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Construction of Arithmetic Teichmuller Spaces II$\frac{1}{2}$: Deformations of Number Fields

Kirti Joshi

TL;DR

This work constructs Arithmetic Teichmuller Spaces for number fields by producing a rich moduli of arithmetically inequivalent avatars (arithmeticoids) of a fixed field and organizing them into adelic Fargues-Fontaine curves. It develops a global Frobenius action and a product-formula-based period mapping that records avatar-dependent arithmetic via hyperplanes in an infinite-dimensional Real vector space, thereby quantifying deformations while preserving the fundamental Galois data. A central theme is the interplay between arithmeticoids, Frobenioids, and Galois cohomology, with heights and stabilized heights capturing avatar-dependent Diophantine information. The framework connects to Mochizuki’s program through precise, avatar-aware structures, provides a mechanism to study deformations of local analytic geometries, and offers a route to arithmetic analogues of Teichmuller theory, including connections to arithmetic loops and knots. The appendices showcase the geometric Szpiro perspective and mutations of p-adic periods, supporting the claim that the arithmetic Teichmuller program yields concrete, quantitative tools for Diophantine problems.

Abstract

This paper lays the foundation of the Theory of Arithmetic Teichmuller Spaces of Number Fields by explicitly constructing many arithmetically inequivalent avatars of a fixed number field. This paper also constructs a topological space of such avatars and describes its symmetries. Notably amongst these symmetries is a global Frobenius morphism which changes the avatar of the number field! The existence of such avatars has been suggested and used by Shinichi Mochizuki in his work on the arithmetic Vojta and Szpiro conjectures. Important to the global aspect of this theory is the fact that the product formula for a number field defines an arithmetic period mapping (Section 5.9). The key advantage of my approach is that one can quantify the difference between two inequivalent avatars and this renders my theory fundamentally and quantitatively more precise than Mochizuki's approach. In the appendix, I provide a discussion of the proofs of the geometric Szpiro Conjectures due to [Bogomolov et. al. 2000] and [Zhang 2001] from the point of view of this paper. I also discuss applications of my theory to the theory of arithmetic loops and arithmetic knots.

Construction of Arithmetic Teichmuller Spaces II$\frac{1}{2}$: Deformations of Number Fields

TL;DR

This work constructs Arithmetic Teichmuller Spaces for number fields by producing a rich moduli of arithmetically inequivalent avatars (arithmeticoids) of a fixed field and organizing them into adelic Fargues-Fontaine curves. It develops a global Frobenius action and a product-formula-based period mapping that records avatar-dependent arithmetic via hyperplanes in an infinite-dimensional Real vector space, thereby quantifying deformations while preserving the fundamental Galois data. A central theme is the interplay between arithmeticoids, Frobenioids, and Galois cohomology, with heights and stabilized heights capturing avatar-dependent Diophantine information. The framework connects to Mochizuki’s program through precise, avatar-aware structures, provides a mechanism to study deformations of local analytic geometries, and offers a route to arithmetic analogues of Teichmuller theory, including connections to arithmetic loops and knots. The appendices showcase the geometric Szpiro perspective and mutations of p-adic periods, supporting the claim that the arithmetic Teichmuller program yields concrete, quantitative tools for Diophantine problems.

Abstract

This paper lays the foundation of the Theory of Arithmetic Teichmuller Spaces of Number Fields by explicitly constructing many arithmetically inequivalent avatars of a fixed number field. This paper also constructs a topological space of such avatars and describes its symmetries. Notably amongst these symmetries is a global Frobenius morphism which changes the avatar of the number field! The existence of such avatars has been suggested and used by Shinichi Mochizuki in his work on the arithmetic Vojta and Szpiro conjectures. Important to the global aspect of this theory is the fact that the product formula for a number field defines an arithmetic period mapping (Section 5.9). The key advantage of my approach is that one can quantify the difference between two inequivalent avatars and this renders my theory fundamentally and quantitatively more precise than Mochizuki's approach. In the appendix, I provide a discussion of the proofs of the geometric Szpiro Conjectures due to [Bogomolov et. al. 2000] and [Zhang 2001] from the point of view of this paper. I also discuss applications of my theory to the theory of arithmetic loops and arithmetic knots.
Paper Structure (92 sections, 52 theorems, 228 equations)

This paper contains 92 sections, 52 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. The Main Results
  3. Table of Analogies
  4. A global Frobenius morphism of a number field
  5. The Product Formula as a global Arithmetic Period Mapping
  6. Relationship with Shinichi Mochizuki's Theory
  7. Outline of this paper
  8. Acknowledgments
  9. Preliminaries
  10. Anabelomorphisms and amphoricity
  11. Some basic assumptions
  12. Valued fields and Archimedean Algebraically closed perfectoid fields
  13. Archimedean Fargues-Fontaine curve
  14. The Gelfand Spectrum and the Archimedean Fargues-Fontaine curve
  15. Some recollections of non-archimedean Fargues-Fontaine curves
  16. ...and 77 more sections

Key Result

Proposition 2.3.1

Any archimedean valuation on ${\mathbb C}$ (in the above sense) is of the form

Theorems & Definitions (172)

  • Proposition 2.3.1: Ostrowski's Theorem
  • Definition 2.3.2
  • Definition 2.3.3
  • Proposition 2.4.3
  • Remark 2.7.1
  • Definition 3.1.1
  • Lemma 3.1.4
  • Proposition 3.2.1
  • Theorem 3.2.2
  • proof
  • ...and 162 more