Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

24 SHINICHI MOCHIZUKI
combinatorial dimensions” of a ring that leave the arithmetic holomorphic structure
fixed — cf. the discussion of the “juggling of ⊞, ⊠ induced by log” in
[AbsTopIII], §I3. The ultimate conclusion of the theory of [IUTchIII] is that
the “a priori unbounded deformations” of the arithmetic holomorphic
structure given by the Θ-link in fact admit canonical bounds, which
may be thought of as a sort of reflection of the “hyperbolicity” of the
given number field equipped with an elliptic curve
— cf. [IUTchIII], Corollary 3.12. Such canonical bounds may be thought of as
analogues for a number field of canonical bounds that arise from differentiating
Frobenius liftings in the context of p-adic hyperbolic curves — cf. the discussion
in the final portion of [AbsTopIII], §I5. Moreover, such canonical bounds are
obtained in [IUTchIII] as a consequence of
the explicit description of a varying arithmetic holomorphic structure
within a fixed mono-analytic “container”
— cf. the discussion of §I2! — furnished by [IUTchIII], Corollary 3.11 [cf. also
the discussion of [IUTchIII], Remarks 3.12.2, 3.12.3, 3.12.4], i.e., a situation that
is entirely formally analogous to the summary of complex Teichmüller theory given
above.
The significance of the log-theta-lattice is best understood in the context of
the analogy between the inter-universal Teichmüller theory developedinthe
present series of papers and the p-adic Teichmüller theory of [pOrd], [pTeich].
Here, we recall for the convenience of the reader that the p-adic Teichmüller theory
of [pOrd], [pTeich] may be summarized, [very!] roughly speaking, as a sort of
generalization, to the case of “quite general” p-adic hyperbolic curves, of
the classical p-adic theory surrounding the canonical representation
π 1 ((P 1 \{0, 1, ∞}) Qp ) → π 1 ((M ell ) Qp ) → PGL 2 (Z p )
—wherethe“π 1 (−)’s” denote the étale fundamental group, relative to a suitable
basepoint; (M ell ) Qp denotes the moduli stack of elliptic curves over Q p ; the first
horizontal arrow denotes the morphism induced by the elliptic curve over the projective
line minus three points determined by the classical Legendre form of the
Weierstrass equation; the second horizontal arrow is the representation determined
by the p-power torsion points of the tautological elliptic curve over (M ell ) Qp . In
particular, the reader who is familiar with the theory of the classical representation
of the above display, but not with the theory of [pOrd], [pTeich], may nevertheless
appreciate, to a substantial degree, the analogy between the inter-universal
Teichmüller theory developed in the present series of papers and the p-adic Teichmüller
theory of [pOrd], [pTeich] by
thinking in terms of the
well-known classical properties of this classical representation.
In some sense, the gap between the “quite general” p-adic hyperbolic curves that
appear in p-adic Teichmüller theory and the classical case of (P 1 \{0, 1, ∞}) Qp may