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Is there any example in the literature where someone has considered the problem of bounding the variation of Faltings height at a place of bad reduction? Specifically, if $A_i$ for $i\in \{1,2\}$ are abelian varieties over a number field $K$ (not necessarily their minimal field of definition) and $\phi:A_1\to A_2$ is an isogeny over $K$ of degree $p^n$ such that for $v\in \mathcal{O}_K$ such that $v|p$ the Neron model $\mathcal{A}_1/\mathcal{O}_K$ has bad reduction at $v$, then has anyone done a computation for this variation?

The only example I am aware of where such a question is addressed is this paper by Ullmo--Szpiro (https://projecteuclid.org/euclid.dmj/1077228502) for calculating the variation of the Faltings height for elliptic curves without CM. However, it evades the computation by the Serre open image theorem.

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    $\begingroup$ I'm not sure if I understand the question correctly, but how about Faltings's isogeny formula (semistable case), which is Lemma 5 of his Endlichkeitsatze paper, and a crucial component of his proof of Tate, Shafarevich and Mordell? $\endgroup$ – Vesselin Dimitrov Oct 31 '17 at 3:19

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