Inter-universal Teichmüller Theory II: Hodge–Arakelov-Theoretic Evaluation
Authors: Mochizuki Shinichi 
Published: March 2021

In the present paper, which is the second in a series of four papers, we study the
Kummer theory
surrounding the 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

[strictly speaking, shifted by a suitable 2-torsion point], for

l\ge 5

a prime number. In the first paper of the series, we studied
"miniature models of conventional scheme theory"
, which we referred to as

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

-Hodge theaters

, that were associated 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

. The underlying

\Theta

-Hodge theaters of these

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

-Hodge theaters were
glued
to one another by means of

"

\Theta

-links"

, that identify the [reciprocal of the

l

-th root of the]
theta function
at primes of bad reduction of

E_F

in one

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

-Hodge theater with [

2l

-th roots of] the

q

-parameter

at primes of bad reduction of

E_F

in another

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

-Hodge theater. The theory developed in the present paper allows one to construct certain new versions of this "

\Theta

-link". One such new version is the

\Theta^{\times\mu}_{\mathrm{gau}}

-link

, which is similar to the

\Theta

-link, but involves the

theta values at

l

-torsion points

, rather than the theta function itself. One important aspect of the constructions that underlie the

\Theta^{\times\mu}_{\mathrm{gau}}

-link is the study of
multiradiality
properties, i.e., properties of the
"arithmetic holomorphic structure"
– or, more concretely, the
ring/scheme
structure – arising from
one

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

-Hodge theater that may be formulated in such a way as to make sense from the point of view of the arithmetic holomorphic structure of
another

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

-Hodge theater which is related to the original

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

-Hodge theater by means of the [
non-scheme-theoretic
!]

\Theta^{\times\mu}_{\mathrm{gau}}

-link. For instance, certain of the various
rigidity
properties of the
étale theta function
studied in an earlier paper by the author may be interpreted as
multiradiality
properties in the context of the theory of the present series of papers. Another important aspect of the constructions that underlie the

\Theta^{\times\mu}_{\mathrm{gau}}

-link is the study of
"conjugate synchronization"
via the

\mathbb{F}^{\rtimes\pm}_l

-symmetry

of a

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

-Hodge theater. Conjugate synchronization refers to a certain system of isomorphisms – which are
free
of any
conjugacy indeterminacies
! – between copies of local absolute Galois groups at the various

l

-torsion points at which the theta function is evaluated. Conjugate synchronization plays an important role in the Kummer theory surrounding the evaluation of the theta function at

l

-torsion points and is applied in the study of
coricity
properties of [i.e., the study of objects left
invariant
by] the

\Theta^{\times\mu}_{\mathrm{gau}}

-link. Global aspects of conjugate synchronization require the resolution, via results obtained in the first paper of the series, of certain technicalities involving
profinite conjugates
of tempered cuspidal inertia groups.