Construction of Arithmetic Teichmuller Spaces I

TL;DR

This work develops Arithmetic Teichmüller Theory by introducing Arithmetic Holomorphic Structures and constructing $v$-adic local and Adelic Arithmetic Teichmüller Spaces $\mathfrak{J}(X,E)$ and $\widetilde{\mathfrak{J}(X/L)}$ that realize a deep analogy between Number fields and Riemann surfaces. It shows that while Berkovich spaces admit isomorphic tempered fundamental groups, their analytic structures can vary, necessitating a holomorphic/arithmetic labeling akin to Teichmüller data. The theory integrates tilt-/untilt- phenomena of perfectoid fields, Fargues–Fontaine curves, diamonds, and Galois cohomology, and aligns with Mochizuki’s IUT program while proceeding from a different, function-theoretic perspective. The framework yields both local and global (adelic) variants, reveals metric distortions and Virasoro-type uniformizations, and provides a platform for relating arithmetic geometry to anabelian geometry, with potential applications to issues such as the abc-conjecture and arithmetic dynamics. Overall, it supplies a robust, arithmetic-analytic Teichmüller landscape that unifies p-adic and archimedean aspects and connects foundational concepts across Teichmüller theory, Galois theory, and modern IUT-inspired approaches.

Abstract

In this paper after proving (in Section 2) the Berkovich analytic space analog of the familiar fact that there exist many non-isomorphic Riemann surfaces of the fixed topological type, I introduce the precise notion of Arithmetic Holomorphic Structures. This leads, for a fixed geometrically connected, smooth quasi-projective variety $X/E$ over a $p$-adic field, to the construction of a category which can be called Arithmetic Teichmuller Space of $X/E$. After establishing the properties of this local i.e. $p$-adic Arithmetic Teichmuller Space, I proceed to the global (adelic) construction, for a geometrically connected, smooth quasi-projective variety $X/L$ over a number field $L$, of the Adelic Arithmetic Teichmuller Space of $X/L$. A fixed number field itself has an Arithmetic Teichmuller Space--this is detailed in Constructions II(1/2) paper in this series of papers. All of these constructions extend the analogy between Number fields and Riemann surfaces and are inspired by (and directly related to) Shinichi Mochizuki's ideas on Inter-Universal Teichmuller Theory and his work on the abc-conjecture. But my approach is based on a completely different set of ideas.

Theorems & Definitions (135)

• Theorem 2.1.1
• Remark 2.1.2
• Theorem 2.3.1
• Remark 2.3.2
• Theorem 2.5.1
• proof
• Remark 2.5.2
• Remark 2.5.3
• Lemma 3.2.1
• proof