Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

84 SHINICHI MOCHIZUKI
the l-torsion points of E F
[i.e., on Δ ab
X ⊗ F l], we obtain a natural homomorphism
Out(Π CK ) → Aut(Δ ab
X ⊗ F l)/{±1}. Thus, relative to a suitable inclusion
Aut(Δ ab
X ⊗ F l)/{±1} ↩→ GL 2 (F l )/{±1}, the images of the groups Aut ɛ (C K ),
Aut(C K ) may be identified with subgroups of the form
{( )} {( )}
∗ ∗
∗ ∗
⊆
⊆ GL 2 (F l )/{±1}
0 ±1
0 ∗
— i.e., “semi-unipotent, up to ±1” and Borel subgroups — of GL 2 (F l )/{±1}. Write
V ±un def
=Aut ɛ (C K ) · V ⊆ V Bor def
=Aut(C K ) · V ⊆ V(K)
for the resulting subsets of V(K). Thus, one verifies immediately that the subgroup
Aut ɛ (C K ) ⊆ Aut(C K )isnormal, and that we have a natural isomorphism
Aut(C K )/Aut ɛ (C K ) ∼ → F l
—sowemaythinkofV Bor as the F l -orbit of V±un . Also, we observe that [in light
of the above discussion] it follows immediately that there exists a group-theoretic
algorithm for reconstructing, from π 1 (D ⊚ ) [i.e., an isomorph of Π CK ] the subgroup
determined by Aut ɛ (C K ).
Aut ɛ (D ⊚ ) ⊆ Aut(D ⊚ )
(ii) Let v ∈ V non . Then the natural restriction functor on finite étale coverings
arising from the natural composite morphism X −→v → C v → C K if v ∈ V good
(respectively, X v
→ C v → C K if v ∈ V bad ) determines [cf. Examples 3.2, (i);
3.3, (i)] a natural morphism φ•,v NF : D v →D ⊚ [cf. §0 for the definition of the term
“morphism”]. Write
: D v →D ⊚
φ NF
v
for the poly-morphism given by the collection of morphisms D v →D ⊚ of the form
β ◦ φ NF
•,v ◦ α
—whereα ∈ Aut(D v ) ∼ = Aut(X −→v ) (respectively, α ∈ Aut(D v ) ∼ = Aut(X v
)); β ∈
Aut ɛ (D ⊚ ) ∼ = Aut ɛ (C K ) [cf., e.g., [AbsTopIII], Theorem 1.9].
(iii) Let v ∈ V arc . Thus, [cf. Example 3.4, (i)] we have a tautological morphism
∼
D v = −→v X → C v → C(D ⊚ ,v), hence a morphism φ NF
•,v : D v →D ⊚ [cf. Definition
4.1, (v)]. Write
: D v →D ⊚
φ NF
v
for the poly-morphism given by the collection of morphisms D v →D ⊚ of the form
β ◦ φ NF
•,v ◦ α