Summary of Introduction to Inter-Universal Teichmüller Theory
1. What is Inter-Universal Teichmüller Theory?
In the first place, IUT theory was created to overcome the obstacle of not being able to directly construct a global theory from a local theory because of the robust structure of the number field.
It bundles an infinite number of prime places and then extends them to the entire number field by symmetry.
However, there are difficulties inherent in this practice.
First, it destroys the holomorphic structure.
In addition, synchronization between prime places must be made so that the addition and multiplication structures are globally connected.
In short, it's a grand theory that breaks apart the structure of numbers and rebuilds it all over again.
(pp. 286-288)
(From いらすとや)
2. Summary of Introduction
(1) Overview
In order to prove a certain Diophantine geometrical theorem, IUT theory asserts that it's sufficient to prove the existence of multiradial expressions of three objects:
(a) $\log$-shell, and
(b) power by some rational number (greater than 1) of the q parameter of an elliptic curve, and
(c) number field.
However, in order to obtain these multiradial expressions, it's necessary to isolate and protect (b) and (c) from the indeterminacy generated by abandoning the ring structure of the setting.
Therefore, a theta function is used for (b) and a $\kappa$-coric function is used for (c).
Then (a) becomes the container for (b) and (c).
(pp. 3-4, 99)
(2) Holomorphic structure
The destruction of the holomorphic structure would be to abandon the ring structure of the setting.
In IUT theory, holomorphic structure means "the ring structure itself, a structure that contains the ring structure, can restore the ring structure, or is essentially prescribed by the ring structure."
(p. 18)
A key aspect of IUT theory is Anabelian geometry.
It leads to certain non-trivial consequences in the setting where we are forced to abandon various ring structures, but the appropriate absolute Galois group is tenable.
"At least, I know no other field of study in arithmetic algebraic geometry, other than that of Anabelian geometry, which is still sufficiently applicable in such setting."
(p. 20)
(3) Multiradial algorithm
Multiradial algorithm is an algorithm that is concurrently executable in a way that is compatible with $\kappa$-coric data identification/sharing.
It can be simultaneously applied to many upper radiant data, and it has meaning for many upper radiant data at the same time.
(p. 25)
(4) Hodge theater
$D$-$\Theta^{\pm\mathrm{ell}}$Hodge theater or $\Theta^{\pm\mathrm{ell}}$Hodge theater can be thought of as a setting for the container (which eventually becomes (a)) for a theta function, its substitution place label management, and its special value (namely (b)).
Also, $D$-$\Theta \mathrm{NF}$Hodge and $\Theta \mathrm{NF}$Hodge theaters can be thought of as multiradial expressions of (c), and the setting for association between (c) and (a) or (b) (which are obtained by the operation of substitution into the theta function in $D$-$\Theta^{\pm\mathrm{ell}}$Hodge theater and $\Theta^{\pm\mathrm{ell}}$Hodge theater).
(pp. 100-101)
The concepts obtained by patching together $D$-$\Theta^{\pm\mathrm{ell}}$Hodge and $\Theta^{\pm\mathrm{ell}}$Hodge theaters, which are constructed based on additive/geometric symmetry, and $D$-$\Theta \mathrm{NF}$Hodge and $\Theta \mathrm{NF}$Hodge theaters, which are constructed based on multiplicative/number-theory symmetry, are $D$-$\Theta^{\pm\mathrm{ell}} \mathrm{NF}$Hodge and $\Theta^{\pm\mathrm{ell}} \mathrm{NF}$Hodge theaters.
Then, by linking two $\Theta^{\pm\mathrm{ell}} \mathrm{NF}$Hodge theaters by a $\log$-link, we can transform a certain singular multiplicative module into a compact additive module called (a).
(p. 4)
3. Picture
