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Arithmetic BF theory and the Cassels-Tate pairing

Jeehoon Park, Junyeong Park

Abstract

We give a systematic treatment of the arithmetic BF theory, introduced by Carlson and Kim. We observe that the Cassels-Tate pairing can be naturally interpreted as an arithmetic BF functional.

Arithmetic BF theory and the Cassels-Tate pairing

Abstract

We give a systematic treatment of the arithmetic BF theory, introduced by Carlson and Kim. We observe that the Cassels-Tate pairing can be naturally interpreted as an arithmetic BF functional.
Paper Structure (17 sections, 11 theorems, 221 equations)

This paper contains 17 sections, 11 theorems, 221 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on cup products
  3. Arithmetic BF theory for local fields
  4. Constructing $\mathbb{Q}/\mathbb{Z}$-torsors
  5. Local unramified BF functionals
  6. Classical arithmetic BF theory
  7. Spaces of fields
  8. Global arithmetic BF functionals
  9. Decomposition formula
  10. Quantum arithmetic BF theory
  11. Hermitian line bundles
  12. Partition functions
  13. Gluing formula
  14. The Cassels-Tate pairing for finite Galois modules
  15. Global arithmetic BF functional and the Cassels-Tate pairing
  16. ...and 2 more sections

Key Result

Lemma 2.3

The pairing AVduality induces a perfect pairing such that the following diagram is commutative:

Theorems & Definitions (51)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Example 2.4
  • Example 2.5
  • Example 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • ...and 41 more