Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 65
(a) the subcategory D ⊢ v ⊆D v may be reconstructed category-theoretically
from D v [cf. [AbsAnab], Lemma 1.3.8];
(b) the category Dv ⊢ (respectively, Dv Θ )maybereconstructed category-theoretically
from Cv
⊢ (respectively, Cv Θ ) [cf. [FrdI], Theorem 3.4, (v); [FrdII], Theorem
1.2, (i); [FrdII], Example 1.3, (i); [AbsAnab], Theorem 1.1.1, (ii)];
(c) the category D v may be reconstructed category-theoretically from F v
= C v
[cf. [FrdI], Theorem 3.4, (v); [FrdII], Theorem 1.2, (i); [FrdII], Example
1.3, (i); [AbsAnab], Lemma 1.3.1].
Note that it follows immediately from the category-theoreticity of the divisor monoid
Φ Cv [cf. [FrdI], Corollary 4.11, (iii); [FrdII], Theorem 1.2, (i)], together with (a),
(c), and the definition of Cv ⊢ ,that
(d) C ⊢ v may be reconstructed category-theoretically from F v
.
Finally, by applying the algorithmically constructed field structure on the image
of the Kummer map of [AbsTopIII], Proposition 3.2, (iii) [cf. Remark 3.1.2; Remark
3.3.2 below], it follows that one may construct the element “p v ”ofOK ⊲ v
category-theoretically from F v
, hence that the characteristic splitting τv
⊢ may be
reconstructed category-theoretically from F v
. [Here, we recall that the curve X F is
“of strictly Belyi type” — cf. [AbsTopIII], Remark 2.8.3.] In particular,
(e) one may reconstruct the split Frobenioids Fv ⊢ , Fv
Θ
from F v
.
category-theoretically
Remark 3.3.1. A similar remark to Remark 3.2.1 [i.e., concerning the phrase
“reconstructed category-theoretically”] applies to the Frobenioids C v , Cv ⊢ constructed
in Example 3.3.
Remark 3.3.2. Note that the p v -adic Frobenioids C v (respectively, Cv ⊢ )ofExamples
3.2, (iii), (iv); 3.3, (i) consist of essentially the same data as an “MLF-
Galois TM-pair of strictly Belyi type” (respectively, “MLF-Galois TM-pair of monoanalytic
type”), in the sense of [AbsTopIII], Definition 3.1, (ii) [cf. [AbsTopIII],
Remark 3.1.1]. A similar remark applies to the p v -adic Frobenioid C v (respectively,
Cv ⊢ ) of Example 3.2 [cf. [AbsTopIII], Remark 3.1.3].
Example 3.4.
Frobenioids at Archimedean Primes. Let v ∈ V arc . Then:
(i) Write
X v , C v , X v , C v , X −→v , C −→v
for the Aut-holomorphic orbispaces [cf. [AbsTopIII], Remark 2.1.1] determined,
respectively, by the hyperbolic orbicurves X K , C K , X K , C K , −→K X , −→K C at v. Thus,