Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

14 SHINICHI MOCHIZUKI
— in essence, a system of Frobenioids — associated to this initial Θ-data, as well as
an associated D-Θ ±ell NF-Hodge theater † HT D-Θ±ell NF — in essence, the system
of base categories associated to the system of Frobenioids † HT Θ±ell NF .
(i) (F ⋊±
l
- and F l -Symmetries) The Θ±ell NF-Hodge theater † HT Θ±ell NF
may be obtained 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)]; a similar statement holds
for the D-Θ ±ell NF-Hodge theater † HT D-Θ±ellNF .Theglobal portion of a D-Θ ±ell -
Hodge theater † HT D-Θ±ell
consists of a category equivalent to [the full subcategory
determined by the connected objects of] the Galois category of finite étale coverings
of the [orbi]curve X K . This global portion is equipped with a F ⋊±
l
-symmetry, i.e.,
a poly-action by F ⋊±
l
on the labels
( −l < ... < −1 < 0 < 1 < ... < l )
— which we think of as elements ∈ F l — each of which is represented in the D-
Θ ±ell -Hodge theater † HT D-Θ±ell
by a D-prime-strip [cf. Fig. I1.3]. The global
portion of a D-ΘNF-Hodge theater † HT D-ΘNF consists of a category equivalent to
[the full subcategory determined by the connected objects of] the Galois category of
finite étale coverings of the orbicurve C K . This global portion is equipped with a
F l -symmetry, i.e., a poly-action by F l
on the labels
(1 < ... < l )
— which we think of as elements ∈ F l
— each of which is represented in the
D-ΘNF-Hodge theater † HT D-ΘNF by a D-prime-strip [cf. Fig. I1.3]. The D-
Θ ±ell -Hodge theater † HT D-Θ±ell
is glued to the D-ΘNF-Hodge theater † HT D-ΘNF
along a single D-prime-strip in such a way that the labels 0 ≠ ±t ∈ F l that arise
in the F ⋊±
l
-symmetry are identified with the corresponding label j ∈ F l
that arises
in the F l -symmetry.
(ii) (Θ-links) By considering the 2l-th roots of the q-parameters “q
v
”of
the elliptic curve E F at v ∈ V bad and extending to other v ∈ V in such a way as
to satisfy the product formula, one may construct a natural F ⊩ -prime-strip
† F ⊩ mod associated to the Θ±ell NF-Hodge theater † HT Θ±ellNF . In a similar vein, by
considering the reciprocal of the l-th root of the Frobenioid-theoretic theta function
“Θ v
” associated to the elliptic curve E F at v ∈ V bad and extending to other v ∈ V
in such a way as to satisfy the product formula, one may construct a natural
F ⊩ -prime-strip † F ⊩ tht associated to the Θ±ell NF-Hodge theater † HT Θ±ellNF .Now
let ‡ HT Θ±ellNF be another Θ ±ell NF-Hodge theater [relative to the given initial Θ-
data]. Then we shall refer to the “full poly-isomorphism” of [i.e., the collection of
all isomorphisms between] F ⊩ -prime-strips
† F ⊩ tht
∼
→
‡ F ⊩ mod
as the Θ-link from [the underlying Θ-Hodge theater of] † HT Θ±ell NF to [the underlying
Θ-Hodge theater of] ‡ HT Θ±ell NF .TheΘ-link induces the full poly-isomorphism
between the F ⊢× -prime-strips
† F ⊢×
mod
∼
→ ‡ F ⊢×
mod