Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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6 SHINICHI MOCHIZUKI degrees ∈ R at v ∈ V ( ∼ → V mod ) that are related to one another by means of the product formula. Throughout the present series of papers, we shall use the notation to denote such prime-strips. ⊩ Type of prime-strip Model at v ∈ V bad Reference D Π v I, 4.1, (i) D ⊢ G v I, 4.1, (iii) F Π v O ⊲ F v I, 5.2, (i) F ⊢ G v O × F v × q N v I, 5.2, (ii) F ⊢× G v O × F v II, 4.9, (vii) F ⊢×μ G v O ×μ F v def = O × F v /O μ F v II, 4.9, (vii) F ⊢◮×μ G v O ×μ F v × q N v II, 4.9, (vii) F ⊢◮ G v q N v III, 2.4, (ii) F ⊢⊥ G v O μ 2l F v × q N v III, 2.4, (ii) F ⊩... = F ⊢... + {global realified Frobenioid associated to F mod } Fig. I1.2: Types of prime-strips In some sense, the main goal of the present paper may be thought of as the construction of Θ ±ell NF-Hodge theaters [cf. Definition 6.13, (i)] † HT Θ±ell NF
INTER-UNIVERSAL TEICHMÜLLER THEORY I 7 — which may be thought of as “miniature models of conventional scheme theory” — given, roughly speaking, by systems of Frobenioids. To any such Θ ±ell NF-Hodge theater † HT Θ±ellNF , one may associate a D-Θ ±ell NF-Hodge theater [cf. Definition 6.13, (ii)] † HT D-Θ±ell NF — i.e., the associated system of base categories. One may think of a Θ ±ell NF-Hodge theater † HT Θ±ellNF as the result of gluing together a Θ ±ell -Hodge theater † HT Θ±ell to a ΘNF-Hodge theater † HT ΘNF [cf. Remark 6.12.2, (ii)]. In a similar vein, one may think of a D-Θ ±ell NF-Hodge theater † HT D-Θ±ellNF as the result of gluing together a D-Θ ±ell -Hodge theater † HT D-Θ±ell to a D-ΘNF-Hodge theater † HT D-ΘNF .AD-Θ ±ell -Hodge theater † HT D-Θ±ell may be thought of as a bookkeeping device that allows one to keep track of the action of the F ⋊± l -symmetry on the labels ( −l < ... < −1 < 0 < 1 < ... < l ) — which we think of as elements ∈ F l — in the context of the [orbi]curves X K , X v [for v ∈ V bad ], and −→v X [for v ∈ V good ]. The F ⋊± l -symmetry is represented in a D-Θ ±ell -Hodge theater † HT D-Θ±ell by a category equivalent to [the full subcategory determined by the connected objects of] the Galois category of finite étale coverings of X K . On the other hand, each of the labels referred to above is represented in a D-Θ ±ell -Hodge theater † HT D-Θ±ell by a D-prime-strip. In a similar vein, a D-ΘNF-Hodge theater † HT D-ΘNF may be thought of as a bookkeeping device that allows one to keep track of the action of the F l -symmetry on the labels (1 < ... < l ) — which we think of as elements ∈ F l — in the context of the orbicurves C K , X v [for v ∈ V bad ], and −→v X [for v ∈ V good ]. The F l -symmetry is represented in a D-ΘNF-Hodge theater † HT D-ΘNF by a category equivalent to [the full subcategory determined by the connected objects of] the Galois category of finite étale coverings of C K . On the other hand, each of the labels referred to above is represented in a D- ΘNF-Hodge theater † HT D-ΘNF by a D-prime-strip. Thecombinatorial structure of D-ΘNF- and D-Θ ±ell -Hodge theaters summarized above [cf. also Fig. I1.3 below] is one of the main topics of the present paper and is discussed in detail in §4 and §6. The left-hand portion of Fig. I1.3 corresponds to the D-Θ ±ell -Hodge theater; the right-hand portion of Fig. I1.3 corresponds to the D-ΘNF-Hodge theater; these left-hand and right-hand portions are glued together along a single D-prime-strip, depicted as “[1 < ... < l ]”, in such a way that the labels 0 ≠ ±t ∈ F l on the left are identified with the corresponding label j ∈ F l on the right. The F ⋊± l -symmetry has the advantange that, being geometric in nature, it allows one to permute various copies of “G v ”[wherev ∈ V non ] associated to distinct labels ∈ F l without inducing conjugacy indeterminacies. This phenomenon, which we shall refer to as conjugate synchronization, will play a key role in the Kummer theory surrounding the Hodge-Arakelov-theoretic evaluation of the
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INTER-UNIVERSAL TEICHMÜLLER THEORY
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completely determined by † D. INT
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conjugation by which maps φ NF IN
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(v) Write INTER-UNIVERSAL TEICHMÜL
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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