An analytic approach to $P$-adic diffeomorphism group and Teichm\"{u}ller theory

TL;DR

This paper develops a $p$-adic Teichmüller theory for infinite-dimensional diffeomorphism groups, connecting it to Mochizuki's IUT and revealing new geometric and hydrodynamic insights.

Contribution

It introduces a $p$-adic Teichmüller framework for diffeomorphism groups and reformulates IUT within this geometric context, highlighting novel structures and applications.

Findings

Establishment of a $p$-adic Teichmüller theory for diffeomorphism groups

Reformulation of Mochizuki's IUT as a Teichmüller theory on automorphisms of group schemes

Identification of hydrodynamic aspects in the $p$-adic setting

Abstract

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