Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

80 SHINICHI MOCHIZUKI
Section 4: Multiplicative Combinatorial Teichmüller Theory
In the present §4, we begin to prepare for the construction of the various
“enhancements” to the Θ-Hodge theaters of §3 that will be made in §5. More
precisely, in the present §4, we discuss the combinatorial aspects of the “D” — i.e.,
in the terminology of the theory of Frobenioids, the “base category” — portion of the
notions to be introduced in §5 below. In a word, these combinatorial aspects revolve
around the “functorial dynamics” imposed upon the various number fields and
local fields involved by the “labels”
∈
F l
def
= F × l /{±1}
— where we note that the set F l
is of cardinality l def
= (l−1)/2 — of the l-torsion
points at which we intend to conduct, in [IUTchII], the “Hodge-Arakelov-theoretic
evaluation” of the étale theta function studied in [EtTh] [cf. Remarks 4.3.1; 4.3.2;
4.5.1, (v); 4.9.1, (i)].
In the following, we fix a collection of initial Θ-data
(F/F, X F , l, C K , V, V bad
mod, ɛ)
as in Definition 3.1; also, we shall use the various notations introduced in Definition
3.1 for various objects associated to this initial Θ-data.
Definition 4.1.
(i) We define a holomorphic base-prime-strip, orD-prime-strip, [relative to the
given initial Θ-data] to be a collection of data
† D = { † D v } v∈V
that satisfies the following conditions: (a) if v ∈ V non ,then † D v is a category which
admits an equivalence of categories † ∼
D v →Dv [where D v is as in Examples 3.2,
(i); 3.3, (i)]; (b) if v ∈ V arc ,then † D v is an Aut-holomorphic orbispace such that
there exists an isomorphism of Aut-holomorphic orbispaces † ∼
D v →Dv [where D v
is as in Example 3.4, (i)]. Observe that if v ∈ V non ,thenπ 1 ( † D v ) determines, in
a functorial fashion, a profinite group corresponding to “C v ” [cf. Corollary 1.2 if
v ∈ V good ; [EtTh], Proposition 2.4, if v ∈ V bad ], which contains π 1 ( † D v )asan
open subgroup; thus, if we write † D v for B(−) 0 of this profinite group, then we
obtain a natural morphism † D v → † D v [cf. §0]. In a similar vein, if v ∈ V arc ,
then since −→v X admits a K v -core, a routine translation into the “language of Autholomorphic
orbispaces” of the argument given in the proof of Corollary 1.2 [cf. also
[AbsTopIII], Corollary 2.4] reveals that † D v determines, in a functorial fashion, an
Aut-holomorphic orbispace † D v corresponding to “C v ”, together with a natural
morphism † D v → † D v of Aut-holomorphic orbispaces. Thus, in summary, one
obtains a collection of data
† D = { † D v } v∈V