Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

conjugation by which maps φ NF
INTER-UNIVERSAL TEICHMÜLLER THEORY I 97
↦→ † φ NF
, φ Θ ↦→ † φ Θ . A(n) [iso]morphism of D-
ΘNF-Hodge theaters is defined to be a pair of morphisms between the respective
associated D-NF- and D-Θ-bridges that are compatible with one another in the
sense that they induce the same bijection between the index sets of the respective
capsules of D-prime-strips. There is an evident notion of composition of morphisms
of D-ΘNF-Hodge theaters.
Proposition 4.7.
Bridges) Let
(Transport of Label Classes of Cusps via Base-
† HT D-ΘNF =( † D ⊚ † φ NF
←− † † φ Θ
D J −→ † D > )
be a D-ΘNF-Hodge theater [relative to the given initial Θ-data]. Then:
(i) The structure at the v ∈ V bad of the D-Θ-bridge † φ Θ determines a bijection
† χ : π 0 ( † D J )=J ∼ → F l
— i.e., determines labels ∈ F l
for the constituent D-prime-strips of the capsule
† D J .
(ii) For each j ∈ J, restriction at the various v ∈ V [cf. Example 4.5] via the
portion of † φ NF
, † φ Θ indexed by j determines an isomorphism of F l -torsors
† φ LC
j : LabCusp( † D ⊚ ) ∼ → LabCusp( † D > )
such that † φ LC
j is obtained from † φ LC
1 [where, by abuse of notation, we write “1 ∈ J”
for the element of J that maps via † χ to the image of 1 in F l
] by composing with
the action by † χ(j) ∈ F l .
(iii) There exists a unique element
[ † ɛ] ∈ LabCusp( † D ⊚ )
such that for each j ∈ J, the natural bijection LabCusp( † D > ) → ∼ F l
of the
second display of Proposition 4.2 maps † φ LC
j ([ † ɛ]) = † φ LC
1 ( † χ(j) · [ † ɛ]) ↦→ † χ(j). In
particular, the element [ † ɛ] determines an isomorphism of F l -torsors
† ζ : LabCusp( † D ⊚ ) ∼ → J ( ∼ → F l )
[where the bijection in parentheses is the bijection † χ of (i)] between “global
cusps” [i.e., “ † χ(j) · [ † ɛ]”] and capsule indices [i.e., j ∈ J → ∼ F l
]. Finally,
when considered up to composition with multiplication by an element of F l ,the
bijection † ζ is independent of the choice of † φ NF
within the F l -orbit of † φ NF
relative to the natural poly-action of F l
on † D ⊚ [cf. Example 4.3, (iii); Fig. 4.4
below].
Proof. Assertion (i) follows immediately from the definitions [cf. Example 4.4,
(i), (ii), (iv); Definition 4.6], together with the bijection of the second display of