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

Question

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!

Answer 1

This is Corollary 2 in Show-Wu Zhang's Equidistribution of points of small heights on abelian varieties, Annals of Math. 147 (1998), no. 1, 159--165.

Observe that $\mathbf Q^{\mathrm{cyc}}=\mathbf Q^t(i)$, where $\mathbf Q^t$ is a totally real extension of $\mathbf Q$ (generated by all $\cos(\pi/n)$, $n>0$). Consequently, your assertion follows from the equidistribution theorem of Szpiro, Ullmo and Zhang, as applied to the Weil restriction of scalars $A'=R_{E(i)/\mathbf Q}A$. Indeed, if the set of torsion points of $A(E(i)\mathbf Q^t)$ were Zariski dense, one would be able to define a dense sequence of torsion points in $A'(\mathbf Q^t)$. The corresponding sequence of probability measures on $A'(\mathbf C)$ supported by their Galois orbits would be supported on $A'(\mathbf R)$ but converge (by the S-U-Z theorem) to the Haar measure on $A'(\mathbf C)$. The general case can be deduced from this one by passing to an appropriate abelian subvariety.

Note that Zhang attributes this result to Zarhin (Duke Math. J. 54 (1987)).