Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

98 SHINICHI MOCHIZUKI
Proposition 4.2. Assertions (ii) and (iii) follow immediately from the intrinsic
nature of the constructions of Example 4.5. ○
Remark 4.7.1. The significance of the natural bijection † ζ of Proposition 4.7,
(iii), lies in the following observation: Suppose that one wishes to work with the
global data † D ⊚ in a fashion that is independent of the local data [i.e., “prime-strip
data”] † D > , † D J [cf. Remark 4.3.2, (b)]. Then
by replacing the capsule index set J by the set of global label classes of
cusps LabCusp( † D ⊚ ) via † ζ , one obtains an object — i.e., LabCusp( † D ⊚ )
— constructed via [i.e., “native to”] the global data that is immune to the
“collapsing” of J → ∼ F l
— i.e., of F l -orbits of V±un — even at primes
v ∈ V of the sort discussed in Remark 4.2.1!
That is to say, this “collapsing” of [i.e., failure of F l
to act freely on] F l -orbits
of V ±un is a characteristically global consequence of the global prime decomposition
trees discussed in Remark 4.3.1, (ii) [cf. the example discussed in Remark 4.2.1]. We
refer to Remark 4.9.3, (ii), below for a discussion of a closely related phenomenon.
Remark 4.7.2.
(i)Attheleveloflabels [cf. the content of Proposition 4.7], the structure of a
D-ΘNF-Hodge theater may be summarized via the diagram of Fig. 4.4 below — i.e.,
where the expression “[ 1 < 2 < ... < (l − 1) < l ]” corresponds to † D > ;the
expression “( 1 2 ... l − 1 l )” corresponds to † D J ; the lower righthand
“F l -cycle of ’s” corresponds to † D ⊚ ;the“⇑” corresponds to the associated
D-Θ-bridge; the“⇒” corresponds to the associated D-NF-bridge; the“/ ’s” denote
D-prime-strips.
(ii) Note that the labels arising from † D > correspond, ultimately, to various
irreducible components in the special fiber of a certain tempered covering of a
[“geometric”!] Tate curve [a special fiber which consists of a chain of copies
of the projective line — cf. [EtTh], Corollary 2.9]. In particular, these labels are
obtained by counting —inanintuitive,archimedean, additive fashion — the number
of irreducible components between a given irreducible component and the “origin”.
In particular, the portion of the diagram of Fig. 4.4 corresponding to † D > may be
described by the following terms:
geometric, additive, archimedean, hence Frobenius-like [cf. Corollary
3.8].
By contrast, the various “’s” in the portion of the diagram of Fig. 4.4 corresponding
to † D ⊚ arise, ultimately, from various primes of an [“arithmetic”!] number
field. These primes are permuted by the multiplicative group F l
= F × l
/{±1}, ina
cyclic — i.e., nonarchimedean — fashion. Thus, the portion of the diagram of Fig.
4.4 corresponding to † D ⊚ may be described by the following terms:
arithmetic, multiplicative, nonarchimedean, hence étale-like [cf.
the discussion of Remark 4.3.2].