Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 55
discussion preceding [EtTh], Definition 2.1, one must in fact assume that the integer
l is odd in order for the quotient Δ X to be well-defined. Since, ultimately, in [EtTh]
[cf. the discussion following [EtTh], Remark 5.7.1], as well as in the present series
of papers, this is the only case that is of interest, this oversight does not affect
either the present series of papers or the bulk of the remainder of [EtTh]. Indeed,
the only places in [EtTh] where the case of even l is used are [EtTh], Remark 2.2.1,
and the application of [EtTh], Remark 2.2.1, in the proof of [EtTh], Proposition
2.12, for the orbicurves “Ċ”. Thus, [EtTh], Remark 2.2.1, must be deleted; in
[EtTh], Proposition 2.12, one must in fact exclude the case where the orbicurve
under consideration is “Ċ”. On the other hand, this theory involving [EtTh],
Proposition 2.12 [cf., especially, [EtTh], Corollaries 2.18, 2.19] is only applied after
the discussion following [EtTh], Remark 5.7.1, i.e., which only treats the curves
“X”. That is to say, ultimately, in [EtTh], as well as in the present series of papers,
one is only interested in the curves “X”, whose treatment only requires the case of
odd l.
Given initial Θ-data as in Definition 3.1, the theory of Frobenioids given in
[FrdI], [FrdII], [EtTh] allows one to construct various associated Frobenioids, as
follows.
Example 3.2. Frobenioids at Bad Nonarchimedean Primes. Let v ∈
V bad = V ⋂ V(K) bad . Then let us recall the theory of the “Frobenioid-theoretic
theta function” discussed in [EtTh], §5:
(i) By the theory of [EtTh], the hyperbolic curve X v
determines a tempered
Frobenioid
F v
[i.e., the Frobenioid denoted “C” in the discussion at the beginning of [EtTh], §5;
cf. also the discussion of Remark 3.2.4 below] over a base category
D v
[i.e., the category denoted “D” in the discussion at the beginning of [EtTh], §5].
We recall from the theory of [EtTh] that D v may be thought of as the category
of connected tempered coverings — i.e., “B temp (X v
) 0 ” in the notation of [EtTh],
Example 3.9 — of the hyperbolic curve X v
. In the following, we shall write
D ⊢ v
def
= B(K v ) 0
[cf. the notational conventions concerning categories discussed in §0]. Also, we
observe that Dv
⊢ may be naturally regarded [by pulling back finite étale coverings
via the structure morphism X v
→ Spec(K v )] as a full subcategory
D ⊢ v ⊆D v
of D v , and that we have a natural functor D v →Dv ⊢ ,whichisleft-adjoint to the
natural inclusion functor Dv
⊢ ↩→ D v [cf. [FrdII], Example 1.3, (ii)]. If (−) isan