Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

56 SHINICHI MOCHIZUKI
object of D v , then we shall denote by “T (−) ”theFrobenius-trivial object of F v
[which is completely determined up to isomorphism] that lies over “(−)”.
(ii) Next, let us recall [cf. [EtTh], Proposition 5.1; [FrdI], Corollary 4.10] that
the birationalization
F ÷ def
= F birat
v v
may be reconstructed category-theoretically from F v
[cf. Remark 3.2.1 below]. Write
Ÿ v
→ X v
for the tempered covering determined by the object “Ÿ log ” in the discussion at the
beginning of [EtTh], §5. Thus, we may think of Ÿ as an object of D v
v [cf. the
object “A ⊚ ”of[EtTh],§5, in the “double underline case”]. Then let us recall the
“Frobenioid-theoretic l-th root of the theta function”, whichisnormalized so as to
attain the value 1 at the point “ √ −1” [cf. [EtTh], Theorem 5.7]; we shall denote
the reciprocal of [i.e., “1 over”] this theta function by
Θ v
∈O × (T ÷ Ÿ
v
)
— where we use the superscript “÷” to denote the image in F ÷ of an object of F .
v v
Here, we recall that Θ v
is completely determined up to multiplication by a 2l-th root
of unity [i.e., an element of μ 2l (T ÷ )] and the action of the group of automorphisms
Ÿ
v
l · Z ⊆ Aut(TŸ
v
) [i.e., we write Z for the group denoted “Z” in [EtTh], Theorem
5.7; cf. also the discussion preceding [EtTh], Definition 1.9]. Moreover, we recall
from the theory of [EtTh], §5 [cf. the discussion at the beginning of [EtTh], §5;
[EtTh], Theorem 5.7] that
TŸ
v
[regarded up to isomorphism] and
Θ v
[regardeduptotheμ 2l (T ÷ Ÿ
v
), l · Z indeterminacies discussed above]
may be reconstructed category-theoretically from F v
[cf. Remark 3.2.1 below].
(iii) Next, we recall from [EtTh], Corollary 3.8, (ii) [cf. also [EtTh], Proposition
5.1], that the p v -adic Frobenioid constituted by the “base-theoretic hull” [cf. [EtTh],
Remark 3.6.2]
C v ⊆F v
[i.e., we write C v for the subcategory “C bs-fld ” of [EtTh], Definition 3.6, (iv)] may
be reconstructed category-theoretically from F v
[cf. Remark 3.2.1 below].
(iv) Write q v for the q-parameter of the elliptic curve E v over K v .Thus,wemay
think of q v as an element q v ∈O ⊲ (T Xv )( ∼ = OK ⊲ v
). Note that it follows from our assumption
concerning 2-torsion [cf. Definition 3.1, (b)], together with the definition