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Construction of Arithmetic Teichmuller spaces II: Proof of a local prototype of Mochizuki's Corollary 3.12

Kirti Joshi

TL;DR

The paper develops a local, $p$-adic version of Mochizuki's Corollary 3.12 by constructing and exploiting Arithmetic Teichmuller spaces for Tate elliptic curves. It introduces a geometric, intrinsic approach to Mochizuki's $\Theta_{gau}$-Links via Mochizuki's Ansatz, and uses the Fontaine ring $B$ to lift Tate-parameter and theta-values, enabling a multi-radial representation of theta-values. A key result is a prototype lower bound for the theta-values set $\widetilde{\Theta}$ in terms of the Tate parameter $q$, formalizing a local analogue of Mochizuki's inequality and offering an intrinsic proof of log-links and log-Kummer phenomena. The framework unifies the Tate-parameter function on Arithmetic Teichmuller spaces, Frobenius- and Galois-stable theta-objects, and $p$-adic Hodge-theoretic lifting, with the potential to extend to higher genus and global (adelic) contexts as discussed in the author’s broader program.

Abstract

This paper deals with consequences of the existence of Arithmetic Teichmuller spaces established in arXiv:2106.11452 and arXiv:2210.11635. Theorem~9.2.1 provides a proof of a local version of Mochizuki's Corollary~3.12. Local means for a fixed $p$-adic field. There are several new innovations in this paper. Some of the main results are as follows. Theorem~3.5.1 shows that one can view the Tate parameter of Tate elliptic curve as a function on the arithmetic Teichmuller space of [Joshi, 2021a], [Joshi, 2022b]. The next important point is the construction of Mochizuki's $Θ_{gau}$-links and the set of such links, called Mochizuki's Ansatz in §6. Theorem~6.9.1 establishes valuation scaling property satisfied by points of Mochizuki's Ansatz (i.e. by my version of $Θ_{gau}$-links). These results lead to the construction of a theta-values set (§8) which is similar to Mochizuki's Theta-values set (differences between the two are in §8.7.1). Finally Theorem~9.2.1 is established. For completeness, I provide an intrinsic proof of the existence of Mochizuki's $\log$-links (Theorem 10.9.1), $\mathfrak{log}$-links (Theorem~10.15.1) and Mochizuki's log-Kummer Indeterminacy (Theorem~10.20.1) in my theory.

Construction of Arithmetic Teichmuller spaces II: Proof of a local prototype of Mochizuki's Corollary 3.12

TL;DR

The paper develops a local, -adic version of Mochizuki's Corollary 3.12 by constructing and exploiting Arithmetic Teichmuller spaces for Tate elliptic curves. It introduces a geometric, intrinsic approach to Mochizuki's -Links via Mochizuki's Ansatz, and uses the Fontaine ring to lift Tate-parameter and theta-values, enabling a multi-radial representation of theta-values. A key result is a prototype lower bound for the theta-values set in terms of the Tate parameter , formalizing a local analogue of Mochizuki's inequality and offering an intrinsic proof of log-links and log-Kummer phenomena. The framework unifies the Tate-parameter function on Arithmetic Teichmuller spaces, Frobenius- and Galois-stable theta-objects, and -adic Hodge-theoretic lifting, with the potential to extend to higher genus and global (adelic) contexts as discussed in the author’s broader program.

Abstract

This paper deals with consequences of the existence of Arithmetic Teichmuller spaces established in arXiv:2106.11452 and arXiv:2210.11635. Theorem~9.2.1 provides a proof of a local version of Mochizuki's Corollary~3.12. Local means for a fixed -adic field. There are several new innovations in this paper. Some of the main results are as follows. Theorem~3.5.1 shows that one can view the Tate parameter of Tate elliptic curve as a function on the arithmetic Teichmuller space of [Joshi, 2021a], [Joshi, 2022b]. The next important point is the construction of Mochizuki's -links and the set of such links, called Mochizuki's Ansatz in §6. Theorem~6.9.1 establishes valuation scaling property satisfied by points of Mochizuki's Ansatz (i.e. by my version of -links). These results lead to the construction of a theta-values set (§8) which is similar to Mochizuki's Theta-values set (differences between the two are in §8.7.1). Finally Theorem~9.2.1 is established. For completeness, I provide an intrinsic proof of the existence of Mochizuki's -links (Theorem 10.9.1), -links (Theorem~10.15.1) and Mochizuki's log-Kummer Indeterminacy (Theorem~10.20.1) in my theory.
Paper Structure (10 sections, 23 theorems, 158 equations)

This paper contains 10 sections, 23 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Fargues-Fontaine curves $\mathscr{Y}_{{\mathbb C}_p^\flat,{\mathbb Q}_p}$ and $\mathscr{X}_{{\mathbb C}_p^\flat,{\mathbb Q}_p}$ and the ring $B$
  3. Arithmetic Teichmuller Space of a Tate elliptic curve
  4. The choice of a theta function
  5. Choice of theta values
  6. Mochizuki's ansatz "$q=q^{j^2}$" and the $\Theta_{gau}$-links
  7. Lifting theta values to $B$--Existence of Teichmuller lifts
  8. Construction of $\widetilde{\Theta}\subset B^{\ell^*}$
  9. A Prototype of Mochizuki's Corollary 3.12
  10. Appendix: Mochizuki's $\log$-links and $\mathfrak{log}$-links

Key Result

Proposition 2.7.1

Let $K_1$, $K_2$ be untilts of ${\mathbb C}_p^\flat$. .

Theorems & Definitions (59)

  • Proposition 2.7.1
  • proof
  • Definition 3.2.1
  • Remark 3.2.2
  • Lemma 3.2.3
  • Theorem 3.5.1
  • proof
  • Remark 3.5.2
  • Remark 4.1.1
  • Proposition 6.2.1
  • ...and 49 more