Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

152 SHINICHI MOCHIZUKI
explicit construction of the number field F mod [cf. Example 5.1]
— i.e., to the construction of an object which is invariant with respect to the
Aut(C K )/Aut ɛ (C K ) → ∼ F l
-symmetries that appear in the discussion of Example
4.3, (iv). That is to say, if one attempts to carry out a similar construction to
the construction of Example 5.1 with respect to the copy of D ⊚± that appears
in a [D-]Θ ell -bridge, then one must sacrifice the crucial ridigity with respect to
Aut(D ⊚± )/Aut ± (D ⊚± ) → ∼ F l
that arises from the structure [i.e., definition] of a
[D-]Θ ell -bridge. Moreover, if one sacrifices this F l
-rigidity, then one no longer has
a situation in which the symmetry under consideration is defined relative to a single
copy of “D v ” at each v ∈ V, i.e., defined with respect to a “single geometric basepoint”.
In particular, once one sacrifices this F l
-rigidity, the resulting symmetries
are no longer compatible with the theory of the Hodge-Arakelov-theoretic evaluation
of the étale theta function to be developed in [IUTchII] [cf. (ii)].
(iv) One way to understand the difference discussed in (iii) between the global
portions [i.e., the portions involving copies of D ⊚ , D ⊚± ]ofa[D-]ΘNF-Hodge theater
and a [D-]Θ ±ell -Hodge theater is as a reflection of the fact that whereas the Borel
subgroup {( )} ∗ ∗
⊆ SL 2 (F l )
0 ∗
is normally terminal in SL 2 (F l ) [cf. the discussion of Example 4.3], the “semiunipotent”
subgroup {( )}
±1 ∗
⊆ SL 2 (F l )
0 ±1
[which corresponds to the subgroup Aut ± (D ⊚± ) ⊆ Aut(D ⊚± ) — cf. the discussion
of Definition 6.1, (v)] fails to be normally terminal in SL 2 (F l ).
(v) In summary, taken as a whole, a [D-]Θ ±ell NF-Hodge theater [cf. Remark
6.12.2, (ii)] may be thought of as a sort of
“intricate relay between geometric and arithmetic basepoints”
that allows one to carry out, in a consistent fashion, both
(a) the theory of the Hodge-Arakelov-theoretic evaluation of the étale theta
function to be developed in [IUTchII] [cf. (ii)], and
(b) the explicit construction of the number field F mod in Example 5.1 [cf.
(iii)].
Moreover, if one thinks of F l as a finite approximation of Z [cf. Remark 6.12.3],
then this intricate relay between geometric and arithmetic — or, alternatively, F ⋊±
l
[i.e., additive!]- and F l
[i.e., multiplicative!]- basepoints — may be thought of as a
sort of
global combinatorial resolution of the two combinatorial dimensions
— i.e., additive and multiplicative [cf. [AbsTopIII], Remark
5.6.1] — of the ring Z.