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Is there a modern reference for NĂ©ron's "Quasi-fonctions et Hauteurs sur les Varietes Abeliennes" i.e. using Grothendieck's language of schemes and in English?

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You might have a look at Bosch-Lorenzini's paper (§ 4) and Cédric Pépin's preprint – Qing Liu Dec 6 '11 at 21:34
I am not sure what topic from that paper you are interested in but you might find some of them in the 'Arithmetic Geometry' volume edited by G. Cornell and J. Silverman:… – Mahdi Majidi-Zolbanin Dec 8 '11 at 21:06
Isn't this pretty much the content of Lang's Fundamentals of Diophantine Geometry, Chapter 11, "Neron Functions on Abelian Varieties"? – Joe Silverman Jan 22 '13 at 4:02

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