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An analytic approach to $P$-adic diffeomorphism group and Teichmüller theory

Yuxiu Lu

Abstract

We consider a specific class of infinite dimensional $p$-adic Lie groups, i.e., a sort of diffeomorphism groups on $p$-adic ball $\operatorname{Diff}^{\operatorname{an}}(B_ε)$. It turns out that this group has a natural logarithmic structure that leads to a $p$-adic version of Teichmüller theory on diffeomorphism groups, which also presents some remarkable hydrodynamic facets. We further apply this framework to Mochizuki's $p$-adic Teichmüller theory and Inter-universal Teichmüller theory (IUT), and give a new reformulation of IUT as a Teichmüller theory on automorphisms of two-dimensional group schemes.

An analytic approach to $P$-adic diffeomorphism group and Teichmüller theory

Abstract

We consider a specific class of infinite dimensional -adic Lie groups, i.e., a sort of diffeomorphism groups on -adic ball . It turns out that this group has a natural logarithmic structure that leads to a -adic version of Teichmüller theory on diffeomorphism groups, which also presents some remarkable hydrodynamic facets. We further apply this framework to Mochizuki's -adic Teichmüller theory and Inter-universal Teichmüller theory (IUT), and give a new reformulation of IUT as a Teichmüller theory on automorphisms of two-dimensional group schemes.
Paper Structure (26 sections, 32 theorems, 157 equations)

This paper contains 26 sections, 32 theorems, 157 equations.

Table of Contents

  1. introduction
  2. $p$-adic analytic manifolds and differential forms
  3. Charts and differential forms
  4. Compact $p$-adic manifolds and Serre's theorems
  5. $p$-adic Lie group and Teichmüller theory
  6. The convenient diffeomorphism group $\operatorname{Diff}^{\operatorname{an}}(B_\epsilon)$
  7. A remark on higher dimensional case
  8. Profinite group $\operatorname{Diff}^{\operatorname{an}}(B_\epsilon)$
  9. The $p$-adic logarithmic function and $p^m$-th power map
  10. The differential structure of $\operatorname{Diff}^{\operatorname{an}}(B_\epsilon)$
  11. Schwarzian derivative and Frobenius approximation
  12. The $p$-adic Bers embedding and Teichmüller theory on $\operatorname{Diff}^{\operatorname{an}}(B_\epsilon)$
  13. $p$-adic Fisher-Rao metric and location-scale family
  14. $p$-adic Teichmüller theory
  15. Teichmüller characters and Witt vectors
  16. ...and 11 more sections

Key Result

Proposition 2.1

If $y=\phi(x)$ is an analytic diffeomorphism of a closed and open set $A\subset \mathbb Q_p^n$ onto a closed and open set $A'\subset \mathbb Q_p^n$, and the Jacobian $\det(D\phi)(x)\neq0$ for any $x\in A$. Then for any integrable function $f:A\to\mathbb C$, we have

Theorems & Definitions (93)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • Example 2.4: Theorem 4.1, bradley2025
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • ...and 83 more