igor V. Nikolaev

Abstract

The Shafarevich-Tate group $W (\mathscr{A})$ measures the failure of the Hasse principle for an abelian variety $\mathscr{A}$. Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove that $W (\mathscr{A})\cong Cl~(Λ)\oplus Cl~(Λ)$ or $W (\mathscr{A})\cong \left(\mathbf{Z}/2^k\mathbf{Z}\right) \oplus Cl_{~\mathbf{odd}}~(Λ)\oplus Cl_{~\mathbf{odd}}~(Λ)$, where $Cl~(Λ)$ is the ideal class group of a ring $Λ$ associated to the K-theory of the non-commutative tori and $2^k $ divides the order of $Cl~(Λ)$. The case of elliptic curves with complex multiplication is considered in detail.

Shafarevich-Tate groups of abelian varieties

Abstract

The Shafarevich-Tate group measures the failure of the Hasse principle for an abelian variety . Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove that or , where is the ideal class group of a ring associated to the K-theory of the non-commutative tori and divides the order of . The case of elliptic curves with complex multiplication is considered in detail.

Paper Structure

This paper contains 12 sections, 6 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Abelian varieties
  4. Shafarevich-Tate group
  5. K-theory of $C^*$-algebras
  6. Higher dimensional non-commutative tori
  7. Definition and properties
  8. Crossed product $\mathcal{A}_{\Theta (k)}\rtimes_{L_v}\mathbf{Z}$
  9. Proof of theorem \ref{['thm1.1']}
  10. Complex multiplication
  11. Abelian varieties
  12. Elliptic curves

Key Result

Theorem \oldthetheorem

The Shafarevich-Tate group of an abelian variety $\mathscr{A}_K$ is a finite group given by the formulas:

Theorems & Definitions (18)

  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • ...and 8 more