Contents

# Contents

## Idea

The notion of initial Θ-theta data was introduced by Shinichi Mochizuki in §3 of Inter-Universal Teichmüller theory I as the starting point for inter-universal Teichmüller theory.

## 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 $F$.

## 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 $(\overline{F} / F, X_{F}, \ell, \underline{C}_{K}, \underline{\mathbb{V}}, \mathbb{V}^{bad}_{mod}, \underline{\epsilon})$ given 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. An auxillary prime $\ell$ (satisfying….?)

(TO BE CONTINUED)