Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

30 SHINICHI MOCHIZUKI
—wherewewriteSL 2 (Z) ∧ for the profinite completion of SL 2 (Z). If one thinks
of SL 2 (Z) ∧ as the geometric étale fundamental group of the moduli stack of elliptic
curves over a field of characteristic zero, then the p-adic Teichmüller theory of
[pOrd], [pTeich] does indeed constitute a generalization of ρ Zp to more general p-
adic hyperbolic curves.
From a representation-theoretic point of view, the next natural direction
in which to further develop the theory of [pOrd], [pTeich] consists of attempting to
generalize the theory of representations into SL 2 (Z p ) obtained in [pOrd], [pTeich]
to a theory concerning representations into SL n (Z p )forarbitrary n ≥ 2. This is
precisely the motivation that lies, for instance, behind the work of Joshi and Pauly
[cf. [JP]].
On the other hand, unlike the original motivating representation ρ R , the representation
ρẐ is far from injective, i.e., put another way, the so-called Congruence
Subgroup Problem fails to hold in the case of SL 2 . This failure of injectivity means
that working with
ρẐ only allows one to access a relatively limited portion of SL 2 (Z) ∧ .
From this point of view, a more natural direction in which to further develop the
theory of [pOrd], [pTeich] is to consider the “anabelian version”
ρ Δ : SL 2 (Z) ∧ → Out(Δ 1,1 )
of ρẐ — i.e., the natural outer representation on the geometric étale fundamental
group Δ 1,1 of the tautological family of once-punctured elliptic curves over the
moduli stack of elliptic curves over a field of characteristic zero. Indeed, unlike the
case with ρẐ, one knows [cf. [Asada]] that ρ Δ is injective. Thus,the“arithmetic
Teichmüller theory for number fields equipped with a [once-punctured] elliptic
curve” constituted by the inter-universal Teichmüller theory developed in
the present series of papers may [cf. the discussion of §I4!] be regarded as a
realization of this sort of “anabelian” approach to further developing the p-adic
Teichmüller theory of [pOrd], [pTeich].
In the context of these two distinct possible directions for the further development
of the p-adic Teichmüller theory of [pOrd], [pTeich], it is of interest to recall
the following elementary fact:
If G is a free pro-p group of rank ≥ 2, then a [continuous] representation
can never be injective!
ρ G : G → GL n (Q p )
Indeed, assume that ρ G is injective and write ...⊆ H j ⊆ ...⊆ Im(ρ G ) ⊆ GL n (Q p )
for an exhaustive sequence of open normal subgroups of the image of ρ G .Thensince
the H j are closed subgroups GL n (Q p ), hence p-adic Lie groups, it follows that the
Q p -dimension dim(Hj
ab ⊗ Q p ) of the tensor product with Q p of the abelianization
of H j may be computed at the level of Lie algebras, hence is bounded by the Q p -
dimension of the p-adic Lie group GL n (Q p ), i.e., we have dim(Hj
ab ⊗ Q p ) ≤ n 2 ,in