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Construction of Arithmetic Teichmuller Spaces III: A `Rosetta Stone' and a proof of Mochizuki's Corollary 3.12

Kirti Joshi

TL;DR

The paper develops Arithmetic Teichmuller Spaces for number fields, introducing holomorphoids as global and local arithmetic-analytic data and fixing Initial Theta Data to enable global theta-values. It builds Mochizuki's Adelic Ansatz and Theta-links across adeles, embeds these into p-adic period rings, and establishes tensor-product structures to realize Mochizuki’s tensor-packet framework. A central achievement is a proof of Mochizuki's Corollary 3.12 in the adelic Joshi setting, together with a detailed Rosetta Stone to translate between Mochizuki's IUT and this arithmetic Teichmuller perspective. The work also clarifies the roles of prime-strips, log-links, and Frobenius-étale duality, and demonstrates how global product formulas constrain local valuations, yielding global-normalization phenomena with potential Diophantine consequences. Overall, it provides a global, explicit, and verifiable framework connecting arithmetic deformation theory, p-adic Hodge theoretic period rings, and a broad, structured approach to Mochizuki’s conjectures, while situating these results within a Rosetta Stone that links IUT to Arithmetic Teichmuller spaces.

Abstract

This is a continuation of my work on Arithmetic Teichmuller Spaces (arXiv:2106.11452, arXiv:2210.11635, arXiv:2303.01662, arXiv:2305.10398). This paper establishes a number of important results including (1) a proof Mochizuki's Corollary 3.12 (2) establishes a `Rosetta Stone' for a parallel reading of Mochizuki's Inter-Universal Teichmuller Theory and my Theory of Arithmetic Teichmuller Spaces, and (3) a proof that Mochizuki's gluing of Hodge-Theaters, Frobenioids along prime-strips as described in his theory is naturally provided by the existence of Arithmetic Teichmuller Spaces. (4) Includes the geometric case of Mochizuki's Corollary 3.12 in §12.

Construction of Arithmetic Teichmuller Spaces III: A `Rosetta Stone' and a proof of Mochizuki's Corollary 3.12

TL;DR

The paper develops Arithmetic Teichmuller Spaces for number fields, introducing holomorphoids as global and local arithmetic-analytic data and fixing Initial Theta Data to enable global theta-values. It builds Mochizuki's Adelic Ansatz and Theta-links across adeles, embeds these into p-adic period rings, and establishes tensor-product structures to realize Mochizuki’s tensor-packet framework. A central achievement is a proof of Mochizuki's Corollary 3.12 in the adelic Joshi setting, together with a detailed Rosetta Stone to translate between Mochizuki's IUT and this arithmetic Teichmuller perspective. The work also clarifies the roles of prime-strips, log-links, and Frobenius-étale duality, and demonstrates how global product formulas constrain local valuations, yielding global-normalization phenomena with potential Diophantine consequences. Overall, it provides a global, explicit, and verifiable framework connecting arithmetic deformation theory, p-adic Hodge theoretic period rings, and a broad, structured approach to Mochizuki’s conjectures, while situating these results within a Rosetta Stone that links IUT to Arithmetic Teichmuller spaces.

Abstract

This is a continuation of my work on Arithmetic Teichmuller Spaces (arXiv:2106.11452, arXiv:2210.11635, arXiv:2303.01662, arXiv:2305.10398). This paper establishes a number of important results including (1) a proof Mochizuki's Corollary 3.12 (2) establishes a `Rosetta Stone' for a parallel reading of Mochizuki's Inter-Universal Teichmuller Theory and my Theory of Arithmetic Teichmuller Spaces, and (3) a proof that Mochizuki's gluing of Hodge-Theaters, Frobenioids along prime-strips as described in his theory is naturally provided by the existence of Arithmetic Teichmuller Spaces. (4) Includes the geometric case of Mochizuki's Corollary 3.12 in §12.
Paper Structure (184 sections, 79 theorems, 471 equations)

This paper contains 184 sections, 79 theorems, 471 equations.

Table of Contents

  1. Introduction
  2. The goals of this paper
  3. Classical Teichmuller Theory as a model for the Arithmetic Case
  4. Mochizuki's Key Principle of Inter-Universality: The role of arbitrary geometric base-points
  5. Number Fields are Deformable
  6. Quantification of Deformations of Number Fields
  7. The role of the Absolute Grothendieck Conjecture
  8. Global aspects: The central role of global arithmetic of number fields
  9. Mochizuki's Corollary 3.12
  10. The geometric case of Mochizuki's Corollary 3.12
  11. Many versions of Mochizuki's Corollary 3.12
  12. Why this work is needed: The incompleteness of mochizuki-iut1mochizuki-iut2mochizuki-iut3
  13. On mochizuki-essential-logic
  14. Other articles on mochizuki-iut1mochizuki-iut2mochizuki-iut3mochizuki-iut4
  15. Two points of departure from mochizuki-iut1mochizuki-iut2mochizuki-iut3
  16. ...and 169 more sections

Key Result

Lemma 2.1.6

Let $(X,L\hookrightarrow \mathfrak{arith}(L))$ be a holomorphoid of $X/L$. Let $v\in\mathbb{V}_L$. Then the local holomorphoid $\mathfrak{hol}_{}(X/L)_{y_{v}}$ at $v$ is a point (i.e. an object) of the local Arithmetic Teichmuller Space $\mathfrak{J}(X/L_v)$ defined in joshi-teich and hence each hol

Theorems & Definitions (237)

  • Definition 2.1.2
  • Remark 2.1.4
  • Definition 2.1.5
  • Lemma 2.1.6
  • proof
  • Remark 2.1.7
  • Remark 2.1.8
  • Remark 2.1.9
  • Theorem 2.3.1
  • Remark 2.3.2
  • ...and 227 more