nLab initial Θ-data

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 $E$ over a number field $F$ such that the Galois theory of $F$ and the geometry of $E$ interact well.

Notation

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

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

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

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

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

5. Given a field $F$ and an elliptic curve $E$ over $F$, the notation $K_E$ denotes the field, a finite Galois extension of $F$, corresponding (by Galois theory) to the kernel of $\phi_l$.

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

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

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 $(\overline{F} / F, X_{F}, l, \underline{C}_{K}, \underline{\mathbb{V}}, \mathbb{V}^{bad}_{mod}, \underline{\epsilon})$ consisting of the following data.

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

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

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

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

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

(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 $E$ is one of the form $y^{2} = x(x-1)(x - \lambda)$ for some $\lambda \in \mathbb{Q}$ satisfying the following conditions:

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

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

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

4. it is defined over a number field $F$ which is isomorphic to that obtained by beginning with the minimal field of definition of $E$ (regarded as defined over $\overline{\mathbb{Q}}$ for example) and then adjoining the fields of definition of the $30$-torsion points, that is, the $2$-torsion, $3$-torsion, and $5$-torsion points, as well as adjoining $\sqrt{-1}$;

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

Remark

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

Remark

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

Related concepts

• Mochizuki's corollary 3.12

References

• Shinichi Mochizuki, Inter-Universal Teichmüller Theory I: Construction Of Hodge Theaters, (2017). Link to paper

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

• 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, Initial Theta Data, video series.

  • Some Prerequisites, and The Easy Part of Initial Theta Data
  • Transvections
  • Local Global l-torsion Compatibility
  • Existence and Examples