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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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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 on equations for the modular curve $Y_1(n)$ over $\mathbf Z[1/n]$.

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