Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 7
— which may be thought of as “miniature models of conventional scheme theory”
— given, roughly speaking, by systems of Frobenioids. To any such
Θ ±ell NF-Hodge theater † HT Θ±ellNF , one may associate a D-Θ ±ell NF-Hodge theater
[cf. Definition 6.13, (ii)]
† HT D-Θ±ell NF
— i.e., the associated system of base categories.
One may think of a Θ ±ell NF-Hodge theater † HT Θ±ellNF as the result of gluing
together a Θ ±ell -Hodge theater † HT Θ±ell
to a ΘNF-Hodge theater † HT ΘNF [cf. Remark
6.12.2, (ii)]. In a similar vein, one may think of a D-Θ ±ell NF-Hodge theater
† HT D-Θ±ellNF as the result of gluing together a D-Θ ±ell -Hodge theater † HT D-Θ±ell
to a D-ΘNF-Hodge theater † HT D-ΘNF .AD-Θ ±ell -Hodge theater † HT D-Θ±ell
may
be thought of as a bookkeeping device that allows one to keep track of the action
of the F ⋊±
l
-symmetry on the labels
( −l < ... < −1 < 0 < 1 < ... < l )
— which we think of as elements ∈ F l — in the context of the [orbi]curves X K ,
X v
[for v ∈ V bad ], and −→v X [for v ∈ V good ]. The F ⋊±
l
-symmetry is represented in a
D-Θ ±ell -Hodge theater † HT D-Θ±ell by a category equivalent to [the full subcategory
determined by the connected objects of] the Galois category of finite étale coverings
of X K . On the other hand, each of the labels referred to above is represented in
a D-Θ ±ell -Hodge theater † HT D-Θ±ell by a D-prime-strip. In a similar vein, a
D-ΘNF-Hodge theater † HT D-ΘNF may be thought of as a bookkeeping device that
allows one to keep track of the action of the F l
-symmetry on the labels
(1 < ... < l )
— which we think of as elements ∈ F l
— in the context of the orbicurves C K ,
X v
[for v ∈ V bad ], and −→v X [for v ∈ V good ]. The F l
-symmetry is represented in a
D-ΘNF-Hodge theater † HT D-ΘNF by a category equivalent to [the full subcategory
determined by the connected objects of] the Galois category of finite étale coverings
of C K . On the other hand, each of the labels referred to above is represented in a D-
ΘNF-Hodge theater † HT D-ΘNF by a D-prime-strip. Thecombinatorial structure
of D-ΘNF- and D-Θ ±ell -Hodge theaters summarized above [cf. also Fig. I1.3 below]
is one of the main topics of the present paper and is discussed in detail in §4 and
§6. The left-hand portion of Fig. I1.3 corresponds to the D-Θ ±ell -Hodge theater;
the right-hand portion of Fig. I1.3 corresponds to the D-ΘNF-Hodge theater; these
left-hand and right-hand portions are glued together along a single D-prime-strip,
depicted as “[1 < ... < l ]”, in such a way that the labels 0 ≠ ±t ∈ F l on the
left are identified with the corresponding label j ∈ F l
on the right.
The F ⋊±
l
-symmetry has the advantange that, being geometric in nature, it
allows one to permute various copies of “G v ”[wherev ∈ V non ] associated to distinct
labels ∈ F l without inducing conjugacy indeterminacies. This phenomenon,
which we shall refer to as conjugate synchronization, will play a key role in
the Kummer theory surrounding the Hodge-Arakelov-theoretic evaluation of the