Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 11
at v ∈ V bad , and, roughly speaking, extending to v ∈ V good in such a way as to
satisfy the product formula, one may construct a natural F ⊩ -prime-strip “F ⊩ mod ”
[cf. Example 3.5, (ii); Definition 5.2, (iv)]. This construction admits an abstract,
algorithmic formulation that allows one to apply it to the underlying “Θ-Hodge
theater” of an arbitrary Θ ±ell NF-Hodge theater † HT Θ±ell NF so as to obtain an F ⊩ -
prime-strip
† F ⊩ mod
[cf. Definitions 3.6, (c); 5.2, (iv)]. On the other hand, by formally replacing the
2l-th roots of the q-parameters that appear in this construction by the reciprocal
of the l-th root of the Frobenioid-theoretic theta function, which we shall denote
“Θ v
”[forv ∈ V bad ], studied in [EtTh] [cf. also Example 3.2, (ii), of the present
paper], one obtains an abstract, algorithmic formulation for the construction of an
F ⊩ -prime-strip
† F ⊩ tht
[cf. Definitions 3.6, (c); 5.2, (iv)] from [the underlying Θ-Hodge theater of] the
Θ ±ell NF-Hodge theater † HT Θ±ell NF .
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 [cf. Corollary 3.7, (i); Definition 5.2, (iv)].
One fundamental property of the Θ-link is the property that it induces a collection
of isomorphisms [in fact, the full poly-isomorphism] between the F ⊢× -prime-strips
† F ⊢×
mod
∼
→ ‡ F ⊢×
mod
associated to † F ⊩ mod and ‡ F ⊩ mod
4.9, (vii)].
[cf. Corollary 3.7, (ii), (iii); [IUTchII], Definition
Now let { n HT Θ±ell NF } n∈Z be a collection of distinct Θ ±ell NF-Hodge theaters
[relative to the given initial Θ-data] indexed by the integers. Thus, by applying the
constructions just discussed, we obtain an infinite chain
...
Θ
−→ (n−1) HT Θ±ell NF Θ
−→ n HT Θ±ell NF Θ
−→ (n+1) HT Θ±ell NF Θ
−→ ...
of Θ-linked Θ ±ell NF-Hodge theaters [cf. Corollary 3.8], which will be referred
to as the Frobenius-picture [associated to the Θ-link]. One fundamental
property of this Frobenius-picture is the property that it fails to admit permutation
automorphisms that switch adjacent indices n, n + 1, but leave the
remaining indices ∈ Z fixed [cf. Corollary 3.8]. Roughly speaking, the Θ-link
n HT Θ±ell NF Θ
−→
(n+1) HT Θ±ellNF may be thought of as a formal correspondence
n Θ v
↦→ (n+1) q
v