nLab
initial Θ-data

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 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.

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,,C̲ K,𝕍̲,𝕍 mod bad,ϵ̲)(\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 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. An auxillary prime \ell (satisfying….?)

(TO BE CONTINUED)

References

Last revised on May 10, 2018 at 02:34:45. See the history of this page for a list of all contributions to it.