Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

28 SHINICHI MOCHIZUKI
Belyi cuspidalization ←→ Lefschetz trace formula for Frobenius
[Here, we note in passing that this correspondence may be related to the correspondence
discussed in [AbsTopIII], §I5, between Belyi cuspidalization and the
Verschiebung on positive characteristic indigenous bundles by considering the geometry
of Hecke correspondences modulo p, i.e., in essence, graphs of the Frobenius
morphism in characteristic p!] It is the hope of the author that these analogies and
correspondences might serve to stimulate further developments in the theory.
§I5. Other Galois-theoretic Approaches to Diophantine Geometry
The notion of anabelian geometry dates back to a famous “letter to Faltings”
[cf. [Groth]], written by Grothendieck in response to Faltings’ work on the
Mordell Conjecture [cf. [Falt]]. Anabelian geometry was apparently originally conceived
by Grothendieck as a new approach to obtaining results in diophantine
geometry such as the Mordell Conjecture. At the time of writing, the author is
not aware of any expositions by Grothendieck that expose this approach in detail.
Nevertheless, it appears that the thrust of this approach revolves around applying
the Section Conjecture for hyperbolic curves over number fields to obtain a contradiction
by applying this Section Conjecture to the “limit section” of the Galois
sections associated to any infinite sequence of rational points of a proper hyperbolic
curve over a number field [cf. [MNT], §4.1(B), for more details]. On the other hand,
to the knowledge of the author, at least at the time of writing, it does not appear
that any rigorous argument has been obtained either by Grothendieck or by other
mathematicians for deriving a new proof of the Mordell Conjecture from the [as
yet unproven] Section Conjecture for hyperbolic curves over number fields. Nevertheless,
one result that has been obtained is a new proof by M. Kim [cf. [Kim]]
of Siegel’s theorem concerning Q-rational points of the projective line minus three
points — a proof which proceeds by obtaining certain bounds on the cardinality
of the set of Galois sections, without applying the Section Conjecture or any other
results from anabelian geometry.
In light of the historical background just discussed, the theory exposed in
the present series of papers — which yields, in particular, a method for applying
results in absolute anabelian geometry to obtain diophantine results such
as those given in [IUTchIV] — occupies a somewhat curious position, relative to
the historical development of the mathematical ideas involved. That is to say, at a
purely formal level, the implication
anabelian geometry =⇒ diophantine results
at first glance looks something like a “confirmation” of Grothendieck’s original
intuition. On the other hand, closer inspection reveals that the approach of the
theory of the present series of papers — that is to say, the precise content of the
relationship between anabelian geometry and diophantine geometry established in
the present series of papers — differs quite fundamentally from the sort of approach
that was apparently envisioned by Grothendieck.
Perhaps the most characteristic aspect of this difference lies in the central role
played by anabelian geometry over p-adic fields in the present series of papers.