Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

26 SHINICHI MOCHIZUKI
is described to a substantial degree in the discussion of [AbsTopIII], §I5, i.e., where
the “future Teichmüller-like extension of the mono-anabelian theory” may be understood
as referring precisely to the inter-universal Teichmüller theory developed
in the present series of papers. The starting point of this analogy is the correspondence
between a number field equipped with a [once-punctured] elliptic curve [in the
present series of papers] and a hyperbolic curve over a positive characteristic perfect
field equipped with a nilpotent ordinary indigenous bundle [in p-adic Teichmüller
theory] — cf. Fig. I4.1 above. That is to say, in this analogy, the number field —
which may be regarded as being equipped with a finite collection of “exceptional”
valuations, namely, in the notation of §I1, the valuations lying over V bad
mod — corresponds
to the hyperbolic curve over a positive characteristic perfect field —which
may be thought of as a one-dimensional function field over a positive characteristic
perfect field, equipped with a finite collection of “exceptional” valuations, namely,
the valuations corresponding to the cusps of the curve.
On the other hand, the [once-punctured] elliptic curve in the present series
of papers corresponds to the nilpotent ordinary indigenous bundle in p-adic Teichmüller
theory. Here, we recall that an indigenous bundle may be thought of as a
sort of “virtual analogue” of the first cohomology group of the tautological elliptic
curve over the moduli stack of elliptic curves. Indeed, the canonical indigenous
bundle over the moduli stack of elliptic curves arises precisely as the first de Rham
cohomology module of this tautological elliptic curve. Put another way, from the
point of view of fundamental groups, an indigenous bundle may be thought of as
a sort of “virtual analogue” of the abelianized fundamental group of the tautological
elliptic curve over the moduli stack of elliptic curves. By contrast, in the
present series of papers, it is of crucial importance to use the entire nonabelian
profinite étale fundamental group — i.e., not just its abelizanization! — of the
given once-punctured elliptic curve over a number field. Indeed, only by working
with the entire profinite étale fundamental group can one avail oneself of the crucial
absolute anabelian theory developed in [EtTh], [AbsTopIII] [cf. the discussion
of §I3]. This state of affairs prompts the following question:
To what extent can one extend the indigenous bundles that appear in classical
complex and p-adic Teichmüller theory to objects that serve as “virtual
analogues” of the entire nonabelian fundamental group of the
tautological once-punctured elliptic curve over the moduli stack of [oncepunctured]
elliptic curves?
Although this question lies beyond the scope of the present series of papers, it is
the hope of the author that this question may be addressed in a future paper.
Now let us return to our discussion of the log-theta-lattice, which, as discussed
above, consists of two types of arrows, namely, Θ-link arrows and log-link arrows.
As discussed in [IUTchIII], Remark 1.4.1, (iii) — cf. also Fig. I4.1 above,
as well as Remark 3.9.3, (i), of the present paper — the Θ-link arrows correspond
to the “transition from p n Z/p n+1 Z to p n−1 Z/p n Z”, i.e., the mixed characteristic
extension structure of a ring of Witt vectors, while the log-link arrows, i.e.,
the portion of theory that is developed in detail in [AbsTopIII], and which will be
incorporated into the theory of the present series of papers in [IUTchIII], correspond
to the Frobenius morphism in positive characteristic. As we shall see in