Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 117
Indeed, recall that since F v
is a Frobenioid of model type [cf. [EtTh], Definition 3.6,
(ii)], it follows [cf. Remark 5.3.3 below] from [FrdI], Corollary 5.7, (i), (iv), that α
preserves base-Frobenius pairs. Thus, once one shows that α induces the identity
on the rational function and divisor monoids of F v
, it follows, by arguing as in the
construction of the equivalence of categories given in the proof of [FrdI], Theorem
5.2, (iv), that the various units obtained in [FrdI], Proposition 5.6, determine [cf.
the argument of the first paragraph of the proof of [FrdI], Proposition 5.6] an
isomorphism between α and the identity self-equivalence of F v
, as desired.
Thus, we proceed to show that α induces the identity on the rational function
and divisor monoids of F v
, as follows. In light of the category-theoreticity
[cf. [EtTh], Theorem 5.6] of the cyclotomic rigidity isomorphism of [EtTh], Proposition
5.5, the fact that α induces the identity on the rational function monoid
follows immediately from the naturality of the Kummer map [cf. the discussion of
Remark 3.2.4; [FrdII], Definition 2.1, (ii)], which is injective by [EtTh], Proposition
3.2, (iii) — cf. the argument of [EtTh], Theorem 5.7, applied to verify the
category-theoreticity of the Frobenioid-theoretic theta function. Next, we consider
the effect of α on the divisor monoid of F v
. To this end, let us first recall that
α preserves cuspidal and non-cuspidal elements of the monoids that appear in this
divisor monoid [cf. [EtTh], Proposition 5.3, (i)]. In particular, by considering the
non-cuspidal portion of the divisor of the Frobenioid-theoretic theta function and
its conjugates [each of which is preserved by α, sinceα has already been shown
to induce the identity on the rational function monoid of F v
], we conclude that α
induces the identity on the non-cuspidal elements of the monoids that appear in
the divisor monoid of F v
[cf. [EtTh], Proposition 5.3, (v), (vi), for a discussion
of closely related facts]. In a similar vein, since any divisor of degree zero on an
elliptic curve that is supported on the torsion points of the elliptic curve admits a
positive multiple which is principal, it follows by considering the cuspidal portions
of divisors of appropriate rational functions [each of which is preserved by α, since
α has already been shown to induce the identity on the rational function monoid of
F v
]thatα also induces the identity on the cuspidal elements of the monoids that
appear in the divisor monoid of F v
. This completes the proof of assertion (iv). ○
Remark 5.3.1.
(i) In the situation of Corollary 5.3, (ii), let
φ : 1 D → 2 D
be a morphism of D-prime-strips [i.e., which is not necessarily an isomorphism!]
that induces an isomorphism between the respective collections of data indexed by
v ∈ V good ,aswellasanisomorphism φ ⊢ : 1 D ⊢ → ∼ 2 D ⊢ between the associated
D ⊢ -prime-strips [cf. Definition 4.1, (iv)]. Then let us observe that by applying
Corollary 5.3, (ii), it follows that φ lifts to a uniquely determined “arrow”
ψ : 1 F → 2 F
—whichwethinkofas“lying over” φ — defined as follows: First, let us recall
that, in light of our assumptions on φ, it follows immediately from the construction