Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 85
—whereα ∈ Aut(D v ) ∼ = Aut( −→v X ) [cf.
Aut ɛ (D ⊚ ) ∼ = Aut ɛ (C K ).
[AbsTopIII], Corollary 2.3, (i)]; β ∈
(iv) For each j ∈ F l ,let
D j = {D vj } v∈V
— where we use the notation v j to denote the pair (j, v) —beacopy of the
“tautological D-prime-strip” {D v } v∈V . Let us denote by
φ NF
1 : D 1 →D ⊚
[where, by abuse of notation, we write “1” for the element of F l
determined by 1]
the poly-morphism determined by the collection {φ NF
v 1
: D v1 →D ⊚ } v∈V of copies of
the poly-morphisms φ NF
v constructed in (ii), (iii). Note that φ NF
1 is stabilized by the
action of Aut ɛ (C K ) on D ⊚ . Thus, it makes sense to consider, for arbitrary j ∈ F l ,
the poly-morphism
φ NF
j
: D j →D ⊚
obtained [via any isomorphism D 1
∼ = Dj ]bypost-composing with the “poly-action”
[i.e., action via poly-automorphisms — cf. (i)] of j ∈ F l
on D ⊚ .Letuswrite
D
def
= {D j } j∈F
l
for the capsule of D-prime-strips indexed by j ∈ F l
denote by
: D →D ⊚
φ NF
[cf. Definition 4.1, (iv)] and
the poly-morphism given by the collection of poly-morphisms {φ NF
j } j∈F
. Thus,
l
φ NF
is equivariant with respect to the natural poly-action of F l
on D ⊚ and the
natural permutation poly-action of F l
, via capsule-full [cf. §0] poly-automorphisms,
on the constituents of the capsule D . In particular, we obtain a natural poly-action
of F l
on the collection of data (D , D ⊚ ,φ NF
).
Remark 4.3.1.
(i) Suppose, for simplicity, in the following discussion that F = F mod . Note
that the morphism of schemes Spec(K) → Spec(F ) [or, equivalently, the homomorphism
of rings F ↩→ K] does not admit a section. This nonexistence of a section
is closely related to the nonexistence of a “global multiplicative subspace” of the
sort discussed in [HASurII], Remark 3.7. In the context of loc. cit., this nonexistence
of a “global multiplicative subspace” may be thought of as a concrete way
of representing the principal obstruction to applying the scheme-theoretic Hodge-
Arakelov theory of [HASurI], [HASurII] to diophantine geometry. From this point
of view, if one thinks of the ring structure of F , K as a sort of “arithmetic holomorphic
structure” [cf. [AbsTopIII], Remark 5.10.2, (ii)], then one may think of
the [D-]prime-strips that appear in the discussion of Example 4.3 as defining, via
the arrows φ NF
j of Example 4.3, (iv),