Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 91
group scheme structure of the elliptic curve determined by X v [i.e., whose origin
is given by the cusp labeled 0 ∈|F l |]. We shall refer to these μ − -translates of the
cusps with labels ∈|F l | as the evaluation points of X v . Note that the value of
the theta function “Θ v
” of Example 3.2, (ii), at a point lying over an evaluation
point arising from a cusp with label j ∈|F l | is contained in the μ 2l -orbit of
{ q j2
v
} j ≡ j
[cf. Example 3.2, (iv); [EtTh], Proposition 1.4, (ii)] — where j ranges over the
elements of Z that map to j ∈|F l |. In particular, it follows immediately from the
definition of the covering X v
→ X v [i.e., by considering l-th roots of the theta
function! — cf. [EtTh], Definition 2.5, (i)] that the points of X v
that lie over
evaluation points of X v are all defined over K v . We shall refer to the points
∈ X v
(K v ) that lie over the evaluation points of X v as the evaluation points of X v
and to the various sections
G v → Π v =Π tp X
v
of the natural surjection Π v ↠ G v that arise from the evaluation points as the
evaluation sections of Π v ↠ G v . Thus, each evaluation section has an associated
label ∈|F l |. Note that there is a group-theoretic algorithm for constructing
the evaluation sections from [isomorphs of] the topological group Π v . Indeed, this
follows immediately from [the proofs of] [EtTh], Corollary 2.9 [concerning the grouptheoreticity
of the labels]; [EtTh], Proposition 2.4 [concerning the group-theoreticity
of Π Cv , Π Xv ]; [SemiAnbd], Corollary 3.11 [concerning the dual semi-graphs of the
special fibers of stable models], applied to Δ tp X
⊆ Π tp X
=Π v ;[SemiAnbd],The-
v v
orem 6.8, (iii) [concerning the group-theoreticity of the decomposition groups of
μ − -translates of the cusps].
(ii) We continue to suppose that v ∈ V bad .Let
D > = {D >,w } w∈V
be a copy of the “tautological D-prime-strip” {D w } w∈V .Foreachj ∈ F l ,write
φ Θ v j
: D vj →D >,v
for the poly-morphism given by the collection of morphisms [cf. §0] obtained by
composing with arbitrary isomorphisms D vj
∼
→B temp (Π v ) 0 , B temp (Π v ) 0 ∼ →D >,v
the various morphisms B temp (Π v ) 0 →B temp (Π v ) 0 that arise [i.e., via composition
with the natural surjection Π v ↠ G v ]fromtheevaluation sections labeled j. Now
if C is any isomorph of B temp (Π v ) 0 , then let us write
π geo
1 (C) ⊆ π 1 (C)
for the subgroup corresponding to Δ tp X
v
⊆ Π tp X
v
=Π v , a subgroup which we recall
may be reconstructed group-theoretically [cf., e.g., [AbsTopI], Theorem 2.6,