Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 93
by considering the closures in π 1 (D ⊚ ) of the images of cuspidal inertia
groups of π 1 (D vj ) [cf. the discussion of Remark 4.5.1 below],
we conclude that these homomorphisms induce a natural isomorphism of F l -torsors
LabCusp(D ⊚ ) → ∼ LabCusp(D vj ). Now let us observe that it follows immediately
from the definitions that, as one allows v to vary, these isomorphisms of F l -torsors
LabCusp(D ⊚ ) → ∼ LabCusp(D vj ) are compatible with the natural bijections in the
first display of Proposition 4.2, hence determine an isomorphism of F l -torsors
LabCusp(D ⊚ ) → ∼ LabCusp(D j ). Next, let us note that the data of the arrow
φ Θ j : D j → D > at the various v ∈ V determines an isomorphism of F l -torsors
LabCusp(D j ) → ∼ LabCusp(D > )[whichmaybecomposed with the previous isomorphism
of F l -torsors LabCusp(D⊚ ) → ∼ LabCusp(D j )]. Indeed, this is immediate
from the definitions when v ∈ V good ;whenv ∈ V bad , it follows immediately from
the discussion of Example 4.4, (ii).
(ii) The discussion of (i) may be summarized as follows:
for each j ∈ F l
, restriction at the various v ∈ V via φNF j , φ Θ j determines
an isomorphism of F l -torsors
φ LC
j : LabCusp(D ⊚ ) ∼ → LabCusp(D > )
such that φ LC
j
j ∈ F l .
is obtained from φ LC
1 by composing with the action by
Write [ɛ] ∈ LabCusp(D ⊚ ) for the element determined by ɛ. Then we observe that
φ LC
j ([ɛ]) ↦→ j; φ LC
1 (j · [ɛ]) ↦→ j
via the natural bijection LabCusp(D > ) ∼ → F l
of Proposition 4.2. In particular,
the element [ɛ] ∈ LabCusp(D ⊚ )maybecharacterized as the unique element ∈
LabCusp(D ⊚ ) such that evaluation at the element yields the assignment φ LC
j ↦→ j.
Remark 4.5.1.
(i) Let G be a group. If H ⊆ G is a subgroup, g ∈ G, then we shall write
H g def
= g · H · g −1 .LetJ ⊆ H ⊆ G be subgroups. Suppose further that each of the
subgroups J, H of G is only known up to conjugacy in G. Put another way, we
suppose that we are in a situation in which there are independent G-conjugacy
indeterminacies in the specification of the subgroups J and H. Thus, for instance,
there is no natural way to distinguish the given inclusion ι : J↩→ H from its γ-
conjugate ι γ : J γ ↩→ H γ ,forγ ∈ G. Moreover, it may happen to be the case that
for some g ∈ G, not only J, but also J g ⊆ H [or, equivalently J ⊆ H g−1 ]. Here, the
subgroups J, J g of H are not necessarily conjugate in H; indeed, the abstract pairs
of a group and a subgroup given by (H, J) and(H, J g ) need not be isomorphic
[i.e., it is not even necessarily the case that there exists an automorphism of H
that maps J onto J g ]. In particular, the existence of the independent G-conjugacy