Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

148 SHINICHI MOCHIZUKI
the F l
-symmetry is reminiscent of the fact that the multiplicative symmetries of
the upper half-plane recalled above act transitively on the entire boundary of the
upper half-plane. That is to say, taken as a whole, the geometry and coordinate
naturally associated to this multiplicative symmetry of the upper half-plane may
be thought of, at the level of “combinatorial prototypes”, as the geometric
apparatus associated to a node, i.e., of the sort that occurs in the reduction modulo
p of a Hecke correspondence [cf. the discussion of [IUTchII], Remark 4.11.4, (iii),
(c); [NodNon], Introduction]. Finally, we note that, just as in the case of the
F ⋊±
l
-, F l
-symmetries discussed in the present paper, the only “coric” symmetries,
i.e., symmetries common to both the additive and multiplicative symmetries of the
upper half-plane recalled above, are the symmetries “{±1}” [i.e., the symmetries
z ↦→ z,−z in the case of the upper half-plane]. The observations of the above
discussion are summarized in Fig. 6.4 below.
Remark 6.12.4.
(i) Just as in the case of the F l
-symmetry of Proposition 4.9, (i), the F⋊±
l
-
symmetry of Proposition 6.8, (i), will eventually be applied, in the theory of the
present series of papers [cf. theory of [IUTchII], [IUTchIII]], to establish an
explicit network of comparison isomorphisms
relating various objects — such as log-volumes — associated to the non-labeled
prime-strips that are permuted by this symmetry [cf. the discussion of Remark
4.9.1, (i)]. Moreover, just as in the case of the F l
-symmetry studied in §4 [cf. the
discussion of Remark 4.9.2], one important property of this “network of comparison
isomorphisms” is that it operates without “label crushing” [cf. Remark 4.9.2, (i)]
— i.e., without disturbing the bijective relationship between the set of indices of
the symmetrized collection of prime-strips and the set of labels ∈ T → ∼ F l under
consideration. Finally, just as in the situation studied in §4,
this crucial synchronization of labels is essentially a consequence of
the single connected component
— or, at a more abstract level, the single basepoint — of the global object [i.e.,
“ † D ⊚± ” in the present §6; “ † D ⊚ ”in§4] that appears in the [D-Θ ±ell -orD-ΘNF-]
Hodge theater under consideration [cf. Remark 4.9.2, (ii)].
(ii) At a more concrete level, the “synchronization of labels” discussed in (i) is
realized by means of the crucial bijections
† ζ : LabCusp( † D ⊚ ) ∼ → J;
† ζ ± : LabCusp ± ( † D ⊚± ) ∼ → T
of Propositions 4.7, (iii); 6.5, (iii). Here, we pause to observe that it is precisely the
existence of these
bijections relating index sets of capsules of D-prime-strips to sets of
global [±-]label classes of cusps