Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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134 SHINICHI MOCHIZUKI ∼ — where we require that the bijection of index sets F l → T induced by the second isomorphism determine an isomorphism of F ± l -torsors — conjugation by which maps φ Θell ± ↦→ † φ Θell ± . We define a(n) [iso]morphism of D-Θ ell -bridges ( † † φ Θell ± D T −→ † D ⊚± ) → ( ‡ D T ′ to be a pair of poly-morphisms ‡ φ Θell ± −→ ‡ D ⊚± ) † D T ∼ → ‡ D T ′; † D ⊚± ∼ → ‡ D ⊚± —where † ∼ D T → ‡ D T ′ is a capsule-+-full poly-isomorphism whose induced morphism on index sets T → ∼ T ′ is an isomorphism of F ± l -torsors; † D ⊚± → ‡ D ⊚± is a poly-morphism which is an Aut csp ( † D ⊚± )- [or, equivalently, Aut csp ( ‡ D ⊚± )-] orbit of isomorphisms — which are compatible with † φ Θell ± , ‡ φ Θell ± . There is an evident notion of composition of morphisms of D-Θ ell -bridges. (iii) We define a base-Θ ±ell -Hodge theater, orD-Θ ±ell -Hodge theater, [relative to the given initial Θ-data] to be a collection of data † HT D-Θ±ell =( † † φ Θ± ± D ≻ ←− † † φ Θell ± D T −→ † D ⊚± ) —whereT is a F ± l -group; † φ Θ± ± is a D-Θ ± -bridge; † φ Θell ± is a D-Θ ell -bridge [relative to the F ± l -torsor structure determined by the F± l -group structure on T ]—such that there exist isomorphisms D ≻ ∼ → † D ≻ ; D ± ∼ → † D T ; D ⊚± ∼ → † D ⊚± conjugation by which maps φ Θ± ± ↦→ † φ Θ± ± , φ Θell ± ↦→ † φ Θell ± . A(n) [iso]morphism of D-Θ ±ell -Hodge theaters is defined to be a pair of morphisms between the respective associated D-Θ ± -andD-Θ ell -bridges that are compatible with one another in the sense that they induce the same poly-isomorphism between the respective capsules of D-prime-strips. There is an evident notion of composition of morphisms of D- Θ ±ell -Hodge theaters. The following additive analogue of Proposition 4.7 follows immediately from the various definitions involved. Put another way, the content of Proposition 6.5 below may be thought of as a sort of “intrinsic version” of the constructions carried out in Examples 6.2, 6.3. Proposition 6.5. Bridges) Let (Transport of ±-Label Classes of Cusps via Base- † HT D-Θ±ell =( † † φ Θ± ± D ≻ ←− † † φ Θell ± D T −→ † D ⊚± ) be a D-Θ ±ell -Hodge theater [relative to the given initial Θ-data]. Then: (i) For each v ∈ V, t ∈ T ,theD-Θ ell -bridge † φ Θell ± induces a [single, welldefined!] bijection of sets of ±-label classes of cusps † ζ Θell v t : LabCusp ± ( † D vt ) ∼ → LabCusp ± ( † D ⊚± )
INTER-UNIVERSAL TEICHMÜLLER THEORY I 135 that is compatible with the respective F ± l the bijection -torsor structures. Moreover, for w ∈ V, † ξ Θell v t ,w t def =( † ζ Θell w t ) −1 ◦ ( † ζ Θell v t ) : LabCusp ± ( † D vt ) ∼ → LabCusp ± ( † D wt ) is compatible with the respective F ± l -group structures. Write LabCusp ± ( † D t ) for the F ± l -group obtained by identifying the various F± l -groups LabCusp± ( † D vt ), as v ranges over the elements of V, via the various † ξv Θell t ,w t . Finally, the various determine a [single, well-defined!] bijection † ζ Θell v t † ζ Θell t : LabCusp ± ( † D t ) ∼ → LabCusp ± ( † D ⊚± ) — which is compatible with the respective F ± l -torsor structures. (ii) For each v ∈ V, t ∈ T ,theD-Θ ± -bridge † φ Θ± ± induces a [single, welldefined!] bijection of sets of ±-label classes of cusps † ζ Θ± v t : LabCusp ± ( † D vt ) ∼ → LabCusp ± ( † D ≻,v ) that is compatible with the respective F ± l the bijections -group structures. Moreover, for w ∈ V, † ξ≻,v,w Θ± def † ξ Θ± v t ,w t =( † ζ Θ± w 0 ) ◦ † ξ Θell v 0 ,w 0 ◦ ( † ζ Θ± v 0 ) −1 : LabCusp ± ( † D ≻,v ) ∼ → LabCusp ± ( † D ≻,w ); def =( † ζ Θ± w t ) −1 ◦ † ξ Θ± ≻,v,w ◦ ( † ζ Θ± v t ) : LabCusp ± ( † D vt ) ∼ → LabCusp ± ( † D wt ) — where, by abuse of notation, we write “0” for the zero element of the F ± l -group LabCusp ± ( † D t ) —arecompatible with the respective F ± l -group structures, and we have † ξ Θ± v t ,w t = † ξ Θell v t ,w t . Write LabCusp ± ( † D ≻ ) for the F ± l -group obtained by identifying the various F± l -groups LabCusp± ( † D ≻,v ), as v ranges over the elements of V, via the various † ξ≻,v,w. Θ± Finally, for any t ∈ T , the various † ζv Θ± t , † ζv Θell t determine, respectively, a [single, well-defined!] bijection † ζ Θ± t : LabCusp ± ( † D t ) ∼ → LabCusp ± ( † D ≻ ); — which is compatible with the respective F ± l -group structures. (iii) The assignment T ∋ t ↦→ † ζ Θell t (0) ∈ LabCusp ± ( † D ⊚± )
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completely determined by † D. INT
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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