Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

12 SHINICHI MOCHIZUKI
[cf. Remark 3.8.1, (i)], which is depicted in Fig. I1.5 below.
In fact, the Θ-link discussed in the present paper is only a simplified version
of the “Θ-link” that will ultimately play a central role in the present series of papers.
The construction of the version of the Θ-link that we shall ultimately be interested
in is quite technically involved and, indeed, occupies the greater part of the theory
to be developed in [IUTchII], [IUTchIII]. On the other hand, the simplified version
discussed in the present paper is of interest in that it allows one to give a relatively
straightforward introduction to many of the important qualitative properties of
the Θ-link — such as the Frobenius-picture discussed above and the étale-picture
to be discussed below — that will continue to be of central importance in the case
of the versions of the Θ-link that will be developed in [IUTchII], [IUTchIII].
n HT Θ±ell NF
n+1 HT Θ±ell NF
...
----
n q
v
n Θ v
----
(n+1) q
v
(n+1) Θ v
----
...
n Θ v
↦→ (n+1) q
v
Fig. I1.5: Frobenius-picture associated to the Θ-link
Now let us return to our discussion of the Frobenius-picture associated to the Θ-
link. The D ⊢ -prime-strip associated to the F ⊢× -prime-strip † F ⊢×
mod
may, in fact, be
naturally identified with the D ⊢ -prime-strip † D ⊢ > associated to a certain F-primestrip
† F > [cf. the discussion preceding Example 5.4] that arises from the Θ-Hodge
theater underlying the Θ ±ell NF-Hodge theater † HT Θ±ellNF . The D-prime-strip
† D > associated to the F-prime-strip † F > is precisely the D-prime-strip depicted
as “[1 < ... < l ]” in Fig. I1.3. Thus, the Frobenius-picture discussed above
induces an infinite chain of full poly-isomorphisms
...
∼
→ (n−1) D ⊢ >
∼
→ n D ⊢ >
∼
→ (n+1) D ⊢ >
∼
→ ...
of D ⊢ -prime-strips. That is to say, when regarded up to isomorphism, the D ⊢ -
prime-strip “ (−) D ⊢ >” may be regarded as an invariant — i.e., a “mono-analytic
core” —ofthevariousΘ ±ell NF-Hodge theaters that occur in the Frobenius-picture
[cf. Corollaries 4.12, (ii); 6.10, (ii)]. Unlike the case with the Frobenius-picture,
the relationships of the various D-Θ ±ell NF-Hodge theaters n HT D-Θ±ell NF to this
mono-analytic core — relationships that are depicted by spokes in Fig. I1.6 below
— are compatible with arbitrary permutation symmetries among the spokes
[i.e., among the labels n ∈ Z of the D-Θ ±ell NF-Hodge theaters] — cf. Corollaries
4.12, (iii); 6.10, (iii), (iv). The diagram depicted in Fig. I1.6 below will be referred
to as the étale-picture.
Thus, the étale-picture may, in some sense, be regarded as a collection of
canonical splittings of the Frobenius-picture. The existence of such splittings
suggests that