claimroutine

Theta-null data after finite covers

Builds theta-null map on finite covers, matching the Nullwerte
sets up theta data for the height comparison
Claim·§4.4 Comparison of base height and Faltings height·inside lemBase height bounded by Faltings height

Statement (paper)

For every fixed even integer r≥2r\ge 2, after passing to a finite cover of each geometric component of YY (a finite extension of its function field), one can choose a symmetric ample line bundle M\mathcal M representing the principal polarization together with the theta data of [Pazuki2012, Section 2.3]. These data spread over a dense open subset of the smooth projective covering curve, where they define an algebraic theta-null map whose value at a fiber is the point Θ(0)\Theta(0) given by Pazuki's good choices of theta data, which after complex uniformization of the fiber is identified, up to the finite coordinate ambiguity among good choices, with the characteristic Nullwerte θ[a,b](τ,0)\theta[a,b](\tau,0). The covers and data may depend on rr.

In plain English

For each fixed even r≥2r\ge 2, after a finite cover of each geometric component of YY, one can choose a symmetric ample line bundle M\mathcal M giving the principal polarization, together with Pazuki's theta data. These spread over a dense open subset of the covering curve and define an algebraic theta-null map whose value on a fiber is Θ(0)\Theta(0), matching the Nullwerte θ[a,b](τ,0)\theta[a,b](\tau,0) up to a finite ambiguity among good choices.

Why it's here

This claim sets up the algebraic theta-null map on a curve cover, so that heights of theta-null values can be compared with Pazuki's theta height in the next claim. Passing to finite covers (possibly depending on rr) is what makes the symmetric bundle and theta data exist globally, and identifying the map with the classical Nullwerte links the algebraic and analytic descriptions.
Classified routine:
A standard spreading-out argument: Pazuki's theta structures (Pazuki 2012, Section 2.3) defined over a finite extension of the function field extend to a dense open subset of the covering curve, and the identification with the classical theta Nullwerte follows from the complex uniformization of the fibers.

Uses

Nothing else in the paper

Used by

claimTheta-null height matches Pazuki theta height

Sources

PaperOpen
Section
§4.4 Comparison of base height and Faltings height
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sections/04-height.tex:373
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