Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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20 SHINICHI MOCHIZUKI in [IUTchII], [IUTchIII]. Once one constructs the Frobenius-picture, one natural and fundamental problem, which will, in fact, be one of the main themes of the present series of papers, is the problem of describing an alien “arithmetic holomorphic structure” [i.e., an alien “conventional scheme theory”] corresponding to some m HT Θ±ell NF in terms of a “known arithmetic holomorphic structure” corresponding to n HT Θ±ell NF [where n ≠ m] — a problem, which, as discussed in §I1, will be approached, in the final portion of [IUTchIII], by applying various results from absolute anabelian geometry [i.e., more explicitly, the theory of [SemiAnbd], [EtTh], and [AbsTopIII]] to the various tempered and étale fundamental groups that appear in the étale-picture. The relevance to this problem of the extensive theory of “reconstruction of ring/scheme structures” provided by absolute anabelian geometry is evident from the statement of the problem. On the other hand, in this context, it is of interest to note that, unlike conventional anabelian geometry, which typically centers on the goal of reconstructing a “known scheme-theoretic object”, in the present series of papers, we wish to apply techniques and results from anabelian geometry in order to analyze the structure of an unknown, essentially non-scheme-theoretic object, namely, the Frobenius-picture, as described above. Put another way, relative to the point of view that “Galois groups are arithmetic tangent bundles” [cf. the theory of the arithmetic Kodaira-Spencer morphism in [HASurI]], one may think of conventional anabelian geometry as corresponding to the computation of the automorphisms of a scheme as H 0 (arithmetic tangent bundle) and of the application of absolute anabelian geometry to the analysis of the Frobeniuspicture, i.e., to the solution of the problem discussed above, as corresponding to the computation of H 1 (arithmetic tangent bundle) — i.e., the computation of “deformations of the arithmetic holomorphic structure” of a number field equipped with an elliptic curve. §I3. Basepoints and Inter-universality As discussed in §I2, the present series of papers is concerned with considering “deformations of the arithmetic holomorphic structure” of a number field — i.e., so to speak, with performing “surgery on the number field”. At a more concrete level, this means that one must consider situations in which two distinct “theaters” for conventional ring/scheme theory — i.e., two distinct Θ ±ell NF-Hodge theaters — are related to one another by means of a “correspondence”, or“filter”, thatfails to be compatible with the respective ring structures. In the discussion so far of the portion of the theory developed in the present paper, the main example of such a “filter” is given by the Θ-link. As mentioned earlier, more elaborate versions
INTER-UNIVERSAL TEICHMÜLLER THEORY I 21 of the Θ-link will be discussed in [IUTchII], [IUTchIII]. The other main example of such a non-ring/scheme-theoretic “filter” in the present series of papers is the log-link, which we shall discuss in [IUTchIII] [cf. also the theory of [AbsTopIII]]. One important aspect of such non-ring/scheme-theoretic filters is the property that they are incompatible with various constructions that depend on the ring structure of the theaters that constitute the domain and codomain of such a filter. From the point of view of the present series of papers, perhaps the most important example of such a construction is given by the various étale fundamental groups — e.g., Galois groups — that appear in these theaters. Indeed, these groups are defined, essentially, as automorphism groups of some separably closed field, i.e., the field that arises in the definition of the fiber functor associated to the basepoint determined by a geometric point that is used to define the étale fundamental group — cf. the discussion of [IUTchII], Remark 3.6.3, (i); [IUTchIII], Remark 1.2.4, (i); [AbsTopIII], Remark 3.7.7, (i). In particular, unlike the case with ring homomorphisms or morphisms of schemes with respect to which the étale fundamental group satisfies well-known functoriality properties, in the case of nonring/scheme-theoretic filters, the only “type of mathematical object” that makes sense simultaneously in both the domain and codomain theaters of the filter is the notion of a topological group. In particular, the only data that can be considered in relating étale fundamental groups on either side of a filter is the étale-like structure constituted by the underlying abstract topological group associated to such an étale fundamental group, i.e., devoid of any auxiliary data arising from the construction of the group “as an étale fundamental group associated to a basepoint determined by a geometric point of a scheme”. It is this fundamental aspect of the theory of the present series of papers — i.e., of relating the distinct set-theoretic universes associated to the distinct fiber functors/basepoints on either side of such a non-ring/scheme-theoretic filter — that we refer to as inter-universal. This inter-universal aspect of the theory manifestly leads to the issue of considering the extent to which one can understand various ring/scheme structures by considering only the underlying abstract topological group of some étale fundamental group arising from such a ring/scheme structure — i.e., in other words, of considering the absolute anabelian geometry [cf. the Introductions to [AbsTopI], [AbsTopII], [AbsTopIII]] of the rings/schemes under consideration. At this point, the careful reader will note that the above discussion of the inter-universal aspects of the theory of the present series of papers depends, in an essential way, on the issue of distinguishing different “types of mathematical object” and hence, in particular, on the notion of a “type of mathematical object”. This notion may be formalized via the language of “species”, which we develop in the final portion of [IUTchIV]. Another important “inter-universal” phenomenon in the present series of papers — i.e., phenomenon which, like the absolute anabelian aspects discussed above,
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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