claimroutine
Theta-null map nonconstant at some level
Shows theta-null map is nonconstant at some even level
nonconstancy needed to bound the base height
Claim·§4.4 Comparison of base height and Faltings height·inside lemBase height bounded by Faltings height
Statement (paper)
For a connected analytic disk with a fixed homology marking on which the period matrix varies, there is a sufficiently divisible even level at which the projective vector of theta-null values in is nonconstant; this nonconstancy persists at all multiples of that level, so a common even multiple works for the finitely many components.
In plain English
On a connected analytic disk with fixed homology marking where the period matrix varies, the projective vector of theta-null values is nonconstant at some sufficiently divisible even level, and it stays nonconstant at all multiples of that level. Hence one common even level works for the finitely many components.
Why it's here
This supplies the nonconstancy needed for the next claim, where the theta-null map on each covering curve must be nonconstant so that it extends across omitted points and can be used to bound the base height by a theta height. Passing to multiples of the level lets a single common even level serve all components at once.
Classified routine:
Follows from the classical fact that theta constants of sufficiently high even level give a projective embedding of the Siegel modular variety (Igusa/Mumford), so they separate distinct period matrices. Nonconstancy is inherited by multiples of the level, which gives a common level for the finitely many components.
Uses
Nothing else in the paper
Used by
Sources
PaperOpen
- Section
- §4.4 Comparison of base height and Faltings height
- Page
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- Label
—- Line
sections/04-height.tex:464
Leannot aligned
Phase 2 aligns this node to a Lean declaration.