Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

70 SHINICHI MOCHIZUKI
log(Θ) ↦→
1
[K v :(F mod ) v ] log Φ(p v ) · log(Θ). On the other hand, if v ∈ V bad , then let us
write
log Φ (Θ v
) ∈ Φ rlf
C Θ v
for the element determined by Θ v
[cf. Example 3.2, (v)] and log Φ (p v ) for the
constant section of Φ Cv determined by p v [cf. the notation “log Φ (q )” of Example
v
3.2, (iv)]; in particular, it makes sense to write log Φ (p v )/log Φ (q ) ∈ Q >0 ;thus,[cf.
v
(i)] ρ Θ v
is given by the assignment
log ⊢ mod(p v ) · log(Θ) ↦→ log Φ(p v )
[K v :(F mod ) v ] ·
log(Θ v
)
log Φ (q
v
)
— cf. Remark 3.5.1, (i), below. Note that, for arbitrary v ∈ V, the various ρ v ,
ρ Θ v are compatible with the natural isomorphisms Cmod
⊩ ∼
→Ctht ⊩ , C⊢ ∼
v →Cv Θ [cf. §0].
This fact may be expressed as a natural isomorphism between collections of data
[consisting of a category, a bijection of sets, a collection of data indexed by V, and
a collection of isomorphisms indexed by V]
F ⊩ mod
∼
→
F ⊩ tht
—wherewewrite
F ⊩ mod
F ⊩ tht
def
= (C ⊩ mod , Prime(C⊩ mod ) ∼ → V, {F ⊢ v } v∈V , {ρ v } v∈V )
def
= (C ⊩ tht , Prime(C⊩ tht ) ∼ → V, {F Θ v } v∈V , {ρ Θ v } v∈V )
[and we apply the natural bijection V ∼ → V mod ]; cf. Remark 3.5.2 below.
(iii) One may also construct a “D-version” — which, from the point of view of
the theory of [AbsTopIII], one may also think of as a “log-shell version” —ofthe
various data constructed in (i), (ii). To this end, we write
D ⊩ mod
for a [i.e., another] copy of Cmod ⊩ . Thus, one may associate to D⊩ mod
various objects
Φ D
⊩ , Prime(D ⊩
mod
mod ) → ∼ V mod , log D mod(p v ) ∈ Φ D
⊩
mod ,v ⊆ Φ D
⊩ [for v ∈ V mod ]
mod
that map to the corresponding objects associated to Cmod ⊩ under the tautological
equivalence of categories Cmod
⊩ ∼
→Dmod ⊩ . Write v ∈ V for the element of V that
corresponds to v. Next, suppose that v ∈ V non ; then let us recall from [AbsTopIII],
Proposition 5.8, (iii), that [since the profinite group associated to Dv ⊢ is the absolute
Galois group of an MLF] one may construct algorithmically from Dv
⊢ a topological
monoid isomorphic to R ≥0
(R ⊢ ≥0) v
[i.e., the topological monoid determined by the nonnegative elements of the ordered
topological group “R non (G)” of loc. cit.] equipped with a distinguished “Frobenius
element” ∈ (R ⊢ ≥0 ) v;ife v is the absolute ramification index of the MLF K v ,thenwe
shall write log D Φ(p v ) ∈ (R ⊢ ≥0 ) v for the result of multiplying this Frobenius element