Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

16 SHINICHI MOCHIZUKI
for the tempered fundamental group π tp
1 (X) [relative to a suitable basepoint]
of X [cf. [André], §4; [SemiAnbd], Example 3.10]; ̂Π X for the étale fundamental
group [relative to a suitable basepoint] of X. Thus, we have a natural inclusion
Π tp X
↩→ ̂Π X
which allows one to identify ̂Π X with the profinite completion of Π tp X
. Then every
decomposition group in ̂Π X (respectively, inertia group in ̂Π X ) associated to
a closed point or cusp of X (respectively, to a cusp of X) iscontainedinΠ tp X
if
and only if it is a decomposition group in Π tp X (respectively, inertia group in Πtp X )
associated to a closed point or cusp of X (respectively, to a cusp of X). Moreover,
a ̂Π X -conjugate of Π tp X contains a decomposition group in Πtp X
(respectively, inertia
group in Π tp X
) associated to a closed point or cusp of X (respectively, to a cusp of
X) if and only if it is equal to Π tp X .
Theorem B is [essentially] given as Corollary 2.5 in §2. Here, we note that
although, in the statement of Corollary 2.5, the hyperbolic curve X is assumed to
admit stable reduction over the ring of integers O k of k, one verifies immediately
that this assumption is, in fact, unnecessary.
Finally, we remark that one important reason for the need to apply Theorem B
in the context of the theory of Θ ±ell NF-Hodge theaters summarized in Theorem A
is the following. The F ⋊±
l
-symmetry, which will play a crucial role in the theory
of the present series of papers [cf., especially, [IUTchII], [IUTchIII]], depends, in an
essential way, on the synchronization of the ±-indeterminacies that occur locally
at each v ∈ V [cf. Fig. I1.1]. Such a synchronization may only be obtained by
making use of the global portion of the Θ ±ell -Hodge theater under consideration.
On the other hand, in order to avail oneself of such global ±-synchronizations
[cf. Remark 6.12.4, (iii)], it is necessary to regard the various labels of the F ⋊±
l
-
symmetry
( −l < ... < −1 < 0 < 1 < ... < l )
as conjugacy classes of inertia groups of the [necessarily] profinite geometric étale
fundamental group of X K . That is to say, in order to relate such global profinite
conjugacy classes to the corresponding tempered conjugacy classes [i.e., conjugacy
classes with respect to the geometric tempered fundamental group] of inertia groups
at v ∈ V bad [i.e., where the crucial Hodge-Arakelov-theoretic evaluation is to be
performed!], it is necessary to apply Theorem B — cf. the discussion of Remark
4.5.1; [IUTchII], Remark 2.5.2, for more details.
§I2. Gluing Together Models of Conventional Scheme Theory
As discussed in §I1, the system of Frobenioids constituted by a Θ ±ell NF-Hodge
theater is intended to be a sort of miniature model of conventional scheme theory.
Onethenglues multiple Θ ±ell NF-Hodge theaters { n HT Θ±ell NF } n∈Z together