Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

18 SHINICHI MOCHIZUKI
If one thinks of the geometry of “conventional scheme theory” as being analogous
to the geometry of “Euclidean space”, then the geometry represented by the
Frobenius-picture corresponds to a “topological manifold”, i.e., which is obtained by
gluing together various portions of Euclidean space, but which is not homeomorphic
to Euclidean space. This point of view is illustrated in Fig. I2.1 above, where the
various Θ ±ell NF-Hodge theaters in the Frobenius-picture are depicted as [twodimensional!
— cf. the discussion of §I1] twice-punctured topological surfaces
of genus one, glued together along tubular neighborhoods of cycles, which
correspond to the [one-dimensional! — cf. the discussion of §I1] mono-analytic
data that appears in the isomorphism that constitutes the Θ-link. The permutation
symmetries in the étale-picture [cf. the discussion of §I1] are depicted in Fig.
I2.1 as the anti-holomorphic reflection [cf. the discussion of multiradiality in
[IUTchII], Introduction!] around a gluing cycle between topological surfaces.
Another elementary example that illustrates the spirit of the gluing operations
discussed in the present series of papers is the following. For i =0, 1, let R i be
acopyofthereal line; I i ⊆ R i the closed unit interval [i.e., corresponding to
[0, 1] ⊆ R]. Write C 0 ⊆ I 0 for the Cantor set and
φ : C 0
∼
→ I1
for the bijection arising from the Cantor function. ThenifonethinksofR 0 and
R 1 as being glued to one another by means of φ, thenitisahighly nontrivial
problem
to describe structures naturally associated to the “alien” ring structure
of R 0 — such as, for instance, the subset of algebraic numbers ∈ R 0 —
in terms that only require the use of the ring structure of R 1 .
A slightly less elementary example that illustrates the spirit of the gluing operations
discussed in the present series of papers is the following. This example is
technically much closer to the theory of the present series of papers than the examples
involving topological surfaces and Cantor sets given above. For simplicity, let
us write
G O × , G O ⊲
for the pairs “G v O × ”, “G
F v
v
O ⊲ ” [cf. the notation of the discussion
F v
surrounding Fig. I1.2]. Recall from [AbsTopIII], Proposition 3.2, (iv), that the
operation
(G O ⊲ ) ↦→ G
of “forgetting O ⊲ ” determines a bijection from the group of automorphisms of
the pair G O ⊲ — i.e., thought of as an abstract topological monoid equipped
with a continuous action by an abstract topological group — to the group of automorphisms
of the topological group G. By contrast, we recall from [AbsTopIII],
Proposition 3.3, (ii), that the operation
(G O × ) ↦→ G
of “forgetting O × ” only determines a surjection from the group of automorphisms
of the pair G O × — i.e., thought of as an abstract topological monoid equipped