nLab inter-universal Teichmüller theory




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.


Pilot objects

Important to the arguments in the latter parts of the IUTT series are 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}.


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.)



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.


  • 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.

Last revised on August 1, 2023 at 21:11:49. See the history of this page for a list of all contributions to it.