Inter-universal Teichmüller Theory III: Canonical Splittings of the Log-Theta-Lattice
Authors: Mochizuki Shinichi 
Published: March 2021

The present paper constitutes the third paper in a series of four papers and may be regarded as the
culmination
of the
abstract conceptual
portion of the theory developed in the series. In the present paper, we study the theory surrounding the
log-theta-lattice
, a
highly noncommutative
two-dimensional diagram of
"miniature models of conventional scheme theory"
, called

\Theta^{\pm\mathrm{ell}}\mathrm{NF}

-Hodge theaters

. Here, we recall that

\Theta^{\pm\mathrm{ell}}\mathrm{NF}

-Hodge theaters were associated, in the first paper of the series, to certain data, called

initial

\Theta

-data

, that includes an

elliptic curve

E_F

over a
number field

F

, together with a
prime number

l\ge 5

. Each
arrow
of the log-theta-lattice corresponds to a certain
gluing operation
between the

\Theta^{\pm\mathrm{ell}}\mathrm{NF}

-Hodge theaters in the domain and codomain of the arrow. The
horizontal arrows
of the log-theta-lattice are defined as certain versions of the

"

\Theta

-link"

that was constructed, in the second paper of the series, by applying the theory of
Hodge–Arakelov-theoretic evaluation
– i.e., evaluation in the style of the
scheme-theoretic Hodge–Arakelov theory
established by the author in previous papers – of the [reciprocal of the

l

-th root of the]
theta function
at

l

-torsion points

. In the present paper, we focus on the theory surrounding the

\mathfrak{log}

-link

between

\Theta^{\pm\mathrm{ell}}\mathrm{NF}

-Hodge theaters. The

\mathfrak{log}

-link is obtained, roughly speaking, by applying, at each [say, for simplicity, nonarchimedean] valuation of the number field under consideration, the

local

p

-adic logarithm

. The significance of the

\mathfrak{log}

-link lies in the fact that it allows one to construct
log-shells
, i.e., roughly speaking, slightly adjusted forms of the image of the local units at the valuation under consideration via the local

p

-adic logarithm. The theory of log-shells was studied extensively in a previous paper by the author. The
vertical arrows
of the log-theta-lattice are given by the

\mathfrak{log}

-link. Consideration of various properties of the log-theta-lattice leads naturally to the establishment of
multiradial algorithms
for constructing
"splitting monoids of logarithmic Gaussian procession monoids"
. Here, we recall that "multiradial algorithms" are algorithms that make sense from the point of view of an
"alien arithmetic holomorphic structure"
, i.e., the ring/scheme structure of a

\Theta^{\pm\mathrm{ell}}\mathrm{NF}

-Hodge theater related to a given

\Theta^{\pm\mathrm{ell}}\mathrm{NF}

-Hodge theater by means of a
non-ring/scheme-theoretic
horizontal arrow of the log-theta-lattice. These logarithmic Gaussian procession monoids, or
LGP-monoids
, for short, may be thought of as the log-shell-theoretic versions of the
Gaussian monoids
that were studied in the second paper of the series. Finally, by applying these multiradial algorithms for splitting monoids of LGP-monoids, we obtain
estimates
for the
log-volume
of these LGP-monoids. Explicit computations of these estimates will be applied, in the fourth paper of the series, to derive various
diophantine results
.