Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 149
Classical
upper half-plane
Θ ±ell NF-Hodge theaters
in inter-universal
Teichmüller theory
Additive z ↦→ z + a, F ⋊±
l
-
symmetry z ↦→ −z + a (a ∈ R) symmetry
“Functions” assoc’d
to add. symm.
q def
= e 2πiz theta fn. evaluated at
l-tors. [cf. I, 6.12.6, (ii)]
Basepoint assoc’d single cusp V ±
to add. symm. at infinity [cf. I, 6.1, (v)]
Combinatorial
prototype assoc’d cusp cusp
to add. symm.
Multiplicative
symmetry
z ↦→ z·cos(t)−sin(t)
z·sin(t)+cos(t) , F l -
z ↦→ z·cos(t)+sin(t)
z·sin(t)−cos(t)
(t ∈ R) symmetry
“Functions”
assoc’d to
mult. symm.
w def
= z−i
z+i
elements of the
number field F mod
[cf. I, 6.12.6, (iii)]
Basepoints assoc’d
( cos(t) −sin(t)
) (
sin(t) cos(t) , cos(t) sin(t)
)
sin(t) −cos(t)
F l
V Bor = F l
· V ±un
to mult. symm. {entire boundary of H } [cf. I, 4.3, (i)]
Combinatorial nodes of mod p nodes of mod p
prototype assoc’d Hecke correspondence Hecke correspondence
to mult. symm. [cf. II, 4.11.4, (iii), (c)] [cf. II, 4.11.4, (iii), (c)]
Coric symmetries z ↦→ z,−z {±1}
Fig. 6.4: Comparison of F ⋊±
l
-, F l -symmetries
with the geometry of the upper half-plane
that distinguishes the finer “combinatorially holomorphic” [cf. Remarks 4.9.1,