Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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66 SHINICHI MOCHIZUKI for □ ∈{X v , C v , X v , C v , −→v X , −→v C },wehaveacomplex archimedean topological field [i.e., a “CAF” — cf. §0] A □ [cf. [AbsTopIII], Definition 4.1, (i)] which may be algorithmically constructed from def □; writeA □ = A □ \{0}. Next, let us write and D v def = X −→v C v for the archimedean Frobenioid as in [FrdII], Example 3.3, (ii) [i.e., “C” ofloc. cit.], wherewetakethebase category [i.e., “D” ofloc. cit.] to be the one-morphism category determined by Spec(K v ). Thus, the linear morphisms among the pseudoterminal objects of C determine unique isomorphisms [cf. [FrdI], Definition 1.3, (iii), (c)] among the respective topological monoids “O ⊲ (−)”—wherewerecall [cf. [FrdI], Theorem 3.4, (iii); [FrdII], Theorem 3.6, (i), (vii)] that these topological monoids may be reconstructed category-theoretically from C. In particular, it makes sense to write “O ⊲ (C v )”, “O × (C v ) ⊆O ⊲ (C v )”. Moreover, we observe that, by construction, there is a natural isomorphism O ⊲ (C v ) →O ∼ K ⊲ v of topological monoids. Thus, one may also think of C v as a “Frobenioid-theoretic representation” of the topological monoid OK ⊲ v [cf. [AbsTopIII], Remark 4.1.1]. ∼ Observe that there is a natural topological isomorphism K v → ADv ,whichmaybe restricted to OK ⊲ v to obtain an inclusion of topological monoids κ v : O ⊲ (C v ) ↩→ A Dv — which we shall refer to as the Kummer structure on C v [cf. Remark 3.4.2 below]. Write def F v =(C v , D v ,κ v ) [cf. Example 3.2, (i); Example 3.3, (i)]. (ii) Next, recall the category TM ⊢ of “split topological monoids” of [AbsTopIII], Definition 5.6, (i) — i.e., the category whose objects (C, −→ C ) consist of a topological monoid C isomorphic to OC ⊲ and a topological submonoid −→ C ⊆ C [necessarily isomorphic to R ≥0 ] such that the natural inclusions C × ↩→ C [where C × , which is necessarily isomorphic to S 1 , denotes the topological submonoid of invertible elements], −→ C ↩→ C determine an isomorphism C × × −→ C → ∼ C of topological monoids, and whose morphisms (C 1 , −→ C 1 ) → (C 2 , −→ C 2 ) are isomorphisms of topological monoids C 1 → C2 that induce isomorphisms −→ ∼ −→ ∼ C 1 → C 2 . Note that the CAF’s K v , A Dv determine, in a natural way, objects of TM ⊢ .Write τ ⊢ v
INTER-UNIVERSAL TEICHMÜLLER THEORY I 67 for the resulting characteristic splitting of the Frobenioid Cv ⊢ def = C v , i.e., so that we may think of the pair (O ⊲ (Cv ⊢ ),τv ⊢ ) as the object of TM ⊢ determined by K v ; D ⊢ v for the object of TM ⊢ determined by A Dv ; F ⊢ v def =(C ⊢ v , D ⊢ v ,τ ⊢ v ) for the [ordered] triple consisting of Cv ⊢ , Dv ⊢ ,andτv ⊢ . Thus, the object (O ⊲ (Cv ⊢ ),τv ⊢ ) of TM ⊢ is isomorphic to Dv ⊢ .Moreover,Cv ⊢ (respectively, Dv ⊢ ; Fv ⊢ )maybealgorithmically reconstructed from F v (respectively, D v ; F v ). (iii) Next, let us observe that p v ∈ K v [cf. §0] may be thought of as a(n) [nonidentity] element of the noncompact factor Φ C ⊢ v [i.e., the factor denoted by a “→” in the definition of TM ⊢ ] of the object (O ⊲ (Cv ⊢ ),τv ⊢ )ofTM ⊢ . This noncompact factor Φ C ⊢ v is isomorphic, as a topological monoid, to R ≥0 ;letuswriteΦ C ⊢ v additively and denote by log(p v ) the element of Φ C ⊢ v determined by p v . Thus, relative to the natural action [by multiplication!] of R ≥0 on Φ C ⊢ v , it follows that log(p v )isa generator of Φ C ⊢ v . In particular, we may form a new topological monoid Φ C Θ v def = R ≥0 · log(p v ) · log(Θ) isomorphic to R ≥0 that is generated by a formal symbol “log(p v )·log(Θ) = log(p log(Θ) v )”. Moreover, if we denote by O × the compact factor of the object (O ⊲ (C ⊢ Cv ⊢ v ),τv ⊢ )of TM ⊢ , and set O × def = O × , then we obtain a new split Frobenioid (C Θ Cv Θ Cv ⊢ v ,τv Θ ), isomorphic to (Cv ⊢ ,τv ⊢ ), such that O ⊲ (C Θ v )=O × C Θ v × Φ C Θ v — where we note that this equality gives rise to a natural isomorphism of split Frobenioids (C ⊢ v ,τ ⊢ v ) ∼ → (C Θ v ,τ Θ v ), obtained by “forgetting the formal symbol log(Θ)”. In particular, we thus obtain a natural isomorphism F ⊢ v ∼ →F Θ v —wherewewriteFv Θ def = (Cv Θ , Dv Θ ,τv Θ ) for the [ordered] triple consisting of Cv Θ , Dv Θ def = Dv ⊢ , τv Θ . Finally, we observe that Fv Θ may be algorithmically reconstructed from F v . Remark 3.4.1. A similar remark to Remark 3.2.1 [i.e., concerning the phrase “reconstructed category-theoretically”] applies to the phrase “algorithmically reconstructed” that was applied in the discussion of Example 3.4.
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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