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For an Abelian scheme over a ring of integers in a number field, Faltings has a theorem that describes how the Faltings' height changes through an isogeny. There are multiple references for this statement and proof. See Proposition 4.1 in the notes Faltings height by Daniele Agostini (pdf), for instance.

I am interested in the analogous statement over curves over a finite field, in particular when the isogeny has degree a power of $p$, the characteristic of the finite field.

Is any such statement true in this case and if so, what's a reference?

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  • $\begingroup$ Quick note: the mathematician's name is "Gerd Faltings", so it should be "Faltings's" or "Faltings'", not "Falting's". The notes you link consistently make that mistake in the references. $\endgroup$ Commented Aug 18, 2020 at 22:18
  • $\begingroup$ And so did I! Thanks for the correction. $\endgroup$ Commented Aug 18, 2020 at 22:32
  • $\begingroup$ I changed the link reference, because a blank 'this' sending the user to a pdf on a personal webpage is not stable. $\endgroup$ Commented Aug 18, 2020 at 23:55
  • $\begingroup$ Thanks, that's a good point! @davidroberts $\endgroup$ Commented Aug 18, 2020 at 23:56

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