Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

36 SHINICHI MOCHIZUKI
acts on Δ E ⊗(Z/lZ) via multiplication by −1. In particular, since l is odd, it follows
that the action by ι on Δ ɛ determines a decomposition into eigenspaces
Δ ɛ
∼
→ Δ
+
ɛ × Δ − ɛ
— i.e., where ι acts on Δ + ɛ (respectively, Δ − ɛ ) by multiplication by +1 (respectively,
−1). Moreover, the natural composite maps
I ɛ
′ ↩→ Δ ɛ ↠ Δ + ɛ ; I ɛ
′′ ↩→ Δ ɛ ↠ Δ + ɛ
∼
determine isomorphisms I ɛ
′ → Δ + ɛ , I ɛ
′′ → Δ + ɛ . Since the natural action of G k
on Δ ɛ clearly commutes with the action of ι, we thus conclude that the quotient
Δ X ↠ Δ ɛ ↠ Δ + ɛ determines quotients
∼
Π X ↠ J X ;
Π C ↠ J C
— where the surjections Π X ↠ G k ,Π C ↠ G k induce natural exact sequences
1 → Δ + ɛ → J X → G k → 1, 1 → Δ + ɛ × Gal(X/C) → J C → G k → 1; we have a
natural inclusion J X ↩→ J C .
Next, let us consider the cusp “2ɛ”ofC — i.e., the cusp whose inverse images in
X correspond to the points of E obtained by multiplying ɛ ′ , ɛ ′′ by 2, relative to the
group law of the elliptic curve determined by the pair (X,ɛ 0 ). Since 2 ≠ ±1 (modl)
[a consequence of our assumption that l ≥ 5], it follows that the decomposition group
associated to this cusp “2ɛ” determines a section
σ : G k → J C
of the natural surjection J C ↠ G k . Here, we note that although, a priori, σ is only
determined by 2ɛ up to composition with an inner automorphism of J C determined
by an element of Δ + ɛ × Gal(X/C), in fact, since [in light of the assumption (∗)!]
the natural [outer] action of G k on Δ + ɛ × Gal(X/C) istrivial, we conclude that σ
is completely determined by 2ɛ, and that the subgroup Im(σ) ⊆ J C determined by
the image of σ is normal in J C . Moreover, by considering the decomposition groups
associated to the cusps of X lying over 2ɛ, we conclude that Im(σ) lies inside the
subgroup J X ⊆ J C . Thus, the subgroups Im(σ) ⊆ J X ,Im(σ) × Gal(X/C) ⊆ J C
determine [the horizontal arrows in] cartesian diagrams
X−→ −→ X Π X−→ −→ Π X Δ X−→ −→ Δ X
⏐ ⏐ ⏐ ⏐ ⏐ ⏐
↓ ↓ ↓ ↓ ↓ ↓
C−→ −→ C Π C−→ −→ Π C Δ C−→ −→ Δ C
of finite étale cyclic coverings of hyperbolic orbicurves and open immersions [with
normal image] of profinite groups; we have Gal( −→ C /C) ∼ = Z/lZ, Gal(X/C) ∼ = Z/2Z,
and Gal(X −→ /C) → ∼ Gal(X/C) × Gal( −→ C /C) ∼ = Z/2lZ.
Definition 1.1. We shall refer to a hyperbolic orbicurve over k that arises, up to
isomorphism, as the hyperbolic orbicurve −→ X (respectively, −→ C ) constructed above
for some choice of l, ɛ as being of type (1,l-tors −−→ ) (respectively, (1,l-tors −−→ ) ±).