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Hello, everyone. I want to ask a question about abelian scheme.

If $X$ is a scheme, proper, finite type, over Dedekind scheme $S$ with a section $e$, and all the fiber are abelian varieties with the identity sections which induced by $e$, does there exists an group structure on $X$ with identity section is $e$? if not could you please show me a counter example?

Thank you very much.

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  • $\begingroup$ I think you need to be a bit careful here: are you looking for a group structure on $X$ which induces the ones you already have on the fibres? Because, unless I'm confused, you can make this impossible even for a trivial family of elliptic curves simply by choosing different base points in each fibre. $\endgroup$ – Martin Bright Jan 16 '12 at 13:32
  • $\begingroup$ Perhaps you should ask if you get a torsor under an $S$-group scheme. $\endgroup$ – S. Carnahan Jan 16 '12 at 13:39
  • $\begingroup$ Yes,Martin.I hope there is a group structure on X which compatible with the group structure on the fibres. $\endgroup$ – kiseki Jan 16 '12 at 13:45
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    $\begingroup$ Martin is pointing out that when one says "elliptic curve" or "abelian variety" one is including the datum of an identity section. In particular, an easy counterexample comes from choosing an elliptic curve $X$ over a trait $S$, and changing the identity section (and hence the group law) in the closed fiber. The identity sections over the two points cannot be glued to a single identity section from $S$ to $X$. You can fix this problem by forgetting about the identity section, and asking for $X$ to be a torsor under some abelian scheme. $\endgroup$ – S. Carnahan Jan 16 '12 at 15:13
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If I understand the question correctly, the answer is Yes. It suffices in fact to have a group structure on the generic fibre $X_K$ of $X$. Since $X$ is smooth and proper, it is then the Néron model of $X_K$, which carries a group structure by its universal property.

By the way, there is only one group structure an an abelian variety with a given element as unit, so not only has $X$ a global group structure, but it induces the given group structure on each closed fibre.

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