nLab
inter-universal Teichmüller theory

Contents

Contents

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” apparently refers to the fact that the 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).

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.

Overview of the proof

Please note: everything in this section is heavily under construction! Contrary to possible current appearances, it is a serious attempt to shed some light on aspects of Mochizuki’s work, and is not in any way intended as a parody or to do any disservice to Mochizuki’s work, quite the contrary. We ask the reader for patience until such a time as what the page is intended to become is more clear.

Dénouement

The logic of the final part of Mochizuki’s approach to the abc conjecture is fairly straightforward in outline, and not, as far we know, a point of contention. We describe it here, deliberately using completely different notation from the IUTT papers to bring out the structure of the argument. Throughout, we shall assume that we have certain data 𝕄\mathbb{M}, which we shall treat as a black box.

Assumption

Given 𝕄\mathbb{M}, there are real numbers t 𝕄t_{\mathbb{M}} and q 𝕄q_{\mathbb{M}} such that t 𝕄q 𝕄-t_{\mathbb{M}} \geq - q_{\mathbb{M}}.

Corollary

Given any real number cc such that t 𝕄cq 𝕄-t_{\mathbb{M}} \leq c \cdot q_{\mathbb{M}}, we have that c1c \geq -1.

Proof

Immediate.

Assumption

For a particular choice of 𝕄\mathbb{M}, one can construct a real number c 𝕄c_{\mathbb{M}} such that:

  1. t 𝕄c 𝕄q 𝕄-t_{\mathbb{M}} \leq c_{\mathbb{M}} \cdot q_{\mathbb{M}};
  2. if c 𝕄1c_{\mathbb{M}} \geq -1, then the abc conjecture holds.

Theorem

(Conditional on Corollary 3.12 of IUTT III) The abc conjecture holds.

Proof

Follows immediately from Corollary and Assumption .

Remark

That Assumption holds is the fundamental result of the IUTT series of papers. It is Corollary 3.12 in IUTT III, and it is this corollary to which the objections raised by Scholze and Stix pertain.

That Assumption holds is the main purpose of IUTT IV. It is established by Theorem 1.10 and Corollary 2.2. Theorem 1.10 establishes the particular inequality of the form c 𝕄1c_{\mathbb{M}} \geq -1 that will be made use of; Corollary 2.2 specialises and refines it to an inequality which directly implies the abc conjecture. As far we know, these parts of the argument are expected to be sound.

The key corollary

As discussed in the previous section, the cornerstone of the IUTT series of papers (at least with regard to the abc conjecture) is Corollary 3.12 in IUTT III. We attempt here to explain certain aspects of this corollary. We shall continue to use completely different notation from the IUTT papers to try to bring out certain key ideas. To begin with, we shall continue to assume that we have a black box of data 𝕄\mathbb{M}. We shall first indicate the logical structure of the corollary, devoid of any significant content.

Assumption

There is a certain subset T 𝕄T_{\mathbb{M}} of {}\mathbb{R} \cup \{ \infty \}, where \mathbb{R} is the set of real numbers.

Assumption

There is a real number q 𝕄q_{\mathbb{M}}.

There are two key aspects to Corollary 3.12 in IUTT III, which we now state.

Theorem

(If correct!) For any t 𝕄T 𝕄t_{\mathbb{M}} \in T_{\mathbb{M}}, we in fact have that t 𝕄t_{\mathbb{M}} \neq \infty, and thus that t 𝕄t_{\mathbb{M}} \in \mathbb{R}. Moreover, t 𝕄q 𝕄-t_{\mathbb{M}} \geq -q_{\mathbb{M}}.

Remark

A major point of confusion in attempts to understand Mochizuki’s work has surrounded what Mochizuki refers to as ‘indeterminacies’. At the level we are working at here, things are clear: all that matters is that t 𝕄t_{\mathbb{M}} may be any one of a set of possible real numbers (what this set is is described precisely by Mochizuki, but this is irrelevant to the question of understanding the logic of the overall argument), whilst q 𝕄q_{\mathbb{M}} is a specific real number.

In particular, the dénouement of Mochizuki’s argument to prove the abc conjecture goes through regardless of the choice of t 𝕄t_{\mathbb{M}} amongst the set T 𝕄T_{\mathbb{M}}.

We now begin working towards explaining what T 𝕄T_{\mathbb{M}} and q 𝕄q_{\mathbb{M}} are.

Pilot objects

Both the set T 𝕄T_{\mathbb{M}} and the real number q 𝕄q_{\mathbb{M}} are constructed ultimately from what Mochizuki refers to as pilot objects. 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 XX, FF, F modF_{mod}, and 𝕍̲\underline{\mathbb{V}} be as at initial Θ-data: XX is a once-puncturing of an elliptic curve EE, FF is a number field, F modF_{mod} is the field of moduli of XX with respect to FF, 𝕍̲\underline{\mathbb{V}} is a certain set of valuations, and all of XX, FF, F¯\overline{F}, and 𝕍̲\underline{\mathbb{V}} satisfy certain conditions which will be referred to here only as needed. Let F mod ×F^{\times}_{mod} denote the multiplicative group of F modF_{mod}.

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

Let β v:F mod ×K v ×/𝒪 v ×\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 modK EF_{mod} \hookrightarrow K_{E} with the canonical ring homomorphisms K EK vK v/𝒪 vK_{E} \hookrightarrow K_{v} \twoheadrightarrow K_{v} / \mathcal{O}_{v}.

Definition

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

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

  2. For every v𝕍̲v \in \underline{\mathbb{V}}, a trivialisation t vt_{v} of the K v ×/𝒪 K v ×K_{v}^{\times} / \mathcal{O}_{K_v}^{\times}-torsor T vT_{v} obtained from TT by change of structure group with respect to β v\beta_{v}. We require that there is a tTt \in T such that, for all but finitely many vv, t vt_{v} is equal to the trivialisation of T vT_{v} determined by β v(t)\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.

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

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

Last revised on June 21, 2021 at 12:08:41. See the history of this page for a list of all contributions to it.