nLab
initial Θ-data

Contents

Idea

The notion of initial Θ-data or initial theta-data was introduced by Shinichi Mochizuki in §3 of Inter-Universal Teichmüller theory I as the starting point for inter-universal Teichmüller theory. In essence, initial Θ-data consists of an elliptic curve EE over a number field FF such that the Galois theory of FF and the geometry of EE interact well.

Notation

In order to understand the definition, we recall a little notation.

  1. Given a field FF, the notation E/FE / F means that EE is a field extension of FF.

  2. Given a field FF, the notation F¯\overline{F} denotes an algebraic closure of FF.

  3. Given a field FF and a once-punctured elliptic curve XX over FF (see below for a precise definition), the notation F modF_{mod} denotes the field of moduli of XX.

  4. Given a field FF, an elliptic curve EE over FF, and a prime l2l \geq 2, let us write ϕ l:Gal(F¯/F)GL 2(𝔽 l)\phi_l: Gal\left(\overline{F} / F\right) \rightarrow GL_{2}\left( \mathbb{F}_{l} \right) for the group homomorphism from the Galois group of FF to the general linear group of 2x2 invertible matrices with values in 𝔽 l\mathbb{F}_{l} determined by the ll-torsion points of EE (see torsion points of an elliptic curve for details of this homomorphism).

  5. Given a field FF and an elliptic curve EE over FF, the notation K EK_E denotes the field, a finite Galois extension of FF, corresponding (by Galois theory) to the kernel of ϕ l\phi_l.

  6. Given a number field FF, the notation 𝕍(F)\mathbb{V}(F) denotes the set of valuations on FF, both archimedean and non-archimedean.

  7. Given a number field FF and a valuation v𝕍(F)v \in \mathbb{V}\left(F\right) on FF, the notation F vF_{v} denotes the completion? of FF with respect to vv.

Definition

The following is Definition 3.1 in Inter-Universal Teichmüller theory I, on pg.61 (currently).

Definition

Initial Θ-data is a 7-tuple (F¯/F,X F,l,C̲ K,𝕍̲,𝕍 mod bad,ϵ̲)(\overline{F} / F, X_{F}, l, \underline{C}_{K}, \underline{\mathbb{V}}, \mathbb{V}^{bad}_{mod}, \underline{\epsilon}) consisting of the following data.

  1. A number field FF such that 1F\sqrt{-1} \in F. In other words, we have a field extension of the quotient ring [X]/(X 2+1)\mathbb{Q}[X] / (X^{2} + 1), which itself is a field because X 2+1X^{2} + 1 is irreducible: see field extension for more details on this.

  2. A scheme X FX_{F} which is obtained by removing a closed point from an elliptic curve E FE_{F} over FF. The scheme structure on X FX_{F} is that inherited from E FE_{F} by virtue of the fact that X FX_{F} is an open subset of (the underlying topological space of) E FE_{F}, as described at open subscheme. We require that X FX_{F} satisfies certain conditions: TODO.

  3. A prime l5l \geq 5 such that ϕ l\phi_l as defined above contains SL 2(𝔽 l)SL_2\left( \mathbb{F}_{l} \right) in its image.

  4. The field extension F/F modF / F_{mod} is Galois.

  5. A subset 𝕍̲\underline{\mathbb{V}} of 𝕍(K E)\mathbb{V}\left(K_E\right) which is isomorphic to 𝕍(F mod)\mathbb{V}\left(F_{mod}\right). In other words, observing that F modF_{mod} is a sub-field of K EK_E, a section of the map V(K E)V(F mod)V\left( K_E \right) \rightarrow V\left( F_{mod} \right) determined by the inclusion of F modF_{mod} into K EK_E.

(TO BE CONTINUED)

Examples of the kind in IUTT

Only one family of examples of initial Θ-data is given in the IUTT series of papers, pertaining directly to the abc conjecture. These examples are described in Corollary 2.2 of IUTT IV. The language of stacks is used, but it seems possible to construct the examples without this.

We shall assume that we have defined the notion of a compactly bounded subset of ¯\overline{\mathbb{Q}}, the algebraic closure of \mathbb{Q} (namely the algebraic numbers). This notion is due to Mochizuki.

Definition

A Mochizuki elliptic curve EE is one of the form y 2=x(x1)(xλ)y^{2} = x(x-1)(x - \lambda) for some λ\lambda \in \mathbb{Q} satisfying the following conditions:

  1. λ\lambda belongs to a compactly bounded subset K VK_V of ¯\overline{\mathbb{Q}} with certain properties (omitted here for the moment);

  2. λ\lambda belongs to an extension of \mathbb{Q} of degree less than or equal to a given positive integer dd (the choice of which comes from the statement of the abc conjecture);

  3. λ\lambda does not belong to a certain subset of K VK_V, denoted ℭ𝔯𝔠 d\mathfrak{Crc}_{d}, where dd is as in 2., which Mochizuki constructs (details omitted for the moment here too);

  4. it is defined over a number field FF which is isomorphic to that obtained by beginning with the minimal field of definition of EE (regarded as defined over ¯\overline{\mathbb{Q}} for example) and then adjoining the fields of definition of the 3030-torsion points, that is, the 22-torsion, 33-torsion, and 55-torsion points, as well as adjoining 1\sqrt{-1};

  5. there is a prime l5l \geq 5 satisfying certain properties for which EE does not admit a sub-group scheme which as a group is isomorphic to the cyclic group /l\mathbb{Z} / l \mathbb{Z}.

Remark

The first part of Corollary 2.2 in IUTT IV precisely concerns the construction, given K VK_{V} (whose construction with the required properties is standard), of ℭ𝔯𝔠 d\mathfrak{Crc}_{d}. The properties of ℭ𝔯𝔠 d\mathfrak{Crc}_{d} are such that can one apply the theory of IUTT to deduce that the abc conjecture holds.

Remark

Given a Mochizuki elliptic curve EE, one can show that one can construct initial Θ-data whose elliptic curve is EE. TODO: details.

References

Last revised on April 5, 2021 at 19:03:23. See the history of this page for a list of all contributions to it.