Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 101
possibilities for the element ∈ F l
to which a given index j ∈ J corresponds [cf.
Proposition 4.7, (i)], prior to the application of this functor.
(iii) (Mono-analytic Capsules) By composing the functor of (ii) with the
mono-analyticization operation discussed in Definition 4.1, (iv), one obtains a
natural functor
category of
D-ΘNF-Hodge theaters
and isomorphisms of
D-ΘNF-Hodge theaters
→
category of l -capsules
of D ⊢ -prime-strips
and capsule-full polyisomorphisms
of l -capsules
† HT D-ΘNF ↦→ † D ⊢ J
whose output data satisfies the same symmetry properties with respect to labels as
the output data of the functor of (ii).
Proof. Assertions (i), (ii), (iii) follow immediately from the definitions [cf. also
Proposition 4.8, (i), in the case of assertion (i)]. ○
Remark 4.9.1.
(i) Ultimately, in the theory of the present series of papers [cf., especially,
[IUTchII], §2], we shall be interested in
evaluating the étale theta function of [EtTh] — i.e., in the spirit of
the Hodge-Arakelov theory of [HASurI], [HASurII]— at the various
D-prime-strips of † D J , in the fashion stipulated by the labels discussed
in Proposition 4.7, (i).
These values of the étale theta function will be used to construct various arithmetic
line bundles. We shall be interested in computing the arithmetic degrees —in
the form of various “log-volumes” — of these arithmetic line bundles. In order to
compute these global log-volumes, it is necessary to be able to compare the logvolumes
that arise at D-prime-strips with different labels. It is for this reason that
the non-labeled output data of the functors of Proposition 4.9, (i), (ii), (iii) [cf. also
Proposition 4.11, (i), (ii), below], are of crucial importance in the theory of the
present series of papers. That is to say,
the non-labeled output data of the functors of Proposition 4.9, (i), (ii),
(iii) [cf. also Proposition 4.11, (i), (ii), below] — which allow one to
consider isomorphisms between the D-prime-strips that were originally
assigned different labels — make possible the comparison of objects
[e.g., log-volumes] constructed relative to different labels.
In Proposition 4.11, (i), (ii), below, we shall see that by considering “processions”,
one may perform such comparisons in a fashion that minimizes the label indeterminacy
that arises.