Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

136 SHINICHI MOCHIZUKI
determines a [single, well-defined!] bijection
( † ζ ± ) −1 : T ∼ → LabCusp ± ( † D ⊚± )
[i.e., whose inverse we denote by † ζ ± ]—whichiscompatible with the respective
F ± l
-torsor structures. Moreover, for any t ∈ T , the composite bijection
( † ζ Θell
0 ) −1 ◦ ( † ζ Θell
t ) ◦ ( † ζ Θ±
t ) −1 ◦ ( † ζ Θ±
0 ) : LabCusp ± ( † D 0 ) ∼ → LabCusp ± ( † D 0 )
coincides with the automorphism of the set LabCusp ± ( † D 0 ) determined, relative to
the F ± l -group structure on this set, by the action of († ζ Θell
0 ) −1 (( † ζ ± ) −1 (t)).
(iv) Let α ∈ Aut ± ( † D ⊚± )/Aut csp ( † D ⊚± ). Then if one replaces † φ Θell
± by α ◦
† φ Θell
± [cf. Proposition 6.6, (iv), below], then the resulting “ † ζt
Θell ” is related to the
“ † ζt
Θell ” determined by the original † φ Θell
± by post-composition with the image of α
via the natural bijection
Aut ± ( † D ⊚± )/Aut csp ( † D ⊚± ) ∼ → Aut ± (LabCusp ± ( † D ⊚± )) ( ∼ = F ⋊±
l
)
determined by the tautological action of Aut ± ( † D ⊚± )/Aut csp ( † D ⊚± ) on the set of
±-label classes of cusps LabCusp ± ( † D ⊚± ).
Next, let us observe that it follows immediately from the various definitions
involved [cf. the discussion of Definition 6.1; Examples 6.2, 6.3], together with the
explicit description of the various poly-automorphisms discussed in Examples 6.2,
(ii), (iii); 6.3, (ii) [cf. also the various properties discussed in Proposition 6.5], that
we have the following additive analogue of Proposition 4.8.
Proposition 6.6. (First Properties of Base-Θ ± -Bridges, Base-Θ ell -Bridges,
and Base-Θ ±ell -Hodge Theaters) Relative to a fixed collection of initial Θ-
data:
(i) The set of isomorphisms between two D-Θ ± -bridges forms a torsor
over the group
{±1} ×
({±1} V)
— where the first (respectively, second) factor corresponds to poly-automorphisms of
the sort described in Example 6.2, (ii) (respectively, Example 6.2, (iii)). Moreover,
the first factor may be thought of as corresponding to the induced isomorphisms of
F ± l
-groups between the index sets of the capsules involved.
(ii) The set of isomorphisms between two D-Θ ell -bridges forms an F ⋊±
l
-
torsor — i.e., more precisely, a torsor over a finite group that is equipped with a
natural outer isomorphism to F ⋊±
l
. Moreover, this set of isomorphisms maps
bijectively, by considering the induced bijections, to the set of isomorphisms of
F ± l
-torsors between the index sets of the capsules involved.