## finiteness of torsion points of an abelian variety over a totally real field？

Ribet has shown the following result: if $A$ is an abelian variety over a number field $E$, then the torsion subgroup of $A(E^{cyc})$ is finite. Here $E^{cyc}$ is the union $\bigcup_nE(\mu_n)$, $\mu_n$ being the set of n-th roots of unity.

Here we have used $E^{cyc}=E\mathbb{Q}^{ab}$. I would like to know if such finiteness result holds for other fields. For example, if $A$ is a CM abelian variety over a number field $E$, and let $F$ a totally real number field, with $F^{ab}$ the maximal abelian extension of $F$, can we expect the torsion subgroup of $A(EF^{ab})$ to be finite, too? Or should one expect the torsion subgroup of $A(EF^{ab})$ to be much smaller that the total torsion part of $A((EF)^{ab})$? The latter is infinite by a theorem of Zarhin, and I wonder if the torsion part can be somehow measured, and cut out some smaller subgroup.

Thanks!

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 I like the non-standard question mark. – darij grinberg Nov 27 2011 at 3:11