Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 9
is well-adapted to the Kummer theory surrounding the Hodge-Arakelov-theoretic
evaluation of the theta function at l-torsion points [cf. the above discussion].
By contrast, the “toral”, or“nodal” [cf. the classical theory of the structure of
Hecke correspondences modulo p], geometry expressed by the multiplicative symmetries
of the upper half-plane admits a natural “associated coordinate”, namely,
the classical biholomorphic isomorphism of the upper half-plane with the unit disc,
which is reminiscent of the way in which the F l
-symmetry is well-adapted to the
Kummer theory surrounding the number field F mod [cf. the above discussion].
For more details, we refer to the discussion of Remark 6.12.3, (iii).
From the point of view of the scheme-theoretic Hodge-Arakelov theory developed
in [HASurI], [HASurII], the theory of the combinatorial structure of a Θ ±ell NF-
Hodge theater — and, indeed, the theory of the present series of papers! — may
be regarded as a sort of
solution to the problem of constructing “global multiplicative subspaces”
and “global canonical generators” [cf. the quotient “Q” and
the cusp “ɛ” that appear in the above discussion!]
—thenonexistence of which in a “naive, scheme-theoretic sense” constitutes the
main obstruction to applying the theory of [HASurI], [HASurII] to diophantine
geometry [cf. the discussion of Remark 4.3.1]. Indeed, prime-strips may be
thought of as “local analytic sections” of the natural morphism Spec(K) →
Spec(F mod ). Thus, it is precisely by working with such “local analytic sections” —
i.e., more concretely, by working with the collection of valuations V, as opposed to
the set of all valuations of K — that one can, in some sense, “simulate” the notions
of a “global multiplicative subspace” or a “global canonical generator”. On the other
hand, such “simulated global objects” may only be achieved at the cost of
“dismantling”, orperforming“surgery” on, the global prime structure
of the number fields involved [cf. the discussion of Remark 4.3.1]
— a quite drastic operation, which has the effect of precipitating numerous technical
difficulties, whose resolution, via the theory of semi-graphs of anabelioids, Frobenioids,
the étale theta function, and log-shells developed in [SemiAnbd], [FrdI],
[FrdII], [EtTh], and [AbsTopIII], constitutes the bulk of the theory of the present
series of papers! From the point of view of “performing surgery on the global prime
structure of a number field”, the labels ∈ F l
that appear in the “arithmetic”
F l
-symmetry may be thought of as a sort of “miniature finite approximation”
of this global prime structure, in the spirit of the idea of “Hodge theory at
finite resolution” discussed in [HASurI], §1.3.4. On the other hand, the labels ∈ F l
that appear in the “geometric” F ⋊±
l
-symmetry may be thought of as a sort
of “miniature finite approximation” of the natural tempered Z-coverings [i.e.,
tempered coverings with Galois group Z] of the Tate curves determined by E F at
v ∈ V bad , again in the spirit of the idea of “Hodge theory at finite resolution”
discussed in [HASurI], §1.3.4.