I am trying to understand the proof of the fact that any morphism $f \colon E_1 \rightarrow E_2$ of elliptic curves over an arbitrary base scheme $S$ satisfying $f(0) = 0$ must respect the group structure. This is part of the claim of Theorem 2.5.1 in the book "Arithmetic moduli of elliptic curves" by Nick Katz and Barry Mazur, but I cannot understand the proof given there. Namely, they have reduced to an Artinian local base $S$ that has a finite residue field, and then on top of p. 80 they claim that $$ f^*(I(f(P))) = I(P + f^{-1}(0)),$$ where $I(D)$ stands for the ideal sheaf of a relative Cartier divisor $D$. How to justify the displayed equality? At first glance it seems that the authors are using that $f$ respects translation, but this is precisely what we are trying to prove! Any help would be greatly appreciated. Thank you!
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5$\begingroup$ Read the proof for abelian schemes in Chapter 6 of GIT. The rigidity technique there is easier to understand. $\endgroup$– user74230Commented Jan 26, 2015 at 5:34
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$\begingroup$ Although it's just over fields, the proof that this is true for abelian varieties given in Mumford's Abelian Varieties via a rigidity argument is also quite enlightening. $\endgroup$– Joe SilvermanCommented Jan 26, 2015 at 11:52
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$\begingroup$ @user74230: The proof in GIT is indeed cleaner, thank you. Though in 6.1 2) there, I think one has to assume in addition that $p$ is locally of finite presentation (to ensure the openness of $p$ used in the proof); this, of course, is harmless in the view of the subsequent discussion. $\endgroup$– Lisa S.Commented Jan 26, 2015 at 22:14
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$\begingroup$ @LisaS.: Note that just above section 6.1 Mumford imposed the standing hypothesis that schemes he considers are all locally noetherian and separated (the archaic distinction of pre-schemes and schemes in original EGA terminology), so in that sense the missing hypothesis was that $p$ is locally of finite type. $\endgroup$– user74230Commented Jan 26, 2015 at 22:30
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