Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 147
a hyperbolic curve in positive characteristic, while the “local system-theoretic”
copy of Z — which, as discussed in (i), also serves as a discrete approximation of
the [geometric portion of the] elliptic curve under consideration — corresponds to
a nilpotent ordinary indigenous bundle over the positive characteristic hyperbolic
curve.
-” and
“F l -”] symmetries of ΘNF-, Θ±ell -Hodge theaters may be thought of as corresponding,
respectively, to the two real dimensions
(iii) Relative to the analogy discussed in (ii) between the “local system-theoretic”
copy of Z of (i) and the indigenous bundles that occur in p-adic Teichmüller theory,
it is interesting to note that the two combinatorial dimensions [cf. [AbsTopIII], Remark
5.6.1] corresponding to the additive and multiplicative [i.e., “F ⋊±
l
· z ↦→ z + a, z ↦→ −z + a;
· z ↦→
z · cos(t) − sin(t)
z · sin(t) + cos(t) ,
z · cos(t)+sin(t)
z ↦→
z · sin(t) − cos(t)
—wherea, t ∈ R; z denotes the standard coordinate on H — of transformations
of the upper half-plane H, i.e., an object that is very closely related to the
canonical indigenous bundles that occur in the classical complex uniformization
theory of hyperbolic Riemann surfaces [cf. the discussion of Remark 4.3.3]. Here,
it is also of interest to observe that the above additive symmetry of the upper
half-plane is closely related to the coordinate on the upper half-plane determined
by the “classical q-parameter”
q
def
= e 2πiz
— a situation that is reminiscent of the close relationship, in the theory of the
present series of papers, between the F ⋊±
l
-symmetry and the Kummer theory
surrounding the Hodge-Arakelov-theoretic evaluation of the theta function on the
l-torsion points at bad primes [cf. Remark 6.12.6, (ii); the theory of [IUTchII]].
Moreover, the fixed basepoint “V ± ” [cf. Definition 6.1, (v)] with respect to which
one considers l-torsion points in the context of the F ⋊±
l
-symmetry is reminiscent of
the fact that the above additive symmetries of the upper half-plane fix the cusp at
infinity. Indeed, taken as a whole, the geometry and coordinate naturally associated
to this additive symmetry of the upper half-plane may be thought of, at the level of
“combinatorial prototypes”, as the geometric apparatus associated to a cusp
[i.e., as opposed to a node — cf. the discussion of [NodNon], Introduction]. By
contrast, the “toral” multiplicative symmetry of the upper half-plane recalled
above is closely related to the coordinate on the upper half-plane that determines
a biholomorphic isomorphism with the unit disc
w
def
= z−i
z+i
— a situation that is reminiscent of the close relationship, in the theory of the
present series of papers, between the F l
-symmetry and the Kummer theory
surrounding the number field F mod [cf. Remark 6.12.6, (iii); the theory of §5 of
the present paper]. Moreover, the action of F l
on the “collection of basepoints
for the l-torsion points” V Bor = F l
· V ±un [cf. Example 4.3, (i)] in the context of