transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
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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 over a number field such that the Galois theory of and the geometry of interact well.
In order to understand the definition, we recall a little notation.
Given a field , the notation means that is a field extension of .
Given a field , the notation denotes an algebraic closure of .
Given a field and a once-punctured elliptic curve over (see below for a precise definition), the notation denotes the field of moduli of .
Given a field , an elliptic curve over , and a prime , let us write for the group homomorphism from the Galois group of to the general linear group of 2x2 invertible matrices with values in determined by the -torsion points of (see torsion points of an elliptic curve for details of this homomorphism).
Given a field and an elliptic curve over , the notation denotes the field, a finite Galois extension of , corresponding (by Galois theory) to the kernel of .
Given a number field , the notation denotes the set of valuations on , both archimedean and non-archimedean.
Given a number field and a valuation on , the notation denotes the completion? of with respect to .
The following is Definition 3.1 in Inter-Universal Teichmüller theory I, on pg.61 (currently).
Initial Θ-data is a 7-tuple consisting of the following data.
A number field such that . In other words, we have a field extension of the quotient ring , which itself is a field because is irreducible: see field extension for more details on this.
A scheme which is obtained by removing a closed point from an elliptic curve over . The scheme structure on is that inherited from by virtue of the fact that is an open subset of (the underlying topological space of) , as described at open subscheme. We require that satisfies certain conditions: TODO.
The field extension is Galois.
A subset of which is isomorphic to . In other words, observing that is a sub-field of , a section of the map determined by the inclusion of into .
(TO BE CONTINUED)
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 , the algebraic closure of (namely the algebraic numbers). This notion is due to Mochizuki.
A Mochizuki elliptic curve is one of the form for some satisfying the following conditions:
belongs to a compactly bounded subset of with certain properties (omitted here for the moment);
belongs to an extension of of degree less than or equal to a given positive integer (the choice of which comes from the statement of the abc conjecture);
does not belong to a certain subset of , denoted , where is as in 2., which Mochizuki constructs (details omitted for the moment here too);
it is defined over a number field which is isomorphic to that obtained by beginning with the minimal field of definition of (regarded as defined over for example) and then adjoining the fields of definition of the -torsion points, that is, the -torsion, -torsion, and -torsion points, as well as adjoining ;
there is a prime satisfying certain properties for which does not admit a sub-group scheme which as a group is isomorphic to the cyclic group .
The first part of Corollary 2.2 in IUTT IV precisely concerns the construction, given (whose construction with the required properties is standard), of . The properties of are such that can one apply the theory of IUTT to deduce that the abc conjecture holds.
Given a Mochizuki elliptic curve , one can show that one can construct initial Θ-data whose elliptic curve is . TODO: details.
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.
Last revised on August 1, 2023 at 21:12:39. See the history of this page for a list of all contributions to it.