Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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22 SHINICHI MOCHIZUKI arises from a “deep sensitivity to particular choices of basepoints” — is the phenomenon of conjugate synchronization, i.e., of synchronization between conjugacy indeterminacies of distinct copies of various local Galois groups, which, as was mentioned in §I1, will play an important role in the theory of [IUTchII], [IUTchIII]. The various rigidity properties of the étale theta function established in [EtTh] constitute yet another inter-universal phenomenon that will play an important role in theory of [IUTchII], [IUTchIII]. §I4. Relation to Complex and p-adic Teichmüller Theory In order to understand the sense in which the theory of the present series of papers may be thought of as a sort of “Teichmüller theory” of number fields equipped with an elliptic curve, it is useful to recall certain basic, well-known facts concerning the classical complex Teichmüller theory of Riemann surfaces of finite type [cf., e.g., [Lehto], Chapter V, §8]. Although such a Riemann surface is one-dimensional from a complex, holomorphic point of view, this single complex dimension may be thought of consisting of two underlying real analytic dimensions. Relative to a suitable canonical holomorphic coordinate z = x + iy on the Riemann surface, the Teichmüller deformation may be written in the form z ↦→ ζ = ξ + iη = Kx + iy —where1
INTER-UNIVERSAL TEICHMÜLLER THEORY I 23 its usual topology], the two combinatorial dimensions of k may also be thought of as corresponding to the · two underlying topological/real dimensions of k. Alternatively, in both the nonarchimedean and archimedean cases, one may think of the two underlying combinatorial dimensions of k as corresponding to the · group of units O × k and value group k× /O × k of k. Indeed, in the nonarchimedean case, local class field theory implies that this last point of view is consistent with the interpretation of the two underlying combinatorial dimensions via cohomological dimension; in the archimedean case, the consistency of this last point of view with the interpretation of the two underlying combinatorial dimensions via topological/real dimension is immediate from the definitions. This last interpretation in terms of groups of units and value groups is of particular relevance in the context of the theory of the present series of papers. That is to say, one may think of the Θ-link † F ⊩ tht ∼ → ‡ F ⊩ mod { † Θ v ↦→ ‡ q v } v∈V bad — which, as discussed in §I1, induces a full poly-isomorphism † F ⊢× mod ∼ → ‡ F ⊢× mod {O × F v ∼ → O × F v } v∈V bad —asasortof“Teichmüller deformation relative to a Θ-dilation”, i.e., a deformation of the ring structure of the number field equipped with an elliptic curve constituted by the given initial Θ-data in which one dilates the underlying combinatorial dimension corresponding to the local value groups relative to a “Θfactor”, while one leaves fixed, up to isomorphism, the underlying combinatorial dimension corresponding to the local groups of units [cf. Remark 3.9.3]. This point of view is reminiscent of the discussion in §I1 of “disentangling/dismantling” of various structures associated to number field. In [IUTchIII], we shall consider two-dimensional diagrams of Θ ±ell NF-Hodge theaters which we shall refer to as log-theta-lattices. The two dimensions of such diagrams correspond precisely to the two underlying combinatorial dimensions of a ring. Of these two dimensions, the “theta dimension” consists of the Frobeniuspicture associated to [more elaborate versions of] the Θ-link. Manyoftheimportant properties that involve this “theta dimension” are consequences of the theory of [FrdI], [FrdII], [EtTh]. On the other hand, the “log dimension” consists of iterated copies of the log-link, i.e., diagrams of the sort that are studied in [AbsTopIII]. That is to say, whereas the “theta dimension” corresponds to “deformations of the arithmetic holomorphic structure” of the given number field equipped with an elliptic curve, this “log dimension” corresponds to “rotations of the two underlying
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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