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On the Drinfeld formal group

Deven Manam

Abstract

We identify Drinfeld's formal group on the prismatization of $\mathrm{Spf}\,\mathbb{Z}_p$ with a formal group arising from homotopy theory, given locally by the Quillen formal group of a decompleted variant of topological periodic cyclic homology. We also prove that this formal group extends to the syntomization of $\mathrm{Spf}\,\mathbb{Z}_p$, and that it admits a certain algebraization conjectured by Drinfeld.

On the Drinfeld formal group

Abstract

We identify Drinfeld's formal group on the prismatization of with a formal group arising from homotopy theory, given locally by the Quillen formal group of a decompleted variant of topological periodic cyclic homology. We also prove that this formal group extends to the syntomization of , and that it admits a certain algebraization conjectured by Drinfeld.
Paper Structure (7 sections, 23 theorems, 94 equations)

This paper contains 7 sections, 23 theorems, 94 equations.

Table of Contents

  1. A variant of TP
  2. Completed $F$-gauges
  3. Graded stacks and graded formal groups
  4. Identification of the Drinfeld and Quillen formal groups
  5. Algebraization of the Drinfeld formal group
  6. Limits of localizations of categories
  7. Congruences

Key Result

Theorem 1

The Quillen formal group is an extension of $\widehat{G}^\mathrm{Dr}$ along the open immersion $\mathbb{Z}_p^{{\mathlarger{\mathbbl{\Delta}}}} \hookrightarrow \mathbb{Z}_p^\mathrm{Syn}$, uniquely determined by the identification of its cotangent space at the identity with the Breuil--Kisin line bund where ${\operator@font{TP}}^{(-1)}$ is the Nygaard-decompleted Frobenius-untwist of ${\operator@fon

Theorems & Definitions (60)

  • Theorem 1: \ref{['drinfeldquillenfg']}, \ref{['drinfeldfgZpSyn']}
  • Remark 2
  • Theorem 3: \ref{['drinfeldfgalg']}, drinfeldfg
  • Remark 4
  • Remark 5
  • Proposition 6: \ref{['Mfgaffine']}, \ref{['affdiagaffineness']}
  • Proposition 1.2
  • Definition 1.3
  • Remark 1.4
  • Remark 1.5
  • ...and 50 more