nLab inter-universal Teichmüller theory

Idea

The generalization of Teichmüller theory to arithmetic geometry has been called inter-universal Teichmüller theory (often abbreviated IUTT) by Shinichi Mochizuki. It is a part of anabelian geometry.

The term “Inter-Universal” refers to Mochizuki’s Key Principle of Inter-Universality [IUTT I, Section I3, Page 25-26] which requires one to work with arbitrary geometric base-points for the tempered fundamental groups.

A choice of a geometric base-point for the tempered fundamental group corresponds to a choice of a fiber functor for the galois category of tempered coverings. Hence Mochizuki asserts [IUTT I, Section I3 Page 25-26] that the idea of working with distinct geometric base-points is tantamount to relating the distinct set-theoretic universes the fiber functors arise from and so theory is referred to as being inter-universal. [The fundamental group is, of course, independent (or agnostic) of the choice of the fiber functor, i.e. it is an object which is oblivious to the choice of the set theoretic universe it arises from.]

Hence Mochizuki has asserted that this theory is meant to formulated explicitly in a way that respects universe enlargement, hence that it is universe polymorphic (IUTT IV, remark 3.1.4, Yamashita 13).

While [IUTT I-IV] offers scant mathematical justification of the Teichmuller aspects of the theory (i.e. [IUTT I-IV] provides no mathematical justification why the theory is a Teichmuller Theory), a clearest proof of this Teichmuller aspect has been provided (in all dimensions) in the works of Kirti Joshi.

It is claimed (IUTT IV) that a proof of the abc conjecture can be given in IUTT. In 2018, a document Why abc is still a conjecture was written by Peter Scholze and Jakob Stix raising objections to the argument. More accurate critiques have appeared as a part of the works of Kirti Joshi on the theory of Arithmetic Teichmuller Spaces which includes, in dimension one and genus one, a precise version of Mochizuki’s IUTT.

In brief, the novel idea due to Mochizuki, is to average over deformations of arithmetic (or an arithmetic holomorphic structure) itself (this notion has been precisely quantified in the works of Kirti Joshi). It should be remarked that Classical Teichmuller Theory should be considered as a model for Mochizuki’s (and Joshi’s) theory at any archimedean prime of a number field.

Details

Pilot objects

Important to the arguments in the latter parts of the IUTT series are what Mochizuki refers to as pilot objects. (A precise construction of Pilot objects has been independently provided in Construction of Arithmetic Teichmuller Spaces III: A Rosetta Stone and a proof of Mochizuki’s Corollary 3.12.)

Though the terminology in IUTT III is very dense, Mochizuki explains in Example 3.6, Remark 3.6.1, and subsequently that these can be understood in quite elementary terms, which we now describe.

Let $X$, $F$, $F_{mod}$, and $\underline{\mathbb{V}}$ be as at initial Θ-data: $X$ is a once-puncturing of an elliptic curve $E$, $F$ is a number field, $F_{mod}$ is the field of moduli of $X$ with respect to $F$, $\underline{\mathbb{V}}$ is a certain set of valuations, and all of $X$, $F$, $\overline{F}$, and $\underline{\mathbb{V}}$ satisfy certain conditions which will be referred to here only as needed. Let $F^{\times}_{mod}$ denote the multiplicative group of $F_{mod}$.

Let $v \in \underline{\mathbb{V}}$. By definition, $v$ is a valuation on $K_E$, where $K_E$ is as defined at initial Θ-data. Denote the completion of $K_E$ with respect to $v$ by $K_{v}$, denote the ring of integers of $K_{v}$ by $\mathcal{O}_{K_{v}}$, and denote the group of units of these two rings by $K^{\times}_{v}$ and $\mathcal{O}_{v}^{\times}$ respectively.

Let $\beta_{v}: F_{mod}^{\times} \rightarrow K_{v}^{\times} / \mathcal{O}^{\times}_{v}$ denote the group homomorphism induced by the composition of the field inclusion $F_{mod} \hookrightarrow K_{E}$ with the canonical ring homomorphisms $K_{E} \hookrightarrow K_{v} \twoheadrightarrow K_{v} / \mathcal{O}_{v}$.

Definition

We denote by $F^{*}_{MOD}$ the category whose objects consist of:

1. An $F^{\times}_{mod}$-torsor $T$, where torsor here is understood with respect to the category of sets.

2. For every $v \in \underline{\mathbb{V}}$, a trivialisation $t_{v}$ of the $K_{v}^{\times} / \mathcal{O}_{K_v}^{\times}$-torsor $T_{v}$ obtained from $T$ by change of structure group with respect to $\beta_{v}$. We require that there is a $t \in T$ such that, for all but finitely many $v$, $t_{v}$ is equal to the trivialisation of $T_{v}$ determined by $\beta_{v}(t)$. (See torsor for the details of trivialisation of a torsor, and for the notion of change of structure group of a torsor.)

TODO: FINISH

Remark

Definition is Example 3.6 in IUTT III, and is also discussed in Remark 3.6.1. Remark 3.1.5 of IUTT I is also relevant.

Related concepts

• anabelian geometry

• étale theta function

• Frobenioid

• initial Θ-data

• Mochizuki's corollary 3.12

• universe polymorphism

• poly-morphism (not to be be confused with polymorphism)

References

• Shinichi Mochizuki, Inter-universal Teichmüller theory I, Construction of Hodge theaters (2012) (pdf)

• Shinichi Mochizuki, Inter-universal Teichmüller theory II, Hodge-Arakelov-theoretic evaluation (2012) (pdf)

• Shinichi Mochizuki, Inter-universal Teichmüller theory III, Canonical splittings of the Log-theta-lattice (2012) (pdf)

• Shinichi Mochizuki, Inter-universal Teichmüller theory IV, Log-volume computations and set-theoretic foundations (2012) (pdf)

Surveys include

• Shinichi Mochizuki, Panoramic overview of inter-universal Teichmuller theory, pdf

• Go Yamashita, FAQ on ‘Inter-Universality’ (pdf)

• Ivan Fesenko, Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, European Journal of Mathematics September 2015, Volume 1, Issue 3, pp 405-440 (publisher, pdf)

• Minhyong Kim, Brief superficial remarks on Shinichi Mochizuki’s Interuniversal Teichmueller Theory (IUTT), version 1, 10/11/2015, (pdf).

• Taylor Dupuy, Hodge Theaters: A First Look at the Big Hodge Theater, Confused Groups and Torsors

• RIMS/Symmetries and Correspondences workshop: Inter-universal Teichmüller Theory Summit 2016

• Jackson Morrow, Kummer classes and Anabelian geometry, notes from Super QVNTS: Kummer Classes and Anabelian Geometry 2017 (pdf)

The following two papers give a statement of Mochizuki's Corollary 3.12 in plain language (not using the elaborate setup in the IUT papers), and assuming the inequality in it as given, derive concrete versions of (weakenings of) Szpiro’s conjecture, again, using conventional terminology and techniques.

• Taylor Dupuy, Anton Hilado, The Statement of Mochizuki’s Corollary 3.12, Initial Theta Data, and the First Two Indeterminacies, arXiv:2004.13228

• Taylor Dupuy, Anton Hilado, Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki’s Corollary 3.12, arXiv:2004.13108