claimroutineinferred_dep

Base height bounded by theta height

Bounds

\hgt(\pi(y))

C(1+h_\Theta(\mathcal A_y))

bounds base height by theta height

Claim·§4.4 Comparison of base height and Faltings height·inside lemBase height bounded by Faltings height

Statement (paper)

At a common even level at which the theta-null map is defined on the covering curves, the theta-null map is nonconstant on each covering curve and extends across the omitted points, and for every

y

in the open set where the theta data are defined one has

\hgt(\pi(y))\le C\bigl(1+h_\Theta(\mathcal A_y)\bigr)

with a constant

C

independent of

y

and its field of definition.

In plain English

At a common even level where the theta-null map is defined on the covering curves, the map is nonconstant on each covering curve and extends across the omitted points. For every

y

where the theta data are defined,

\hgt(\pi(y))\le C(1+h_\Theta(\mathcal A_y))

with

C

independent of

y

and its field of definition.

Why it's here

This step bounds the height of the base point

\pi(y)

by the theta height of the fiber. It does so by using the theta-null map, which is nonconstant at a suitable common even level by the previous claim, and whose height is Pazuki's theta height up to a bounded error. Combined with the next claim, which compares theta height and Faltings height, it gives the base height comparison of the parent lemma.

Classified routine:

Follows from the two preceding claims via a standard height comparison for a nonconstant morphism of curves: a finite map extending across the omitted points bounds the height on one side linearly by the height on the other, and the theta-null height matches Pazuki's theta height. Here this is applied to the theta-null map at a common even level.

Uses

claimTheta-null height matches Pazuki theta heightclaimTheta-null map nonconstant at some level

Used by

claimTheta height bounded by Faltings height

Sources

PaperOpen

Section: §4.4 Comparison of base height and Faltings height
Page: —
Label: —
Line: sections/04-height.tex:490

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Base height bounded by theta height" rests on claim-hei-base-height-2
info
inferred_dep
outline: "Base height bounded by theta height" rests on claim-hei-base-height-3