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][1]), 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? [1]: https://www2.mathematik.hu-berlin.de/~bakkerbe/faltings6.pdf