Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 151
— cf. the theory of étale theta functions “of standard type”, as discussed in [EtTh],
Theorem 1.10; the theory to be developed in [IUTchII].
(ii) Whereas the F ⋊±
l
-symmetry of the theory of the present §6 has the advantage
that it allows one to relate zero-labeled and non-zero-labeled prime-strips,
it has the [tautological!] disadvantage that it does not allow one to “insulate”
the non-zero-labeled prime-strips from confusion with the zero-labeled prime-strip.
This issue will be of substantial importance in the theory of Gaussian Frobenioids
[to be developed in [IUTchII]], i.e., Frobenioids that, roughly speaking, arise from
the theta values
{ q j2
} j
v
[cf. the discussion of Example 4.4, (i)] at the non-zero-labeled evaluation points.
Moreover, ultimately, in [IUTchII], [IUTchIII], we shall relate these Gaussian Frobenioids
to various global arithmetic line bundles on the number field F . This will
require the use of both the additive and the multiplicative structures on the number
field; in particular, it will require the use of the theory developed in §5.
(iii) By contrast, since, in the theory of the present series of papers, we shall
not be interested in analogues of the Gaussian Frobenioids that involve the zerolabeled
evaluation points, we shall not require an “additive analogue” of the portion
[cf. Example 5.1] of the theory developed in §5 concerning global Frobenioids.
Remark 6.12.6.
(i) Another fundamental difference between the F l
-symmetry of §4 andthe
-symmetry of the present §6 lies in the geometric nature of the “single basepoint”
[cf. the discussion of Remark 6.12.4] that underlies the F ⋊±
l
-symmetry. That
F ⋊±
l
is to say, the various labels ∈ T → ∼ F l that appear in a [D-]Θ ±ell -Hodge theater correspond
— throughout the various portions [e.g., bridges] of the [D-]Θ ±ell -Hodge
theater — to collections of cusps in a single copy [i.e., connected component] of
-symmetry
of the [D-]Θ ell -bridge [cf. Proposition 6.8, (i)] without permuting the collection of
valuations V ± (⊆ V(K)) [cf. the discussion of Definition 6.1, (v)]. This contrasts
sharply with the arithmetic nature of the “single basepoint” [cf. the discussion
“D v ” at each v ∈ V; these collections of cusps are permuted by the F ⋊±
l
of Remark 6.12.4] that underlies the F l
-symmetry of §4, i.e., in the sense that
the F l -symmetry [cf. Proposition 4.9, (i)] permutes the various F l
-translates of
V ± = V ±un ⊆ V Bor (⊆ V(K)) [cf. Example 4.3, (i); Remark 6.1.1].
(ii) The geometric nature of the “single basepoint” of the F ⋊±
l
-symmetry of
a[D-]Θ ±ell -Hodge theater [cf. (i)] is more suited to the theory of the
Hodge-Arakelov-theoretic evaluation of the étale theta function
to be developed in [IUTchII], in which the existence of a “single basepoint” corresponding
to a single connected component of “D v ”forv ∈ V bad plays a central
role.
(iii) By contrast, the arithmetic nature of the “single basepoint” of the F l -
symmetry of a [D-]ΘNF-Hodge theater [cf. (i)] is more suited to the