Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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