Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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50 SHINICHI MOCHIZUKI Section 3: Chains of Θ-Hodge Theaters In the present §3, we construct chains of “Θ-Hodge theaters”. Each “Θ-Hodge theater” is to be thought of as a sort of miniature model of the conventional scheme-theoretic arithmetic geometry that surrounds the theta function. This miniature model is formulated via the theory of Frobenioids [cf. [FrdI]; [FrdII]; [EtTh], §3, §4, §5]. On the other hand, the link [cf. Corollary 3.7, (i)] between adjacent members of such chains is purely Frobenioid-theoretic, i.e., it lies outside the framework of ring theory/scheme theory. It is these chains of Θ-Hodge theaters that form the starting point of the theory of the present series of papers. Definition 3.1. We shall refer to as initial Θ-data any collection of data that satisfies the following conditions: (F/F, X F , l, C K , V, V bad mod, ɛ) (a) F is a number field such that √ −1 ∈ F ; F is an algebraic closure of F . def Write G F = Gal(F/F). (b) X F is a once-punctured elliptic curve [i.e., a hyperbolic curve of type (1, 1)] over F that admits stable reduction over all v ∈ V(F ) non .WriteE F for the elliptic curve over F determined by X F [so X F ⊆ E F ]; X F → C F for the hyperbolic orbicurve [cf. §0] over F obtained by forming the stacktheoretic quotient of X F by the unique F -involution [i.e., automorphism of order two] “−1” of X F ; F mod ⊆ F for the field of moduli [cf., e.g., def [AbsTopIII], Definition 5.1, (ii)] of X F ; V mod = V(F mod ). Then V good mod V bad mod ⊆ V mod is a nonempty set of nonarchimedean valuations of F mod of odd residue characteristic over which X F has bad [i.e., multiplicative] reduction. Write def = V mod \ V bad def mod ; V(F )□ = V □ mod × V mod V(F )for□ ∈{bad, good}; Π XF def def = π 1 (X F ) ⊆ Π CF = π 1 (C F ) Δ X def = π 1 (X F × F F ) ⊆ Δ C def = π 1 (C F × F F ) for the étale fundamental groups [relative to appropriate choices of basepoints] of X F , C F , X F × F F , C F × F F . [Thus, we have natural exact sequences 1 → Δ (−) → Π (−)F → G F → 1 for “(−)” taken to be either “X” or“C”.] Here, we suppose further that the extension F/F mod is Galois, and that the 2 · 3-torsion points of E F are rational over F . (c) l is a prime number ≥ 5 such that the outer homomorphism G F → GL 2 (F l )
INTER-UNIVERSAL TEICHMÜLLER THEORY I 51 determined by the l-torsion points of E F contains the subgroup SL 2 (F l ) ⊆ GL 2 (F l ); write K ⊆ F for the finite Galois extension of F determined by the kernel of this homomorphism. Also, we suppose that l is prime to the elements of V bad mod ,aswellastotheorders of the q-parameters of E F [i.e., in the terminology of [GenEll], Definition 3.3, the “local heights” of E F ] at the primes of V(F ) bad . (d) C K is a hyperbolic orbicurve of type (1,l-tors) ± [cf. [EtTh], Definition 2.1] over K, withK-core [cf. [CanLift], Remark 2.1.1; [EtTh], the discussion def at the beginning of §2] given by C K = C F × F K. [Thus, by (c), it follows that C K is completely determined, up to isomorphism over F ,byC F .] In particular, C K determines, up to K-isomorphism, a hyperbolic orbicurve X K of type (1,l-tors) [cf. [EtTh], Definition 2.1] over K, together with natural cartesian diagrams X K −→ X F Π XK −→ Π XF Δ X −→ Δ X ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ↓ ↓ ↓ ↓ ↓ ↓ C K −→ C F Π CK −→ Π CF Δ C −→ Δ C of finite étale coverings of hyperbolic orbicurves and open immersions of profinite groups. Finally, we recall from [EtTh], Proposition 2.2, that Δ C admits uniquely determined open subgroups Δ X ⊆ Δ C ⊆ Δ C ,which may be thought of as corresponding to finite étale coverings of C F C × F F by hyperbolic orbicurves X F , C F of type (1,l-tors Θ ), (1,l-tors Θ ) ± , respectively [cf. [EtTh], Definition 2.3]. (e) V ⊆ V(K) is a subset that induces a natural bijection def = V ∼ → V mod — i.e., a section of the natural surjection V(K) ↠ V mod . Write V non def = V ⋂ V(K) non , V arc def = V ⋂ V(K) arc , V good def = V ⋂ V(K) good , V bad def = V ⋂ V(K) bad . For each v ∈ V(K), we shall use the subscript v to denote the result of base-changing hyperbolic orbicurves over F or K to K v . Thus, for each v ∈ V(K) lying under a v ∈ V(F ), we have natural cartesian diagrams X v −→ X v −→ X v Δ X −→ Π Xv −→ Π Xv ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ↓ ↓ ↓ ↓ ↓ ↓ C v −→ C v −→ C v Δ C −→ Π Cv −→ Π Cv of profinite étale coverings of hyperbolic orbicurves and injections of profinite groups. Here, the subscript v denotes base-change with respect to F↩→ F v ; the various profinite groups “Π (−) ”admitnatural outer surjections onto the decomposition group G v ⊆ G K = Gal(F/K) determined, def up to G K -conjugacy, by v. If v ∈ V bad , then we assume further that the
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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