Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 61
[cf. [FrdI], Proposition 1.11, (iv)]. In particular, one may pull-back sections of the
monoid O ⊲ (−) on D Θ Cv
Θ v of Example 3.2, (v), to B. Such pull-backs are useful, for
instance, when one considers the roots of Θ v
, as in the theory of [EtTh], §5.
Remark 3.2.4. Before proceeding, we pause to discuss certain minor oversights
on the part of the author in the discussion of the theory of tempered Frobenioids
in [EtTh], §3, §4. Let Z log
∞ be as in the discussion at the beginning of [EtTh], §3.
Here, we recall that Z log
∞ is obtained as the “universal combinatorial covering” of
the formal log scheme associated to a stable log curve with split special fiber over
the ring of integers of a finite extension of an MLF of residue characteristic p [cf.
loc. cit. for more details]; we write Z log for the generic fiber of the stable log curve
under consideration.
(i) First, let us consider the following conditions on a nonzero meromorphic
function f on Z log
∞ :
(a) For every N ∈ N ≥1 , it holds that f admits an N-th root over some
tempered covering of Z log .
(b) For every N ∈ N ≥1 which is prime to p, it holds that f admits an N-th
root over some tempered covering of Z log .
(c) The divisor of zeroes and poles of f is a log-divisor.
It is immediate that (a) implies (b). Moreover, one verifies immediately, by considering
the ramification divisors of the tempered coverings that arise from extracting
roots of f, that (b) implies (c). When N is prime to p, iff satisfies (c), then
it follows immediately from the theory of admissible coverings [cf., e.g., [PrfGC],
§2, §8] that there exists a finite log étale covering Y log → Z log whose pull-back
Y∞
log → Z∞
log to the generic fiber Z∞
log of Z log
∞ is sufficient
(R1) to annihilate all ramification over the cusps or special fiber of Z log
∞
might arise from extracting an N-th root of f, aswellas
that
(R2) to split all extensions of the function fields of irreducible components of
the special fiber of Z log
∞ that might arise from extracting an N-th root of
f.
That is to say, in this situation, it follows that f admits an N-th root over the
tempered covering of Z log given by the “universal combinatorial covering” of Y log .
In particular, it follows that (c) implies (b). Thus, in summary, we have:
(a) =⇒ (b) ⇐⇒ (c).
On the other hand, unfortunately, it is not clear to the author at the time of writing
whether or not (c) [or (b)] implies (a).
(ii) Observe that it follows from the theory of [EtTh], §1 [cf., especially, [EtTh],
Proposition 1.3] that the theta function that forms the main topic of interest of
[EtTh] satisfies condition (a) of (i).