Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 5
v ∈ V non ,theportionofaD ⊢ -prime-strip labeled by v is given by a category equivalent
to [the full subcategory determined by the connected objects of] the Galois
category associated to G v ;whenv ∈ V arc , an analogous definition may be given.
In some sense, D-prime-strips may be thought of as abstractions of the “local
arithmetic holomorphic structure” of [copies of] F mod — cf. the discussion of
[AbsTopIII], §I3. On the other hand, D ⊢ -prime-strips may be thought of as “monoanalyticizations”
[i.e., roughly speaking, the arithmetic version of the underlying
real analytic structure associated to a holomorphic structure] of D-prime-strips —
cf. the discussion of [AbsTopIII], §I3. Throughout the present series of papers, we
shall use the notation
⊢
to denote mono-analytic structures.
Next, we recall the notion of a Frobenioid over a base category [cf. [FrdI]
for more details]. Roughly speaking, a Frobenioid [typically denoted “F”] may
be thought of as a category-theoretic abstraction of the notion of a category of
line bundles or monoids of divisors over a base category [typically denoted “D”]
of topological localizations [i.e., in the spirit of a “topos”] suchasaGalois category.
In addition to D- andD ⊢ -prime-strips, we shall also consider various types
of prime-strips that arise from considering various natural Frobenioids — i.e., more
concretely, various natural monoids equipped with a Galois action —atv ∈ V. Perhaps
the most basic type of prime-strip arising from such a natural monoid is an
F-prime-strip. Suppose, for simplicity, that v ∈ V bad .Thenv and F determine,
up to conjugacy, an algebraic closure F v of K v .Write
· O F v
for the ring of integers of F v ;
· O ⊲ F v
⊆O F v
for the multiplicative monoid of nonzero integers;
· O × F v
⊆O F v
for the multiplicative monoid of units;
· O μ F v
⊆O F v
for the multiplicative monoid of roots of unity;
· O μ 2l
F v
⊆O F v
for the multiplicative monoid of 2l-th roots of unity;
· q
v
∈O F v
for a 2l-th root of the q-parameter of E F at v.
Thus, O F v
, O ⊲ , O × , O μ ,andO μ 2l
are equipped with natural G
F v F v F v F v -actions. The
v
portionofaF-prime-strip labeled by v is given by data isomorphic to the monoid
O ⊲ , equipped with its natural Π
F v (↠ G v )-action [cf. Fig. I1.2]. There are various
v
mono-analytic versions of the notion an F-prime-strip; perhaps the most basic is the
notion of an F ⊢ -prime-strip. TheportionofaF ⊢ -prime-strip labeled by v is given
by data isomorphic to the monoid O × × q N, equipped with its natural G F v
v-action
v
[cf. Fig. I1.2]. Often we shall regard these various mono-analytic versions of an
F-prime-strip as being equipped with an additional global realified Frobenioid,
which, at a concrete level, corresponds, essentially, to considering various arithmetic