Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 31
contradiction to the well-known fact since G ∼ = Im(ρ G )isfree pro-p of rank ≥ 2, it
holds that dim(Hj
ab ⊗ Q p ) →∞as j →∞. Note, moreover, that
this sort of argument — i.e., concerning the asymptotic behavior of
abelianizations of open subgroups —ischaracteristic of the sort of proofs
that typically occur in anabelian geometry [cf., e.g., the proofs of
[Tama1], [pGC], [CombGC]!].
On the other hand, the fact that ρ G can never be injective shows that
so long as one restricts oneself to representation theory into GL n (Q p )
for a fixed n, one can never access the sort of asymptotic phenomena
that form the “technical core” [cf., e.g., the proofs of [Tama1], [pGC],
[CombGC]!] of various important results in anabelian geometry.
Put another way, the two “directions” discussed above — i.e., representationtheoretic
and anabelian —appeartobeessentially mutually alien to one
another.
In this context, it is of interest to observe that the diophantine results derived
in [IUTchIV] from the inter-universal Teichmüller theory developed in the present
series of papers concern essentially asymptotic behavior, i.e., they do not concern
properties of “a specific rational point over a specific number field”, but rather properties
of the asymptotic behavior of “varying rational points over varying number
fields”. One important aspect of this asymptotic nature of the diophantine results
derived in [IUTchIV] is that there are no distinguished number fields that occur
in the theory, i.e., the theory — being essentially asymptotic in nature! — is
“invariant” with respect to passing to finite extensions of the number field involved
[which, from the point of view of the absolute Galois group G Q of Q, corresponds
precisely to passing to smaller and smaller open subgroups, as in the above discussion!].
This contrasts sharply with the “representation-theoretic approach to
diophantine geometry” constituted by such works as [Wiles], where specific rational
points over the specific number field Q — or, for instance, in generalizations
of [Wiles] involving Shimura varieties, over specific number fields characteristically
associated to the Shimura varieties involved — play a central role.
Acknowledgements:
I would like to thank Fumiharu Kato, Akio Tamagawa, Go Yamashita, and
Mohamed Saïdi for many helpful discussions concerning the material presented in
this paper. Also, I feel deeply indebted to Go Yamashita and Mohamed Saïdi for
their meticulous reading of and numerous comments concerning the present paper.