Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

122 SHINICHI MOCHIZUKI
(respectively, D-NF-bridge portion † φ NF
)oftheD-ΘNF-Hodge theater † HT D-ΘNF
[cf. Remark 5.1.2].
n ·
◦ ...
· v
. ..
n ′ · ◦ ... · v ′ . ..
n ′′ · ◦ ... · v ′′
Fig. 5.2: Pivotal distribution
Remark 5.4.2.
(i) At this point, it is useful to consider the various copies of † F ⊚ mod
and its realifications
introduced so far from the point of view of “log-volumes”, i.e., arithmetic
degrees [cf., e.g., the discussion of [FrdI], Example 6.3; [FrdI], Theorem 6.4;
Remark 3.1.5 of the present paper]. That is to say, since † F ⊚ j may be thought of as
a sort of “section of † F ⊚ J
over † F ⊚ mod ” — i.e., a sort of “section of K over F mod” [cf.
the discussion of prime-strips in Remark 4.3.1] — one way to think of log-volumes
of † F ⊚ 〈J〉 is as quantities that differ by a factor of l — i.e., corresponding, to the
cardinality of J → ∼ F l
— from log-volumes of † F ⊚ j . Put another way, this amounts
to thinking of arithmetic degrees that appear in the various factors of † F ⊚ J
as being
averaged over the elements of J and hence of arithmetic degrees that
appear in † F ⊚ 〈J〉
as the “resulting averages”.
This sort of averaging may be thought of as a sort of abstract, Frobenioid-theoretic
analogue of the normalization of arithmetic degrees that is often used in the theory
of heights [cf., e.g., [GenEll], Definition 1.2, (i)] that allows one to work with heights
in such a way that the height of a point remains invariant with respect to change
of the base field.
(ii) On the other hand, to work with the various isomorphisms of Frobenioids
—suchas † F ⊚ ∼
j → † F ⊚ 〈J〉
— involved amounts [since the arithmetic degree is an
intrinsic invariant of the Frobenioids involved — cf. [FrdI], Theorem 6.4, (iv);
Remark 3.1.5 of the present paper] to thinking of arithmetic degrees that appear
in the various factors of † F ⊚ J
as being
summed [i.e., without dividing by a factor of l ] over the elements of J
and hence of arithmetic degrees that appear in † F ⊚ 〈J〉
as the “resulting
sums”.