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Let $S$ be an integral 1-dimensional scheme with function field $K$.

Let $E$ be an elliptic curve over $K$. The torsion of $E$ over $K$ is not necessarily finite. As an example consider an elliptic curve over $\mathbf C(t)$.

Now, assume the residue field of each closed point of $S$ to be finite.

Is the torsion of $E(K)$ finite?

Note that I'm not assuming $S$ to be noetherian.

Note that we may assume $S$ to be affine. To answer the above question positively, it would suffice to show the torsion embeds into the rational points of a special (not geometric) fibre of $E$.

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I have a vague idea that this might be wrong.

First, it seems to me that you can even assume that $S$ is the spectrum of a local ring. Then, if $p$ is the resicual characteristic, the reduction map will in general not be injective on $p$--torsion. So, if I had to produce couterexamples to this, I would start with an $E$ over $\mathbb Z_p$ with supersingular reduction, and then base change $E$ to some big, totally ramified extension of $\mathbb Z_p$ where lots of $p$--power torsion points of $E$ are defined.

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    $\begingroup$ I think you can work backwards, choosing the subfield $K$ of an algebraic closure of $\mathbb{Q}_p$ by adjoining the coordinates of all $p$-power torsion points to $\mathbb{Q}_p$. $\endgroup$
    – S. Carnahan ♦
    Commented Dec 7, 2012 at 3:59

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