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Construction of Arithmetic Teichmuller Spaces IV: Proof of the abc-conjecture

Kirti Joshi

TL;DR

The work addresses the abc-conjecture by embedding number-field arithmetic in a Teichmuller-inspired framework of Arithmetic Teichmuller Spaces. It develops a quantitative deformation theory of additive structures (arithmeticoids) and anabelomorphy via holomorphoids, enabling averaging of arithmetic data across a space of deformations. Central technical pillars include the log-different/log-conductor framework, Tate divisors, and initial Theta Data, culminating in two major bounds that yield Vojta’s inequality on compact subsets and hence abc and the Arithmetic Szpiro conjecture. The approach provides a constructive, geometric interpretation of Mochizuki’s ideas, addresses prior criticisms, and offers a scalable path toward higher-genus and higher-dimensional arithmetic geometry implications. If correct, it represents a substantial advance in Arakelov-like height theory and Teichmuller-type parameterization of arithmetic objects within a fixed universe.

Abstract

This is a continuation of my work on Arithmetic Teichmuller Spaces developed in the present series of papers. In this paper, I show that the Theory of Arithmetic Teichmuller Spaces leads, using Shinichi Mochizuki's rubric, to a proof of the $abc$-conjecture (as asserted by Mochizuki).

Construction of Arithmetic Teichmuller Spaces IV: Proof of the abc-conjecture

TL;DR

The work addresses the abc-conjecture by embedding number-field arithmetic in a Teichmuller-inspired framework of Arithmetic Teichmuller Spaces. It develops a quantitative deformation theory of additive structures (arithmeticoids) and anabelomorphy via holomorphoids, enabling averaging of arithmetic data across a space of deformations. Central technical pillars include the log-different/log-conductor framework, Tate divisors, and initial Theta Data, culminating in two major bounds that yield Vojta’s inequality on compact subsets and hence abc and the Arithmetic Szpiro conjecture. The approach provides a constructive, geometric interpretation of Mochizuki’s ideas, addresses prior criticisms, and offers a scalable path toward higher-genus and higher-dimensional arithmetic geometry implications. If correct, it represents a substantial advance in Arakelov-like height theory and Teichmuller-type parameterization of arithmetic objects within a fixed universe.

Abstract

This is a continuation of my work on Arithmetic Teichmuller Spaces developed in the present series of papers. In this paper, I show that the Theory of Arithmetic Teichmuller Spaces leads, using Shinichi Mochizuki's rubric, to a proof of the -conjecture (as asserted by Mochizuki).
Paper Structure (56 sections, 42 theorems, 210 equations, 1 table)

This paper contains 56 sections, 42 theorems, 210 equations, 1 table.

Table of Contents

  1. Introduction
  2. The goal of this paper
  3. Why is this paper needed?
  4. Archival Note
  5. Suggested reading order and logical dependency of various results
  6. A brief outline of the proof
  7. What is new in my approach?
  8. Where is Anabelian Geometry in all this?
  9. Comparison of the arithmetic and the geometric cases
  10. The Scholze-Stix Report
  11. Acknowledgments
  12. The conjectures and the theorems
  13. The $abc$-conjecture
  14. Arithmetic Szpiro Conjecture
  15. Functions of bounded discrepancy class
  16. ...and 41 more sections

Key Result

Proposition 2.6.2

The strong $abc$-conjecture (joshi-teich-abc-conj-arxiv:con:abc-vojta) implies the $abc$-conjecture (joshi-teich-abc-conj-arxiv:con:abc).

Theorems & Definitions (101)

  • Conjecture 2.1.1: The $abc$-conjecture
  • Remark 2.1.2
  • Conjecture 2.2.1: The Arithmetic Szpiro Conjecture
  • Remark 2.2.2
  • Remark 2.2.3
  • Conjecture 2.5.1: Vojta's Height Inequality for curves: Mochizuki's formulation
  • Conjecture 2.6.1: The strong $abc$-conjecture: Mochizuki's formulation
  • Proposition 2.6.2
  • proof
  • Theorem 2.7.1
  • ...and 91 more