Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The cyclosyntomic regulator of a number field

Tess Bouis, Quentin Gazda

Abstract

We construct a q-deformation of the p-adic regulator of a number field, called the cyclosyntomic regulator, building on the Habiro ring of Garoufalidis-Scholze-Wheeler-Zagier. The key new ingredient in our construction is a refinement of Sulyma's norm maps in prismatic cohomology, which interpolate between classical powers and Frobenius maps at various prime numbers p. Furthermore, we compute the values of the cyclosyntomic regulator at units of the form $1-ζ$, where $ζ$ is a root of unity.

The cyclosyntomic regulator of a number field

Abstract

We construct a q-deformation of the p-adic regulator of a number field, called the cyclosyntomic regulator, building on the Habiro ring of Garoufalidis-Scholze-Wheeler-Zagier. The key new ingredient in our construction is a refinement of Sulyma's norm maps in prismatic cohomology, which interpolate between classical powers and Frobenius maps at various prime numbers p. Furthermore, we compute the values of the cyclosyntomic regulator at units of the form , where is a root of unity.
Paper Structure (14 sections, 21 theorems, 127 equations)

This paper contains 14 sections, 21 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Norm maps on $q$-Witt vectors
  3. Review of Witt vectors
  4. Review of $q$-Witt vectors
  5. Alternative approach to $q$-ghost maps
  6. Norm maps on $q$-Witt vectors
  7. Cyclotomic rings
  8. The cyclotomic logarithm
  9. Frobenius maps on the Habiro ring of a number field
  10. The cyclotomic logarithm
  11. Cyclosyntomic regulator
  12. The cyclosyntomic cohomology of a number field
  13. The first Chern class
  14. The first $q$-polylogarithm

Key Result

Theorem 1

Let $K$ be a number field, and $R$ be the étale $\mathop{\mathrm{\mathbb{Z}}}\nolimits$-algebra $\mathcal{O}_K[\Delta_K^{-1}]$. For every prime number $p$, there is a natural $p$-adic realisation map to the syntomic cohomology of $R$ relative to the $q$-prism $(\mathop{\mathrm{\mathbb{Z}}}\nolimits_p[\![q-1]\!],[p]_q)$. Moreover, for every integer $d \geqslant 2$, there exists a natural cyclosynto

Theorems & Definitions (84)

  • Definition 1.1: Cyclosyntomic cohomology; see Definition \ref{['definitioncyclosyntomiccohomology']}
  • Theorem 1: First Chern class; see Sections \ref{['subsectionthecyclosyntomiccohomologyofanumberfield']} and \ref{['subsec:the-first-chern-class']}
  • Theorem 2: See Corollary \ref{['corollaryformulaforfirstChernclass']}
  • Theorem 3: First $q$-polylogarithm as cyclosyntomic Chern class; see Theorem \ref{['theoremmainLi1cyclosyntomic']}
  • Lemma 2.1: Dwork's lemma
  • Definition 2.2: Big Witt vectors
  • Remark 2.4
  • Remark 2.5
  • Definition 2.7: Big $q$-Witt vectors
  • Remark 2.10
  • ...and 74 more