Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

4 SHINICHI MOCHIZUKI
[cf. the discussion at the beginning of §4; Definitions 6.1, 6.4] will play an important
role in the discussion to follow. The natural action of the stabilizer in Gal(K/F) of
the quotient E K [l] ↠ Q on Q determines a natural poly-action of F l
on C K , i.e.,
a natural isomorphism of F l
with some subquotient of Aut(C K ) [cf. Example 4.3,
(iv)]. The F l -symmetry constituted by this poly-action of F l
may be thought
of as being essentially arithmetic in nature, in the sense that the subquotient of
Aut(C K ) that gives rise to this poly-action of F l
is induced, via the natural map
Aut(C K ) → Aut(K), by a subquotient of Gal(K/F) ⊆ Aut(K). In a similar vein,
the natural action of the automorphisms of the scheme X K on the cusps of X K
determines a natural poly-action of F ⋊±
l
on X K , i.e., a natural isomorphism of F ⋊±
l
with some subquotient of Aut(X K ) [cf. Definition 6.1, (v)]. The F ⋊±
l
-symmetry
constituted by this poly-action of F ⋊±
l
may be thought of as being essentially geometric
in nature, in the sense that the subgroup Aut K (X K ) ⊆ Aut(X K ) [i.e., of
K-linear automorphisms] maps isomorphically onto the subquotient of Aut(X K )
that gives rise to this poly-action of F ⋊±
l
. On the other hand, the global F l -
symmetry of C K only extends to a “{1}-symmetry” [i.e., in essence, fails to extend!]
of the local coverings X v
[for v ∈ V bad ]andX −→v [for v ∈ V good ], while the global
F ⋊±
l
-symmetry of X K only extends to a “{±1}-symmetry” [i.e., in essence, fails to
extend!] of the local coverings X v
[for v ∈ V bad ]andX −→v [for v ∈ V good ] — cf. Fig.
I1.1 below.
{±1}
{X v
or X −→v } v∈V
F ⋊±
F l l
X K C K
Fig. I1.1: Symmetries of coverings of X F
We shall write Π v for the tempered fundamental group of X v
,whenv ∈ V bad
[cf. Definition 3.1, (e)]; we shall write Π v for the étale fundamental group of −→v X ,
when v ∈ V good [cf. Definition 3.1, (f)]. Also, for v ∈ V non , we shall write Π v ↠ G v
for the quotient determined by the absolute Galois group of the base field K v .Often,
in the present series of papers, we shall consider various types of collections of data
— which we shall refer to as “prime-strips” — indexed by v ∈ V ( → ∼ V mod )that
are isomorphic to certain data that arise naturally from X v
[when v ∈ V bad ]orX −→v
[when v ∈ V good ]. The main types of prime-strips that will be considered in the
present series of papers are summarized in Fig. I1.2 below.
Perhaps the most basic kind of prime-strip is a D-prime-strip. When v ∈
V non ,theportionofaD-prime-strip labeled by v is given by a category equivalent
to [the full subcategory determined by the connected objects of] the category of
tempered coverings of X v
[when v ∈ V bad ]orfinite étale coverings of X −→v [when
v ∈ V good ]. When v ∈ V arc , an analogous definition may be obtained by applying
the theory of Aut-holomorphic orbispaces developed in [AbsTopIII], §2. One
variant of the notion of a D-prime-strip is the notion of a D ⊢ -prime-strip. When