Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

62 SHINICHI MOCHIZUKI
(iii) In [EtTh], Definition 3.1, (ii), a meromorphic function f as in (i) is defined
to be “log-meromorphic” if it satisfies condition (c) of (i). On the other hand, in the
proof of [EtTh], Proposition 4.2, (iii), it is necessary to use property (a) of (i) —
i.e., despite the fact that, as remarked in (i), it is not clear whether or not property
(c) implies property (a). The author apologizes for any confusion caused by this
oversight on his part.
(iv) The problem pointed out in (iii) may be remedied — at least from the
point of view of the theory of [EtTh] — via either of following two approaches:
(A) One may modify [EtTh], Definition 3.1, (ii), by taking the definition of a
“log-meromorphic” function to be a function that satisfies condition (a) [i.e., as
opposed to condition (c)] of (i). [In light of the content of this modified definition,
perhaps a better term for this class of meromorphic functions would be “temperedmeromorphic”.]
Then the remainder of the text of [EtTh] goes through without
change.
(B) One may modify [EtTh], Definition 4.1, (i), by assuming that the meromorphic
function “f ∈ O × (A birat )” of [EtTh], Definition 4.1, (i), satisfies the following
“Frobenioid-theoretic version” of condition (a):
(d) For every N ∈ N ≥1 , there exists a linear morphism A ′ → A in C such
that the pull-back of f to A ′ admits an N-th root.
[Here, we recall that, as discussed in (ii), the Frobenioid-theoretic theta functions
that appear in [EtTh] satisfy (d).] Note that since the rational function monoid of
the Frobenioid C, as well as the linear morphisms of C, arecategory-theoretic [cf.
[FrdI], Theorem 3.4, (iii), (v); [FrdI], Corollary 4.10], this condition (d) is categorytheoretic.
Thus, if one modifies [EtTh], Definition 4.1, (i), in this way, then the
remainder of the text of [EtTh] goes through without change, except that one must
replace the reference to the definition of “log-meromorphic” [i.e., [EtTh], Definition
3.1, (ii)] that occurs in the proof of [EtTh], Proposition 4.2, (iii), by a reference to
condition (d) [i.e., in the modified version of [EtTh], Definition 4.1, (i)].
(v) In the discussion of (iv), we note that the approach of (A) results in a
slightly different definition of the notion of a “tempered Frobenioid” from the original
definition given in [EtTh]. Put another way, the approach of (B) has the advantage
that it does not result in any modification of the definition of the notion of a
“tempered Frobenioid”; that is to say, the approach of (B) only results in a slight
reduction in the range of applicability of the theory of [EtTh], §4, which is essentially
irrelevant from the point of view of the present series of papers, since [cf. (ii)] theta
functions lie within this reduced range of applicability. On the other hand, the
approach of (A) has the advantage that one may consider the Kummer theory
of arbitrary rational functions of the tempered Frobenioid without imposing any
further hypotheses. Thus, for the sake of simplicity, in the present series of papers,
we shall interpret the notion of a “tempered Frobenioid” via the approach of (A).
(vi) Strictly speaking, the definition of the monoid “Φ ell
W ” given in [EtTh],
Example 3.9, (iii), leads to certain technical difficulties, which are, in fact, entirely
irrelevant to the theory of [EtTh]. These technical difficulties may be averted by