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As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic curves over fields. Could anyone suggest me any references(books, papers, lecture notes, etc)?

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2 Answers 2

Try the first chapters of Arithmetic Moduli of Elliptic Curves, by N. Katz and B. Mazur, Annals of Math Studies 108

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You could try reading (the relevant parts of) Qing Liu's book on Algebraic geometry or the book on Neron models by Bosch-Lutkebohmert-Raynaud to get a feeling for elliptic curves over one-dimensional schemes. You could also try reading some papers where abelian schemes are used, e.g., Szpiro's asterisque (1985) on the Mordell conjecture, or Jinbi Jin's paper http://arxiv.org/abs/1303.4327 on equations for the modular curve $Y_1(n)$ over $\mathbf Z[1/n]$.

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